Prologue MAM 2080W is a demanding subject.

By the time you are done with the subject you should have resolved 25 tutorials, 12 exercise sheets, 4 tests and 2 exams as part of the subjects program.

This means that regardless of how intelligent you are, you must dedicate yourself to at least finishing all tutorials in time, so that you can get all your doubts cleared up with time.

I also advise you to, after each lecture, spend 20 min revising it. It ends up saving you dozens of hours of study.

I was always the intelligent type, never doing much work for subjects and cruising through them.

I tried the same approach with MAM2080W and found myself repeating the course twice until I accepted that I had to dedicate myself to work.

Most of us learn maths 20% through theory and 80% through resolving exercises.

To make it simple just imagine MAM2080W as an old steam powered locomotive. It starts very slowly and slowly picks up speed.

You are a passenger of that locomotive, and you do not want to fall off. If you try to board the train slightly after it has departed you can run for a bit and still catch the slowly accelerating train. But after a month or so this train will be cruising at such high speed and accelerating that it will require you to run in a struggle for very long until you can catch the train.

MAM 2080W’s course convener Alan Rynhoud is the most arrogant lecturer I have ever met at UCT. Do not get put off by his claims since the very first day of lectures, that you will fail MAM2080W. Do not give him the pleasure. The course is as pre-historic as he is.

All exercise sheets only change when the pre-described book gets updated, and even then only the exercise numbers change. Check it for yourself.

The tutorials simply do not change!

I found the tests to be repetitive throughout the years, so have a good look at past papers to see if you are prepared for the test ahead.

This book is a result of my struggle with the subject, and now I am sharing it with you. With it you will not need to go scavenge for past tests to confirm how well prepared you are, or go hunting for the exercise sheet that was handed in the day you could not make it to the lecture, I have done that for you!

So I strongly advise you to take benefit of this book and with it prepare yourself to cruise through the subject!

Log on to http://mam2080w.blogspot.com/ to support your fellow students with 2009 past papers, tutorials, exercise sheets….

I plan on making this resource book free from 2010 onwards, so it is up to you to keep it updated.

Enjoy!

Chukas

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS

MAM2080W -INFORMATION SHEET - 2008

GENERAL MAM2080W is a core course taken by all second year Engineering students. The main topics to be covered in the course are (a) functions of several variables, and the integration and differentiation of such functions; (b) basic matrix methods, including the process of diagonalisation; (c) solutions of differential equations (both ordinary and partial) , and an introduction to the method of Laplace transforms.

These topics build on the material covered in MAM 1 003W and thus it is essential that you start with a good understanding of all the material in the first year mathematics course.

LECTURES The course consists of 96 lectures, four lectures per week in the first period on Tuesdays, Wednesdays, Thursdays and Fridays. Students will be divided into three groups for lectures. The lecturing team together with their office numbers are listed below: Mr Alan Rynhoud (Convenor) Mi Pieter Moster Prof John Webb

Room 416 Room 109 Room 322

Students are encouraged to visit the lecturers in their offices if they have any problems or queries concerning the course.

EXERCISE SHEETS Lists of problems (exercise sheets) will be handed out In lectures at the start of each section. In order to understand and master the ideas discussed in the lectures, students need to practice and wrestle with problems. These exercise sheets will provide the necessary problems. Note that the answers to the problems will be given, but worked solutions will NOT be provided. Should you be unable to start a problem, or be uncertain as to whether or not your attempt is correct, take the attempt to one of the lecturers. It is intended that the exercise sheets be completed in your own- time, that is, outside the allocated tutorials times. You should do the problems in parallel with the lectures and not leave them until a section has been completed.

TUTORIALS Each student registered for MAM2080W will be required to attend a two-period tutorial per week in 6th and ih periods as follows: Tuesdays Chemical Wednesdays - Electrical Thursdays Civil , Mechanical and Geomatics

You will be informed of the venues in lectures. Attendance is compulsory and a register will be taken. These tutorials will be run in a similar way to those in MAM1003W last year. Students will be required to work in groups on a set of problems, which will be handed out at the tutorials. Staff and tutors will be available to assist the groups during these sessions. The tutorial questions set will provide a link between the material discussed in lectures and the problems set on the exercise sheets, and thus in our view form a vital part of the course. Ensure that you have read your lecture notes before attending a tutorial if you wish to benefit from the tutorial.

EXAMINATIONS The final examination for MAM2080W consists of two papers. The fi rst !s writt~n in June Arc! the ~econd paper !s written in November. Each of these papers will

be not longer than 2% hours.

CLASS TESTS Four class tests will be held during the year. They wi!! be written as

follows: Test 1 Test 2 Test 3 Test 4

Monday, 31 March Monday 12 May Monday, 25 August Monday 6 October

18hOO -19h15 18hOO -19h15 18hOO -19h15 18hOO -19h15

Venues will be announced during lectures. Class test results and related notices will be cjisplayed on the departmental notice board outside Room 206. Please consult th is notice board regularly. In both the class tests and the examinations, marks will be allocated to the presentation (logic and clarity) of your solutions as well as to the correctness.

CALCULATORS Only calculators which have been approved by the Depat1ment of Mathematics and Applied Mathematics may be used in class tests and examinations. A list of the approved calculators is posted on the relevant departmental notice board. This is the same list of calculators which was allowed for MAM1003W in 2007.

TEXTBOOK The prescribed textbook for MAM2080W (carried over from MAM1003W) is "Calculus: Concepts and Contexts" by James Stewart 3

rd edition (long

version) . It will be assumed that each student registered for the course has a copy of the prescribed text.

CLASS RECORD, DP, SUPPLEMENTARY EXAMINATIONS Your class record for MAM2080W is the average of your four class test marks. Students will be awarded a DP if (i) they have a class record of at least 35%, and (ii) they attend at least 80% of the tutorials for the course. The class record (CR) counts 40% towards the final mark for the course. Thus, if you obtain x% for paper 1 in June and y% for paper 2 in November, your final mark will be calculated as follows

FINAL MARK = 4CR + 3x+3v% 10

Please note that NO supplementary examinations will be offered for MAM2080W.

ABSENCE FROM CLASS TESTS Absence from a class test will mean that the student scores 0% for that test. Medical certificates must state that the student is unable to write the class test on that particular day, and must be presented personally to the course convenor, who will then discuss the matter with the student. If a medical certificate is accepted the student's class record will be the average of the remaining three class tests. If more than one medical certificate is presented, the student may be required to sit an alternative test or an oral at a time specified by the course convenor.

MATHEMATICS 2080W

TUTORIAL 1 19,20,21 FEBRUARY 2008

This tutorial revi.ses those sections of MAMlO03W which will be used and built on in the first semester of MAM2080W. If, after completing this tutorial, you still have difficulty with .ANY of these sections, please consult your first-year notes and get the matter addressed BEFORE it

. becomes a problem during this year!

l. (i)

(ii)

(iii.)

(iv)

(v)

(vi)

(vii)

Find a vector equation of the line L which passes through the points P( - 2, 1, 1) and Q(I, 3, -1).

Describe the line segment of L joining F and Q by adding a condition to your answer in (i) .

Does the point (4,5,3) lie on L?

Find the angle between OP and OQ, where 0 is the origin.

Find a non-zero vector perpendicular to both OP and OQ. Find a cartesian equation for the plane ¢ which passes through F,Q and O.

Find the point of intersection of if> and the line x = (1, -1, 3) + t(1 , -1, 3) , t E R. (Hint: think, don't just hash it out!)

2. (i) Find the distance from the origin to the line y = I, z = 2.

(ii) Find a cartesian equation of the plane which contains the line in (i) and which is farthest from the origin.

3. (i) Obtain a vector equation of the plane which passes through the points A(l , 2, -I), B(3, -1,4) and C(O, 3, 2).

(ii) Find a vector equation of a line through the origin, parallel to the plane in (i).

4. Give a rough sketch of the graph of each of the following functions, showing clearly all asymptotes. (Do NOT use the derivatives, nor calculate local maxima or minima.)

. . 1 .. x... x2 • .

(1) Y = -1--2 (11) Y = -1 2 (m) y = -1- (IV) Y = e- xsmx, x 2: O. +x +x +x

5. (a) Find a cartesian equation for the curve. in R2 defined parametrically by

x=2sint, y=3cost, O~t~7r.

Sketch the given curve. Does the given pair of parametric equations define y as a function of x?

(b) Find parametric equations for the curve y = 1 - 4x2 , 0 ~ x ~ 1. Calculate the length of this curve.

6. Let f(x) = { a ~ bx if x < 0 if x ~ 0

(i) Sketch the graph of y = f(x ) in the case where a = 0 and b = -1.

1

(ii) For which value(s) of a and b is f a differentiable function? Explain.

] 7. Find the equation of the tangent line to y = x cos x + -( --)-3 + tan(x2) at the point. where

x-I X= O.

8. Find the maximum and minimum values of f(x) = x2 - x3 (i) on the interval (-00,00), and (ii) on the interval [0, 2].

9. IntegTate each of the following:

(1) xex , (ii) x2ex3, (iii) x

2 + 1 x+ 1 '

10. Evaluate eac..h. of the fo1l0wing:

(i) I: cos x dx, 4

( .. ) 13 arctan x d . 11 4 X ,

-3 1 +x

(Have your answers and reasons checked!)

x +1 (iv) ;2 + l'

(iii) J~ .J1 - cos2 x dx. 2

11. Suppose f(x) 2: 0 for a ~ x ~ b. Give at least three different physieal interpretations for

.lab

f(3:) dx.

2

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MATHEMATICS 2080W

TUTORIAL 2 26,27,28 FEBRUARY 2008

1. For each of the following surfaces, sketch and label the family of level curves (contours). Use your family of level curves to sketch the graphs of the surfaces and where possible "name" the surfaces:

(i) z = x2 + y2 - 1

(iii) z = VI3,- x2 - y2

(v) z = V x2 + y2 - 1.

(ii) z = 2X2 - 1

(iv) z = v''-6---2-x'''-2 _-3y-2

2. (i) Sketch the surfaces z = x2 + y2 and 2y + z = 0 on the same set of axes.

(ii) Do the surfaces in (i) intersect? If so, describe and sketch the curve of intersection.

(iii) What does the set of points {(x, y, z) : x2 + (y + 1)2 = I} represent (in relation to the surfaces in (i))?

(iv) Give the equation(s) of the curve of intersection 'found in (ii).

3. The region within the circle x2 + y2 = 1 can be described in polar co-ordinates by o ~ r < 1, 0 ~ (} ~ 21!'.

(a) Describe the region lying within the circle x2 + (y - 1)2 = 1 in polar co-ordinates.

(b) Describe the region lying within the circle (x - 1)2 + y2 = I, and to the right of the line x = 1, in polar co-ordinates.

(c) Describe the region lying within the circle (x - 1)2 + (y -1)2 = 1 in polar co-ordinates.

4. (a) A point is given in spherical co-ordinates as (vs,~,~). Express this point in both

rectangular and cylindrical co-ordinates.

(b) Express the circle x2 + y2 + Z2 = 32, z = 4 in

(i) spherical co-ordinates, and

(ii) cylindrical co-ordinates.

5. In each of the following cases sketch the region enclosed by the two given surfaces and express the region in terms of spherical co-ordinates:

{n ,., '" - . /,...2..L 0.2 - - V...., 1::J, :; = y'4 x2

(ii) Z=VX2+y2, z=l.

•• 2 :J

1

6. In each of the following cases sketch the curve given by r(t):

(i) r(t) = (cost,sint), 0 ~ t ~ ~

(ii) r(t) = (cos2t,sin2t), 0 ~ t ~ ~

(iii) r(t) = (t, v'f=t2) , 0 ~ t ~ 1

(iv) r(t) = (In t, VI - (In t)2), 1 :::; t ~ e.

vVhat did you observe? Could this curve also 'have been given paral:netrically by r(t) = (et , VI - e2t )? Explain.

2

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6-. e ~C

MATHEMATICS 2080W

TUTORIAL 3

1. Let ret) = (2 sin t, 2 cos t, 5t), t ~ O.

(i) Sketch the curve given by r(t).

4, 5, 6 MARCH 2008

(ii) Does the curve intersect the cylinder x2 + y2 = 4? IT so, at which points?

(iii) Does this curve intersect the upper half cone z = J x2 + y2? If so, at which points?

2. Let rl(t) = (cost, - cost, 2 cost)

r2(t) = (t, -t, t2)

rs(t) = (cost,-cost,cos2 t)

r4(t) = (cost,sint,cos2 t).

(i) Sketch and describe each of the given curves.

(ii) Is there any difference between r2 and ra? Explain.

(iii) Calculate the length of rl.

(iv) Does the point (-2,2,4) lie on r2?

(v) Obtain a cartesian equation of a plane containing rl, r2 and ra. Does r4lie on this plane?

3. The position of a moving particle at time t is given by r(t) = ti + t 2j + t3k.

(i) How far from the origin is the particle after 2 seconds?

(ii) Obtain the position of the particle after 2 + 6.t seconds.

(iii) Compute r(2 + 6.t) - r(2) . Call this vector a.

(iv) What happens to a as 6.t -t O? Was this expected? Draw a picture to illustrate.

(v) Compute ~t (r{2 + .6.t) - r(2». Call this vector h. Compare the directions of a and b.

(vi) Find lim h. What does this represent? Draw a picture to illustrate. ~t ..... o

(vii) Find the direction of the tangent line to the curve given by ret) at t = 2.

(viii) Find the velocity of the particle at t = 2.

(ix) What is the speed of the particle at t = 27

(x) Find the acceleration of the particle at t = 2.

4. A particle moves along a curve C so that its position at time t is given by ret) = (tcost, tsint, t).

(i) Find the co-ordinates of the point A on C which corresponds to t = o.

1

(ii) Does the point (i, i, i) lie on C? Why/why not?

(iii) At what time does the particle reach the point B(O, i, i)?

(iv) Observe that C lies on the cone Z2 = x2 + y2. Can you verify this?

(v) Use your observation in (iv) to sketch the curve C.

(vi) Find ret) and hence the speed of the particle at the point B.

(vii) Write down the vector equation of the tangent line to C at the point B.

(viii) Calculate the distance (correct to 2 decimal places) that the particle has travelled in moving from A to B. The following formula may be useful:

! u a2

Va2 + u2 du = '2va2 + u2 + 2'ln lu + Va2 + u2 1 + K.

(ix) Calculate the angle between ret) and r"(t) at the point where t = o. (x) Calculate the angle between r'(t) and r"(t) at the point where t = 1r.

(xi) Is r/(t) always orthogonal to r"(t)?

5. Let C be the curve ret) = (sint,cost,sin2 t) where 0 ~ t ~ 27r.

(i) Show that P(l, 0, 1) lies on C.

(ii) Q is the point on C corresponding to t = ~. Find the co-ordinates of Q.

(iii) Show that C lies on the cylinder x2 + y2 = 1.

(iv) Show that C also lies on the "parabolic gutter" z = x 2 .

(v) Use the results of (iii) and (iv) to sketch C. Plot P and Q on your sketch of C.

(vi} Find r'(t) and r"(t).

(vii) Use the fact that r'(t) x r"(t) is a normal to the osculating plane to find the cartesian equation of the osculating plane to C at Q.

(viii) The curvature K, is defined by K, = ';;glI1 .

We proved in lectures that the curvature can also be computed using

/r'(t) x rl/(t)1 K,= •

Ir'(t)j3

Find the curvature of C at the point Q.

2

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MATHEMATICS 2080W

TUTORIAL 4

1. Let C be the curve r(t) = (cos t, sin t - 1,2 - 2 sin t), 0 ~ t ~ 211".

(i) Find three different surfaces on which C lies.

(ii) Sketch the curve C.

11, 12, 13 MARCH 2008

(iii) Find a vector equation for the tangent line to C at the point (1, -I, 2).

(iv) Find the curvature of C at the point (I, -I, 2).

(v) Find a cartesian equation for the osculating plane to C at the point (I, -I , 2).

2. Find parametric equations for the curve of intersection of the surfaces, and sketch the curve of intersection:

(i) z = x 2 + y2 - I,

(ii) z = x2 + y2 - I,

z = V13 - x2 _ y2

Z = 2x2-1.

3. (a) By finding two different paths along which the function approaches different values, show y2 _x2

that lim -2--2 does not exist. (x,y)-t(O ,O) x + y

4x2y3 (b) (i) Show that the function -3--9 approaches 0 as (x, y) tends to (0,0) along all

x +y straight lines through the origin.

(ii) 4X2y3

Show that lim -3--9 does not exist. (x,y)-t(O,O) x + y

{

I . (c) Is the fun~tion I(x, y) = sin(x + y2)

x+Y. not?

4. Let I(x, y) = 8 - x2 - 2y2.

(i) Sketch the surface z = I(x, y) .

ifx+y=O

otherwise

(ii) Which level curve passes through the point (2, I)?

continuous at (0, O)? Why/why

(iii) Sketch and describe the curve of intersection C of the surface z = I(x, y) with the plane x = 2.

(iv) Use partial differentiation to find a vector equation of the tangent line to C at the point (2, I, 2) .

(v) Find parametric equations for the curve C and hence find a vector equation of the tangent line to C at the point (2, I, 2).

5. Let f(x,y) = { l~X if y =1= 0 if y = 0

(i) Sketch the surface z = f(x , y) .

(ii) Use your sketch in (i) to find 1:1:(0, O}, ly(O, 0) , /:1:(1 ,0) and fy(l, 0).

(iii) Now use the definition to find Ix(O, 0), ly(O, 0), Ix(l , 0) and ly(l,O), and compare your answers with those found in (ii) .

2

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MATHEMATICS 2080W

TUTORIAL 5 18, 19, 20 MARCH 2008

1. In each of the following cases decide whether or not the given statement is true. If the statement is false give an example to illustrate.

(i) If f(x , y) is not continuous at (a, b) then f x(a, b) and fy(a, b) cannot exist.

(ii) If fAa, b) and fy(a, b) both exist, then z = f(x, y) has a tangent plane at the point where x = a, y = b.

(iii) If f(x, y) is continuous at (a, b) then fx(a, b) and f ll (a, b) both exist.

2. Let g(x, y) = { x2 ! y2 if x = 0 or y = 0

otherwise

(i) Find g(O, 0) ,9(1,0) and g(l , 1).

(ii) Sketch the surface z = g(x, V).

(iii) Find lim g(x, y) if it exists. (x,y)-t(O,O)

(iv) Does lim g(x, y) exist? (x,1/)-t(I,I)

(v) Is 9 continuous at (O,O)? Explain.

(vi) Find gx(O, 0).

(vii) Find gx(1, 0), 9y(1, 0), 9x(1, 1) and gy(2, 0) .

(viii) Is 9 differentiable at (O, O)? Explain.

(ix) Does 9 have a tangent plane at (O, O)?

3. Let f(x, y) = x 1/ 3yl/3, and let C be the curve of intersection of z = f(x, y) with the plane y =x.

(i) Find f(8,8), f(O , y) and lex, 0).

(ii) Sketch the family of level curves of z = lex, v) . (iii) Sketch the graph of the curve C.

(iv) Does the curve C have a tangent line at the origin?

(v) Find parametric equations for C.

(vi) Find the vector equation of the tangent line to C at the point (8,8,4) .

(vii) Find the curvature of C at the point (8,8,4).

(viii) Is y = x the osculating plane to C at the point (8, 8, 4)?

(ix) Sketch the surface z = f( x, y).

(x) Find lx(8 , 8). (xi) Does fx exist at all points (x,y)? Why/why not?

(xii) Find f:z;(O, 1) .

1

(xiii) What is the value of fx(O, O)?

(xiv) Is f differentiable at (O,O)? Why/why not?

(xv) Is f continuous at (O, O)?

(xvi) Does f have a tangent plane at (O,O)?

{I -1 x2 + y'2 if Y = 0

4. Let f(x , y) = 1 x 'f ...,L 0 . .- y 1 Y r

(i) Does lim f(x , y) exist? Expl~in. (x,1I)--+(O,O)

(ii) Find fx(O, 1) and fy(l, 0).

(iii) Is f differentiable at (0: O)? Explain.

(iv) Is f differentiable at (1,0)? Explain.

(v) Find the directional derivative of f at (0, 0) in the direction (1,2) .

(vi) Is f differentiable at (O,l),? Explain.

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( vi; .Fe;.)"," (K'/ ~ t,.II A (f" ~ f ~ A c-R . ( iI i;~/ ~.r'

I ( x /, / j)~ jO .. £; /

(k'v}/ tJo .

£Ii) IIj(,If J.I-(fo.l) -~(.o,'l--'&(O.l) ... J2A- _) a a.o.-1./-A' ! Jl. ~~A v ' t I'll )Ul.f.!t- v

s'" I ,¢ 0<.1f.eN --+. 01/<: 4.f'- I' 0, I ) .

MATHEMATICS 2080W

T UTORIAL 6 1, 2, 3 April 2008

1. The ideal gas law PV = nRT involves a constant R, the number n of moles of the gas, the volume V, the Kelvin temperature T, and the pressure P.

(a) Express each of P, T , Vasa function of the remaining two variables. av or ap

(b) Find or' ap and avo

(c) Calculate (:) (~) (~~). Did you expect this result?

2. Let z = feu , v) have continuous second order partial derivatives, and suppose

u(x, y) = xy, vex, y) = ~. x

( az i) Find -a in terms of the partial derivatives of z with respect to u and V.

x

Oi) Find an expression for x2 ~:~ in terms of u , v and the partial derivatives of z with respect

to ~ and 'U.

3. Show that the curve ret) = W, 3t, 2d) pierces the surface 2X2 + y2 + ~z2 = 13 at right angles

at the point (1 , 3, 2) .

4. A rectangular co-ordinate system is chosen so that the height (in meters) at a point (x, y) on a mountain is given by hex, y) = 1400 - x2 - y2. A mountaineer is positioned on the mountain above the point (10,20). Find the direction in which the mountaineer must travel so that his height

(i) increases most rapidly,

(ii) decre'ases most rapidly,

(iii) does not change.

Draw these directions on a contour map of the mountain.

5. Let f(x, y, z) = yz2 + zv'l+X and let P be the point (3,2, 1).

(i) Find the level surface of f on which Plies.

(ii) Find grad f at P.

(iii) Find a cartesian equation for the tangent pJane to the surface f (x, y, z) = 4 at P .

(iv) What is the rate of change of f at P in the direction (-5,0, I )'?

(v) In what direction is the rate of change of f at P a maximum? What is the greatest rate of change?

(vi) In which direction is the rate of change of f at P equal to 7'1 Explain.

(vii) Find the directional derivative of f at P in the direction (0, 0, 1).

6. Letg(x,y, z )=x+2y+3z.

(i) Show that the point Q(3, -1 , 1) lies on the surface g(x, y , z) = 4.

(ii) Find V g( Q). Did you expect this?

(iii) Find the directional derivative of 9 at Q in the direction of the vector (1,1, -1). Was this result expected?

2

5:.(i) 0 (jl-+ !; J il-A... .:< ~

(it; V ( -- (;-.1;. '6 ~ 02ffJ of J f+ JL):- v yciJ) -- (-{" I, r:).

(Ii<) -: (X-J) ~ f(J -.J).H;(j -Iro

(I v:J V( tfJ) ~ If[ {-S"~ 0, I} = 4-~

l") (j, II U . /-rQ,. ~ ~ J!c..~..c. /J:ri/ . {"'J A>o ~~~~t~ . I~ (c'!)! ~ 7 . (vv :;g:: (;. " (,j; v(j ((D).; @:l ,Y J (i;;Y (1,';/,1), d ~ I, -I) -=- D ·

MATHEMATICS 2080W

TUTORIAL 7 8, 9, 10 April 2008

1. Let R be the region in the first quadrant bounded by the graphs of the parabolas y = 2x2 , Y = 9 - x2 and the line x = O.

(i) Find the points of intersection of the parabolas.

(ii) Sketch the region R.

(iii) Express the area of the region R as a repeated integral, integrating first with respect to V, and then with respect to x.

(iv) Express the area of the region R as a sum of repeated integrals, integrating first with respect to x, and then with respect to y.

(v) Use either (iii) or (iv) to calculate the area of the region R.

(vi) Let R' be the region bounded by the parabolas y = 2x2 and y = 9 - x 2 • Find the area of the region R'.

(vii) Evaluate f f xy2 dA. (Hint: Think carefully about the function xy2!)

R'

2. We want to evaluate r214 y cos x2dxdV. Proceed as follows: Jo yZ

(i) Realize t~at you cannot integrate cos x2 with respect to x.

(ii) Sketch the region over which you are integrating.

(iii) Change the order of integration and evaluate the given repeated integral.

3. In each of the following cases describe and sketch a solid whose volume is given by the repeated integraL

(i) 121~ -2 _~ (5 - x - 2y)dxdy.

tr~ (ii) io It: (2 - x2

- y2)dydx.

1

4. Consider the solid bounded by the graphs of x2 + y2 ::: 9, z = 0 and z ::: y3 . Choose the correct integral representing the volume of the solid.

(i) 413 10

y3 dy dx o -~ 1

3 1v'9-X! (ii) - 3 -V9=X2 y3 dy dx

(iii) 213 1~ 3 - 3 0 Y dx dv (iv) 4 r r~ 3 io io y elx dy

Evaluate each of the above integrals.

t rx 5. Let 1 = io ix f( x ,y)dydx.

(i) Express I as a sum of repeated integrals, integrating first with respect to x and then with respect to y.

(ii) Give two possible physical interpretations of I if f(x, y) is non-negative over the region under discussion.

(iii) Evaluate I if f(x, y) = )1 - x2 •

6. Calculate the surface area of that portion of the cylinder y2 + Z2 = 1, lying in the first octant, and cut off by the planes x = 0, y = 0, z = 0 and y = 2 - x.

7. Let S be that portion of the surface y = 1 - 4x2 which lies in the first octant and between the planes z = 0 and z = 3. Find the mass of S if the density at any point is equal to the distance from the yz-plane.

2

I' 'J ilii 0 '-

4.(£-) - i/~IA;.<- (i~) 0

((i~ 0 (ill) t/c~~. tZ - f-o j " Ck4n::_. L.-tH. t/s.

(,(2)

+.

iL

Ho IS, ai f2 -i If,.,l L.o C..t<""t:; ~~ 6;>0 ~~ J ~ If~'J):

3 , }

of A =- -.-- = Ii-I ,

MATHEMATICS 2080W

TUTORIAL 8 17, 18, 19 APRIL 2007

(vi) Write down parametric equations for the curve of intersection of the two surfaces de-l. Let F(x, y, z) = (--y, x, 6) represent the velocity field of a fluid. scribed in (i).

(i) If S is that portion of the surface of the hemisphere z = -/4 - x2 - y2 lying above t he plane z = I, find the volume of fluid passing through S per unit time in the upward direction.

(ii) If S is the entire surface of the sphere x 2 + y2 + z2 = 4,find the volume of fluid passing through S per unit time in the outward direction.

2. Sketch a region whose volume is given by the repeated integral

~ (6-y)/3 r3jY 18-

Y-l dz dx dy. Jo y

Express the volume of the solid as a repeated integral of the form

/ J J dx dy dz.

3. (a) Find the mass of the solid region in the first octant which is enclosed by z = 0, y = 0, x = 3, Y = x and z = 4 - y2, if the density is given by p(x,y,z) = x2

.

(b) Express the volume of the solid enclosed by z = x 2 + 2y2 and z = 4 - x 2 - y2 as a repeated integral of the form

III dz dy dx.

Do not evaluate.

4. (i) Express the volume of the region enclosed by the hemisphere z = J4 - x2 - y2 and the paraboloid 3z = x2 + y2 as a triple integral with respect to the variables x , y and z. Which order of integration do you think is best?

(ii) Express the volume of the region described in (i) as a double integral with respect to the variables x and y.

(iii) Express the surface area of that part of the hemisphere z =~ x2 - y2 that lies within the paraboloid 3z = x2 + y'2, as a rcpeated integral.

(iv) Find the mass of the portion of the hemisphere described in (iii) if the density is given by p(x, y, z) = z.

(v) Express the mass of the solid described in (i) as a repeated integral if the density is given by p(x, y, z) = Iyl.

5. Decide which of the following are linear transformations. Give reasons.

(i) T: ]R2 -+ ~2 defined by T ( x ) = ( x + Y4 ), y I . Y -- '

(ii) T IR3 ~ R' defined by T ( n ~ ( x x: y ),

ir+y ) (iii) T R' -> IR3 defined by T ( ~ ) = ~ ;: 2x ,

(iv) T:}R2 - t ]R2 satis(ying T ( ~ ) = ( ~ ), T ( ~ )

( 1 2 3)

6. Let A = 0 - 1 0 . 1 2 3

( 1' (1\ ~ ), T 1)

Find the images, under the linear map represented by A, of each of the planes

(i) x - y + 3t: = 3

(ii) x - y + z = 3.

( ~ )-

/-?rJ--.;'.f 0)-0 00 \,y - 41\.:t.rAJ~ 70 7f.,.t",~~L ~ - ~~l,t- c-?-Co8 " ---

I, (f:; r~)( .= J~r: !1 ~1. S' ; /lS:1 , ~ s :: .// J ~ - .I~-y '" . ~ -11/ .; ('. 1)" (!f~ ~ IS> //. J s d R?t'y ~v

( " LY' L/) teL: ~/ / ,-V n4-'\tS fl." • .L~ ~O' ae.e. oil y~ /4 0 .

I .1 r 13,l-- ~ '-JJ- [i;.~yi..-

tI= f j /l.oIx1J7 r I j j'.~h7~ oJ a 1

0 :J

MATHEMATICS 2080W

TUTORIAL 9 22, 23, 24 APRlL 2008

( 1 2 3)

1. Let A = 0 -1 2 . 145

(i) Find the image of the pJane z = 1 under the linear map represented by A.

(ii) Find A ( ~ ) and the normal to the plane onto which A maps z ~ 1.

(iii) Does the normal to the plane z = 1 get mapped to the nornlal of the image plane?

(iv) Does A preserve lengths and angles?

(v) Find detA.

2. Let F: ~.2 -+ lR? be defined by F ( ~ ) = ( ~ ) where u(x, y) = x+3y and vex, y) = 3x-y,

and let R be the triangular region with vertices at (0,0), (1,0) and (0,2).

(i) Find the image R' of R under F.

(ii) Calculate the area of R' .

(iii) Calculate ~~~: -~~ and compare it with the ratio of the areas of R' and R.

(iv) Find the affine approximation for F about the point ( ~ ). How good -is your approxi-

mation? Why? .

- 3. Let F: ]R2 -1 R2 be defined by F ( ~ ) = ( ~ ) ~here u(x, y) = x2 - y2 and vex, y) = xy.

(i) Find the region into which F maps the square with vertices at (0,0), (1,0), (0, 1) and (1 ,1).

(ii) Find the approximate area of the image under F of a circle of area 10-3 centered at

(~, ~).

4. Let F : ]R2 -+ R2 be defined by F ( : ) = ( ~ ) where u(x, y) = x3 + y3 and v(x, y) =

x2 + 2y2.

(i) Find the derivative matrix F' ( : ).

1

(ii) Use the affine approximation for F about the point ( ~I ) to estimate the value of

( -0 9 ) ( -0 9 ) F 0,9 . Compare your estimate with the actual value of F 0,9 .

(iii) Find the approximate area of the image under the mapping F of a circle of radius -~, 10

with centre ( ~ 1 ).

(iv) What would the approximate-area of the image of the circle be if its centre was at ( ~ )? ( ) 'P' d 8(x, y) v .. ill 8{-u,v\ '

5. R is the region in the first quadrant bounded by the graphs of xy = 3, xy = 5, y = x and y =2x.

(i) The region R is transformed into the region R' under the mapping u(x, y) = xy, vex, y) = '!!... Sketch the region R'.

x

(oo) F' d &(u, v) 11 ill 8(x, y)'

(iii) Express the double integral / / (;r sin (;r dA as a repeated integral with respect R -

to u and v .

(iv) Evaluate the double integral in (iii).

141~ 6. Express yeX dxdy as a repeated- integral using polar co-ordinates. Do NOT 2 -v'4y-y2

evaluate the integral.

2

.t<-

V (l) ,~

I

(~J ~ ( ;)<>~ 7) '. C ;; (I<A~ /I~e { ~ (~::):- ~,/,:s)

(ii'tj t?,.."" (~J'" .: I,J, .- (-/)2. i,/ . -.L. /(l . ~fJ) D.

f

,

·r :;

4- ~"~o

Ja

/f 4-1.,'~6) , "€

< --o

MATHEMATICS 2080W

TUTORIAL 10 1, 2, 3 MA.Y 2007

1. Let R be the region enclosed by both the sphere x 2 + y 2 + Z2 = 9 and the cone z = vi x2 + y2.

(i) Sketch the region R and describe it using spherical co-ordinates.

(ii) Calculate the volume of the region R. 1

(iii) Find the mass of the region R if the density at the point (x, y, z) is ---;:.==::::: vlX2 + y2'

2. Use cylindrical co-ordinates to evaluate J J J y2 dV where R is the region lying below the

R

hemisphere z = vi 4 - x2 - y2, inside the cylinder x2 + y2 = 1 and above the plane z = o.

3. Let R be the region lying between the surfaces z = J4~ x 2 - y2 and z = 0 and within x 2 + y2 - 2y = 0.

(i) Sketch the region R.

(ii) Find the equation of the circle x 2 + y2 - 2y = 0, z = 0 in polar co-ordinates.

(iii) Write down the volutne of half of the given hemisphere .

(iv) Use cylindrical co-ordinates to find the volume of the region R. Compare your answer with that obtained in (iii) , and decide whether or not your answer is plausible.

(v) Find the volume of R correctly if your answers to (iii) and (iv) were the same.

4. Let R be the region in the first octant lying inside the sphere x 2 + y2 + Z2 = 8.

(i) Find parametric equations for the intersection of the surfaces x 2 + y2 + Z2

2z = x 2 + y2.

(ii) Find the mass of the region R if the density is given by e(x2+y2+z2)3/2.

(iii) Find the volume of that part of R that lies within the paraboloid 2z = x 2 + y2.

8 and

(iv) Find the surface area of that part of the sphere x 2 + y2 -+ Z2 = 8 lying in the first octant and within the cylinder x·2 -+ y2 = 2-Jiy.

5. Let S1 be the paraboloid z = x2 -+ (y + 1)2 and S2 the plane z = 5 + 2y.

(i) Calculate the volume of the region R enclosed by Sl and 52'

(ii) Find the flux of F(x,y, z ) = (y,-x , 3z) through the portion of S1 t hat lies below S2 , in the downwards direction.

1

f . (0/

MATHEMATICS 2080W

TUTORlAL 11 6,7,8 May 2008

1. (a) Evaluate L (x2 + y2)ds where C is

(i) the straight line joining (0, 0) to (2,2) , (ii) the portion of the circle x 2 + (y - 2)2 = 4 in the first quadrant joining (0, 0) to (2,2) .

(b) Give two possible physical interpretations of each of the line integrals in (a) .

Itty to,' ) 2. Evaluate the line integral

I ~

fc (x2 + 2)dx + xy dy c

(0.' J --;> oX..

(:1. d) along the closed path C as shown in the sketch.

Give a physical interpretation of this line integral.

3. Let f(x, y) = x 2y and let S be that part of the cylinder x 2 + y2 = 9 which lies in the first octant between the surfaces z = a and z = I(x, y).

(i) Make a rough sketch of S.

(li) Use a line integral to calculate the area of S.

(iii) A piece of wire has the shape x = 3 cos t, Y = 3 sin t, a ::; t ::; ~. Find the mass of this wire if the density at the point (x, y) is given by x2y.

(iv) Calculate the area of the projection of S onto the yz-plane.

4. Let C be the upper half of the ellipse

1 ( 2 1 2 4x-5) +g(y-l) =1.

(i) Sketch the curve C.

(li) Find a parametric representation for C.

(iii) Evaluate [(3X + 4y)dx + (4x + 2y2)dy using your parametric representation from (li) .

(iv) Is there a quicker method that can be used to evaluate the line integral in (iii)? If so, use your quicker method to evaluate this line integral..

1

5. Let C be that part of the helix

x = cos 21ft, Y = sin 21ft, Z = 4t,

joining A(l, 0, 0) to B(l, 0, 4), and let F(x, y, z) = (x, -y, z) be a given force field .

(i) Sketch the curve C.

(ii) Calculate L F.dr.

(iii) What does the line integral in (ii) represent physically?

(iv) Calculate curl F. What can you conclude from this result?

(v) Let D be the closed curve which consists of the path C followed by the straight line joining B to A. Wha.t is the work done in moving a particle through the force field F along D?

(vi) Find a potential function f for F, and then calculate feB) - I(A) . Compare your answer with that in (li). Was this expected?

'2

To 1/ -

.: /(;{ f -I) .

3 . (t~

= - .7-,

(!u) PCr-, 1/ ~ J.J. +4: % & (~. 7): 4A '"I-~ ~ • I? :: 't-) (til ~ f.

~~_ ~ L q~. ) ().J> ~ 4i /~ ~a{.~--'l~~lttf-.. C~-~~ J ~j .?~..l~~ : 7- /

t.J]) ~ j(IM-Vd.J.. +-0 ~ _.... = +-~. 3 J

t>~; Q. pJ<e-<f,'oo.( Jf..~--- .... -!-(I\ "J o-f ~ <'r 4-:1:.;; -1-1 J . lv~o~ ~ 1ft) - f(4): -.. - ::. .~

i

(Ii.) W·J. -' ktHTf(-J-i' 'P< '''~ff6) -Y-:"J-lTt6-i;C",~ )f-lf 6. If) dt-o

: --- - ; 8 ~

(v) 0 -

MATHEMATICS 2080W

TUTORIAL 12 15, 16, 17 MAY 2007

1. Evaluate / (2X3 - y3)dx + (x 3 + y3)dy where C is the unit circle. Try to choose the most

e efficient method.

2. Evaluate --dx+-- dy J - y x x2 +y2 x2 + y2

e

(i) where C is the ellipse x2 + 2y2 = 8;

(ii) where C is the ellipse (x - 4)2 + 2(y - 3)2 = 8.

3. Sketch the curve C' which is given by the parametric equations x = t2, Y = t3 - 3t, t E R Use the result that t he area of a region R bounded by a simple closed curve C is given by

l f x dy - y dx, to calculate the area enclosed by the loop of the curve C'.

e

4. Let 5 be the surface of the sphere x2 + y2 + Z2 = 4 lying above the plane z = 0, and let F(x, y , z) = (2x - z, y + 3x, x 2 + z) .

(i) Comment on the following calculation: J J F· nd5 = J J J div FdV where n is a

8 R unit outer normal to 5, and R is the region bounded by S. Thus

J J F . nd5 = J f f 4dV = 4 (Volume of the region R)

S R

= 4 x H~1f8) = 6~7f.

(ii) Now evaluate J J F· nd5.

S

5. Let S be the surface of the region R which is bounded by x2 + y2 = 4, z = 0 and z = 3, and let F(x , y,z) = (2X+y2,2y ,2z +xy).

(i) Sketch the region R

(ii) Calculate J f F . nd5 directly, where n is a unit outer normal to 5 .

s

(iii) Calculate f J F . nd5 using the Divergence theorem .

s

6. Let F(x, y, z) = (x2 +' y2, y2 + Z2, 1 - 2xz - 2yz ) represent the flow of a fluid. Find the flux of F through the surface S, where S is the upper half of the sphere x 2 + y2 + Z2 = 1 orientated upwards.

7. Let F (x, y , z) = (y , -x, z), and let S b~ the surface of the paraboloid z = x 2 +y2 1ying between z = 0 and z = 4.

(i) Evaluate J J curl F . ndS ,,,,,here n is a unit outer normal to S. s

(ii) Evaluate J F . dr where C is the curve of intersection of S with z = 4.

c

(iii) Compare your a.nswer to (i) and (ii) .

8. Calculate f ! curl F . nd5 where F(x , y, z) = (6yz, 5x, yzeX2) and 5 is that portion of the

s paraboloid z = ~X2 + y2 for 0 ~ z ~ 4 orientated upwards.

2

/' (t' )

l\IIATHEMATICS 2080W

TUTORIAL 13 23, 24, 25 MAY 2006

1. (a) Let f(x, y) = x3 + y3 + 3x2 - 3y2 - 8.

(i) Find the Taylor expansion, up to the second degree terms, for f about the poillt (0 , 0). What did you observe?

(ii) Write down (without any further calculations) the TaY'.·)f expansion, up to the third degree terms, for f about the point (0,0).

(iii) Find the Taylor expansion , up to the second degree terms, for f about the point (1,2).

1 (b) Let g(x,y) = -. -2--2'

1 +x +y

(i) Find the second-order partial derivatives of g(x, V).

(ii) Write down the Taylor expansion, up to the second degree terms, for 9 about the point (0,0).

(iii) Use the result from (ii) to classify the stationary point (0,0).

(iv) Sketch the surface z = g(x, V).

2. Find and classify the stationary points of f (x, y) = 2x3 - x2 y + y2.

3. Let g(x, y) = 2(x2 + y2)e-(x2+y2).

(i) Find all the stationary points of g . Do NOT try to classify them using the second derivative test.

(ii) For which (x, y) is g(x, y) < O?

(iii) What happens to g(x, y) as x and y both tend to infinity? Explain.

(iv) Use your results from (ii) and (iii) to classify the stationary points of g.

(v) Try to sketch (and describe) the graph of the surface z = g(x, V).

4. Let f(x, y) = x3 + y2 - 3xy + y.

(i) .Locate the stationary points of f. (ii) Use the second derivative test to classify the stationary points of f.

(iii) Let R be the region {(x,y) : 0 S x S; 2, a S; y S; 2}. Find the (global) maximum and minimum of f over R.

5. (i) Sketch the graphs of x3 + y3 = 1 and x 2 + 2y2 = 4 on the same set of axes.

(ii) Use your sketch in (i) to decide on the number of solutions to the system

~C3 + y3 1

x2 + 2y2 4.

Is ( ~ 1 ) a rea;' ..Jnable approximatlOn to one of these roots?

(iii) Use the two vari;,bk Newton method to find a second estimate X2 to a solution of the

system of equations in (ii), starting with first estimate Xl = ( ~ 1 ).

(iv) Could you have started with Xl = ( ~ ) as the first estimate? Explain.

2

MATHEMATICS 2080W

TUTORIAL 14 31 July, 1, 2 August 2007

THIS TUTORIAL IS REVISION OF MATERIAL FROM MAMlO03W

1. Which of the following matrix products are defined, and where possible find the product?

(i) (! ~ !)( ~1 ! n (~) (! ~ n (T D (iil) (~l :) (! ; !).

2. (a) Find a vector equation for the plane with cartesian equation 2x - y + 3z = 4.

(b) Obtain a vector equation of a line contained in the plane in (a).

( 2 ° 3) 3. Let A = 0 3 1 1 2 -1

(i) Find A-l and check your answer.

(ii) Calculate det A.

(iii) Find the image of t he plane 2x - y + 3z = 4 under the linear map represented by A. (Did you expect your answer to be a plane?) - --

4. (a) Let T : ]R2 ---+ ]R3 be defined by T ( x ) = ( ,xx: 2; ). y 3y - 4x

Is T a linear map? Can T be represented by a matrix?

(b) Let S : ]R2 ---+ R.2 be a linear map defined by S ( ~ ) = ( ~ ) ,

s ( !1 ) = ( 161 ).

Find the 2 x 2 matrix that represents S.

Find a cartesian equation for the image under S of the line y = 3x - 4.

5. Let A and B be n x n matrices. Prove the following: If A and B are invertible matrices, then (AB)-l = B-1A-I.

6. (a) Find all solut ions to the system of equations

x + 2y - 3z +4w

2x+4y + z+w

o O.

(b) Show that (1 ,0, -1) and (1,3, 2) are both solutions to the system of equations

x-y+z 0

x2 + y - Z2 0 .

(c) Decide whether the sum of solutions to each of the systems in (a) and (b) are again solutions to the given systems of equations.

. dy x 7. (a) Solve the separable differential equation T - - = O.

ux y

(b) Solve the linear first-order differential equation dy - y = xeX•

dx (c) Decide whether the sum of solutions to the differential equations in (b) is again a solution

to the given differential equation.

(d) Could the function on the right-hand side of the differential equation in (b) be changed, so that the sum of solutions to this new differential equations is now a solution?

8. Find the solut ion to each of the initial-value problems:

9.

10.

(i) y"(t) + 2y'(t) + y(t) = 0, y(O) = I, y'(O) = 0

(ii) yl/(t) + 2y'(t) + 5y(t) = 0, y(O) = 0, y'(O) = 2.

Sketch the graphs of the solutions.

(? 3y (i) Find the imaginary part of ~(- t.) .

z 1 - z

(ii) Find the real part of e(2+3i)x + e(2-3i)X, if x is real.

(iii) Given that Izi = 2 and that z lies in the second quadrant, plot a possible position for z

and hence the positions of z and ~ on an Argand diagram. Z

(iv) Express ](x) = 3x3 - llx2 + 16x - 10 as the product of real linear and real irreducible

quadratic factors, given that ](1 - i) = O.

(i) Does ] et~' dt converge'! (HINT, Use a suit.ble comparison.)

1

1

(ii) Does J et~t dt converge?

o

(iii) Does] et~' dt converge?

o

2

( i//" ttl) (1.1. 4~

(:;~~j (;18 IS

;. • ( Ct; J .!- ::. (~o, 0) .,. ~ ~~., 4) .~ (01 3/ I) I

(}J ~ (~ oJ ,,) +. ~CI,;J .. ()J A G-i!... .

.3. f.J ,4 -" -, .r :2.. . ' - I _ -' (S: -,. :)

3 If. -,

(5' -/<7

f."-) -,.7.0 X t- S""J- ~.26 + -76 = 0

4 " Crv /l.J f.)

(~j (: ~;)/ .j Y :: ;<. -ut

~ I .. f< G- t?. .

6. (q) ..t..:- r;(.(-~~ ~ 0" 0} ~ (:.~ 0') ~ /}" ~ (f G- R _ (c) ~s fr-rcy ~ No FOi-· l~).

,- (aJ 1/= x~ C

(tv 11~ Ce~+ ~~~ (C:J ;V'o

(e>l) 0

.8·(0 11 ll ) ; 6e -lr -6-.,..e

-I- t-tv 1} N) : e M;"~

MATHEMATICS 2080W

TUTORIAL 15 29,30,31 July 2008

1. (a) In each of the following cases describe geometrically the set of all linear combinations of the given set of vectors:

(i) {(3, - 1, 1), (4,2,3)}, (ii) {(3 , -1,1) , (-6,2,-2)}

(b) Let a, b and c be three vectors in R3.

(i) Describe geometrically the set of all linear combinations of a, b and c. (Consider all cases.)

(ii) Can we generate the whole or 1R3 by linear combinations of any set of threE vectors in JR3? Explain.

(c) Express every point on the plane X + 2y - 3z = 0 as a linear combination of vectors.

2. (a) Construct a 3 x 3 matrix A and a vector b so that the equation Ax = b has no solution.

(b) Either find a 3 x 3 matrix A and a vector b such ·that Ax = b has exactly three solutions, or provide an argument to show that it is not possible.

(c) You are given that (1,3,4) and (2, -1, 5) are both solutions of the same linear system Ax=b.

(i) Can you find a non-zero solution of Ax = 0 ? Explain.

(ii) What can you say about the number of solutions of Ax = O?

(iii) How many solutions does the equation Ax = b have?

(iv) Write down as many solutions of Ax = b as possible. Are these the only solutions?

3. For each of the following augmented matrices, circle the pivot elements, decide on the num­ber of free variables (parameters), and hence solve the system represented by the augmented matrix. Give your solutions in vector form.

(

1 3 2· 4 (i) 0 0 2 1

o 0 0 3 ID ( ... ) (1 1 -1 3 0 III 000 10 I ~)

(132411) (ii) 0 0 2 1 3 o 0 0 0 6

Write down all the solutions of the homogeneous system associated with the given augmented matrices.

1

4. For which value(s) of k does the following system have (i) no solution, (ii) a unique solution, (iii) infinitely many solutions'! Find all solutions (in terms of k) whenever possible.

Xl + 2X2 - X3 3

2Xl + 5X2 + X3 7 Xl + X2 - k2X3 -k

5. Which of the following subsets of JR2 are dosed under (i) addition of vectors, and (ii) scalar multiplication?

(a) {(x,y): x 2: O,y 2: O}

to) {(x, y) : xy 2: O}

Are either of (a) or (b) linear subspaces of R2? Why/why not?

( 1 2 -1 1 - 1)

6. Let A = 1 1 -3 1 2 . 3 7 -1 3 -;-6

(i) Express the solutions of Ax = 0 as a linear combination of vectors.

(ii) Is the set of all solutions of the matrix equation Ax = 0 closed under addition of vectors and under scalar multiplication? Give reasons for your answer.

(iii) Is the set of all solutions of the matrix equation Ax = ( ~ ) closed under addition of

vectors and under scalar multiplication? Give reasons for your answer.

7. Solve the system Ax = 0 where

(

1 - i 2i A = 1 + i -2

1 i-I

Does the system Ax = 0 have any real solutions? (Reminder: i 2 = -1)

2

-I-i) 1- i .

i

I 'r ~ ({.~) I - J (~+-Y)

I£)--~

~j-f,e

jLt-lf

S {c..) ~ .M...c4r dtofo.e.th"":' hwl IUO, SC....e~ ;4.J~Y.I '~b.-.. . ) -~ ~ ~ \

(6) ~ .-u-4..,.. 5:~)t,Uu£I./ 6wt NJe"Y- """,cAr-- .,..otctt..t,,~

~~~~~0tA.. ~rOCA4.

b.(~ ~(t) ~i (f) ~ 6 (1) ) V<i,dG-~' C0 y .... ~) Qtu P o .

7- . (A /.9 )At · --- N { ~ /2-a< ~\~ k .x..;::-o -- .-

MATHEMATICS 2080W

TUTORIAL 16 5, 6, 7 August 2008

1. (a) Express (0, -3,3) as a linear combination of the vectors (1, 2, ~"!, (2,1, -1) and (1, -1, -2).

(b) Express (0,0,0) as a linear combination of the vectors (1, -1,1), (3,1,5) and (1,1,2), in at least two different ways.

(c) For which value(s) of A is (l,A - 6,>.) a linear combination of the vectors (-1,2,4), (1 , 1,3) and (2, -1, -I)?

(d) Are there any vectors in lR3 which cannot be expressed as a linear combination of the vectors (1 , 1, -2), (1,0, -1) and (-2,3, -1)':' If so, describe the set geometrically.

(e) Let S be a given set of vectors. If one element of S can be expressed as a linear combi­nation of the other elements, does it follow that each element of S can be written as a linear combination of the other elements of S? Give full reasons for your answer.

2. Explain geometrically what it means to say that

(i) two non-zero vectors in lR3 are linearly dependent,

(ii) three non-zero vectors in lR3 are linearly dependent.

3. Are the following sets of vectors linearly dependent? Give full reasons for your answers.

~) {(I ,O,I),(2,O, 0) ,(1, 0, 3), (1, 1,0)}

(ii) {(I, 0,1), (0, 1,0) , (0,0, O)}

(iii) {(I, 2, 3), (1,0,1), (0, 1, I)}

0v) {(0,0,1,1,0),(0,1,1,0,0),(0,1,0,1,O),(O,2,3,1,2)}.

4. In each of the following cases find (all) the values of k for which the given set is linearly independent .

(i) {(I, 3, 4), (2,2 + k, 4 + k), (k, 6 + k, I2)}

(ii) {(I, 1, k), (1, k, 1), (k, 1, I)}.

5. "rJa) Can you find a linearly dependent subset of]R3 that generates the whole of ]R3? Explain.

\(b1 Which of the sets in question 3(i), (ii), (iii) are generating sets for R3? Explain. If the J set does not generate ]R3 j find the linear subspace of R3 generated by the set.

6. (i) Write down a system of linear equations in the unknowns Xl! X2, X3, X.!, Is which must be satisfied in order to balance the cbemical equation

x1Cu + X2H N03 --7 x3Cu(N03h + X4H20 + xsNO.

1

7.

(ii) Find all solutions of the system you found in (i) and hence balance the chemical equation.

(i) Use Kirchhoff's laws to set up a system of linear equations for the currents in the branches of the network

T 1 r 20 i' ~~ T~ tIn 1 i Ut -____ ~~_~___.l 6V 3V

(ii) Solve the system you obtained in (i). How many solutions did you expect?

(iii) Write down the solution of the associated homogeneous system without solving the sys­tem. Did you expect this?

8. The following diagram shows a road network where all the streets are one-way. The flow of traffic in and out of the network is measured in vehicles per hour, and is indicated on the diagram. Let Xl, X2, X3 and X4 denote the number of vehicles flowing along the various

branches per hour. 100 ~o

1S rA~ .... - - - I.lS 1:,

~l r~ 3.'; ( ~ _ C :J ~ .... '1S"

5'0 100

(i) Construct a system of linear equations that describes the traffic flow in this road network.

(ii) Solve the system and find the maximum and minimum values of Xl·

2

H~~ ~'v·:...,·,Oa t,.y - . ~S~<f 7b TZi. ... 7ts/4~ / 6 - -~@.Ct r:/-' a~g

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e!J ,POT.'~ o?( ~-rC/''''- x 7' CJ =-0. (e) /fJ~. M~ /' a) (v!

;2 .Ct) ~I:" c.Af"-~ (i;.; 71"'-1 J..e ~a.,,-Cc.<. .

:; ~{c / .~~ {' t AX ~.A-"' RJ )

(iV '-:: '/J . ( ' CC)M.~~ g) tr:,; L]J . ( I~ ~ ,3J ~, ~ ~J!/t--a ("> ~ /)

C; /I 'J .L....2'. - 1J~ ~~""- n. 7.e.r- t!(. ~;dd~_e.-t4

i-f{'-7 /.L~ ~ ,I£ $~.

(~ /L-F / /( :F -.2. . /

s:- ~I Y ... ~ [("t;o , o) (o.J~o){oj:>.JI! (i/~ .-)J (6J (v Y..& (~luo - ~ ?-2MAQ.'-:Ji- . A~""- ~ d .

t) Ati> - ~· 7 ~~~ '!zI{~AQ. X7-lj ;j"""o. 6. (~ J c~. +-S rrIJC:J -- :I G.t. {'hojol 7- ¥. ~o 7"-0:>. N O.

~,@ it (~ ~II ~ ~ /J)

(tiL) 9

8 ,fly .'!: ;( I") '--j (00 of- 0<-'1-s- -0( ,

oL "

A1~ 0 ~ tJt ~-7-S' ,

. MATHEMATICS 2080W

TUTORIAL 17 12,13,14 August 2008

1. Find a basis for the set of all solutions of the matrix equation

(1234) 3 4 9 2 x = O ·

Find the dimension of the subspace which consists of all solutions of the matrix equation. How many solutions are there for the given matri.x equation? Find another basis for the set of all solutions to the matrix equation.

2. Find a basis for and the dimension of the linear subspace

{(x, y, z): X - 3y + 5z = O}

of ]R3. Give two other bases for this subspace.

3. Are the following statements true or false? Either give an example to show that the statement is false, or provide a proof if the statement is true.

(a) Every subset of a linearly dependent set is linearly dependent.

(b) Every set which contains a linearly dependent subset is linearly dependent,

4. Which of the following sets are bases for ]R3?

(i) { ( D ' ( D ' ( D ' ( D } (il) { 0) , ( n ,( ~1 ) }

(iii) { ( D ' 0 ) , (:1

) }

(iv) { UJ ' ( D } If the set is not a basis for lR3 , either delete elements from the given set so that the remaining su bset is a basis, or extend the set to form a basis for lR3 .

1

(1 2 -1 3) 4 5 x 1

5. Let A = 0 3 2 4 .

\ -1 -3 -4 0

(i) Find the cofactor of the entry x in the matrix A.

(ii) Expand det A by cofactors of the second row.

6. Evaluate

11 2 3 4

~ I o 2 1 3 o 0 ,! I 12 4 7 61 (i) 0 0 5 1 (ii) 0 2 71 (iii) I! ~ -; ! I 0 0 0 3 -1 I 538

o 0 0 0 4 I 8 9 -2 2 I

C bed) 7. Let A = a (3 'Y 6 1 2 3 4 . 7 5 3 1

Express the determinant of each of the following matrices in terms of det A. Give reasons for your claims.

(

b a Cd) n (3a"{o 1 2 1 3 4

5 7 3 1 (

a bed) (ii) :; ~ !

1 234

(

a bed) ( ... ) a: (3 'Y 0 lil 1 2 3 4

8 7 6 5 (

a b Cd) , a {3 'Y 0

(IV) 1 2 3 4

9 9 9 9

(

a bed) a {3 'Y 6

(v) 3 '6 9 12

14 10 6 2 (

a+a b+/3 c+"{ d+O) ( ') a + a f3 + b 'Y + C 0 + d Vl 1 2 3 4

7 5 3 1

2

n/9-·.I-1 ~ D ?o W - ~~.S' To ~P-c/f!Lr4L /7- - ·~B--u-('r- - cge0.~

I. 8~.ia I{!) (fJ J. ),~..,.~'-- ~ :2. J-.{ .~ A~ '<..J.

0-« --. &l:. <./ ! (-~ ) I tV f . :2. 8~ -'" 1 (f)/ t{)J. j)~~<~~ La ~. o~ ~ : !(1)/;;J 0.-<

} (t) f-lJJ. . ' J( .. ) ~-1'." .' J ( !) /{!J; (Vf;., LJJ. - 6~ (( i) j ~ ~.J. (6/ s~·~~~" '6- h ~&;.

4. t> AJ-a - 4......""b

(y Y-u, - ~~ Or~. CJ --e~~_J' £~ C5J /OCJ _ ~ ~ v

(fv) Nt. - ~ ..... at J 'I2~~ i<~ ( ~ O~"C- (U! (tit) ap(~ ( i).

s.(~ (~3 : C-:/;J+-J I.: } ! /: __ . = _1:.7

/-1 -3 vi (-

(V ~~ 4 ,- -'f!f ~I ;/ -f S 10' -; ~/- 13.1 . - J ,.. f' 0 / -I -~ D

- - .. , - ; 13 -13.x..

t -(i/ IJ-e

(V - 4<0

et~..J f)c,.1

~, If.; - ~. A ( C; C-J c~ ) (iV ~ A- ('~~ ~ O$"..{ p~ (-J ~-e..c) P~)

(i~~ ~ A- (4 '~ J{~. )2.1)

fi/J ~4 {J2~ -:J ~ 1- ~~}

{VJ 604.!! A ( :1 -,;. /).l) .2 (. Rr,.) (II!; 0 { 'R, ~ R.ol- J

r I , 0 3 J..' f l .;,f --'I ,,-, '~J -'-I-

MATHEMATICS 2080W

TUTORIAL 18 19, 20, 21 AUGUST 2008

1. Let A = ! I! x ~ ~ . Find, by inspection, three values of x for which (

2 1 3 2X)

-2 - 1 -3-x det A = O. Give reasons for your answers, but do not evaluate det A.

2. Let A be a 4 x 4 matrix with det A = 7. Find

(i) det AT,

(ii) det A-1,

(iii) det A2,

(iv) det(2A),

(v) det( -A) .

3. Find all the values of x for which

1+x 1 1 1

4. Solve the equation

-8- oX

( 1 2 3)

5. Let A = 4 2. 3 -1 -2

(i) Find the value of det A

(ii) Show that A-I = t ( 104 -7

(iii) Find the adjoint of A.

(iv) Evaluate A(adjA )

0 -9 1

1 -11

7

1+x 1 1

10 2-oX

-9 1

1 1

l+x 1

7 0

8-oX -1

1~ ). -7

1

1 1 1

l+x

-9 0 -9

2-oX

=0.

1= O.

( 21 3 4) 4 1 -2 6

6. Let A = I 1 3 0 x .

\ 1 1 3

(i) Evaluate det A.

(ii) For which value(s) of x is A an invertible matrix?

(iii) Find the cofactor of the element a21 in A .

(iv) Put x = 0 in A. Now write down the element in the first row and second column of A -1 .

Do NOT use Gauss-reduction.

( 3 1 4 2) ( 4 ) ( Xl ) -1 3 2 4 2 X2

7. Let B = 5 oX 0 3 ,b = 0 and x = Xa .

2 1 1 5 1 X 4

(i) Evaluate det B . Record the operations you use.

(ii) For which value(s) of oX does the system Bx = b have a unique solution? Use Cramer's rule to find the value of X 4 for these values of oX .

8. Use a determinant to determine whether the following sets are linearly dependent or linearly

:~:Inl ( T ), ( )5) ,( ~1 ) } (ii) { ( ~) ( T ) ( ~~ ) ( ~1 ) }

2

J1 A-H cJ-O 8"0 iY - .4..1J'S' W~ ~ / u~"4'--. /8 - .~ .. ~srr p2~ . '

l ~;.s CRJ.;'-Py.). :,(::'0 (I?=--p~) - 1~1 (Ci--;z.C z ) ,/ /

~. (f; ~, r:V ~ ~ (iii) * '7 UII).J 'to ~ (I/J L-!J~ ~ . ) /

3. ).3(4-1-.x) ':::'0 •

it . - (~- "')~( of- )t.){/- ,,) ':6rD.

{, . (tj '1 (x -=1L) (:r,) :l. r 1

(i~) IS'- OJ L

r£l) (A2 : - %1 .

7a. (~) -- 6o{ A -(-.:1- )

(i~ A -=f ' -~ ; /4~ ~¥ .: 0 1

~ a (~J ~.I,

(hJ 1.... ... 0,

MATHEMATICS 2080W

TUTORIAL 19

1. Consider the differential equations

- y'I(X) + (yl(X))2

ylll(X) - y(x) + x'l

xy'(x)

2 +4x2

o 2y(x}.

26,21,28 August 2008

(i) Which of these differential equations have y(x) = x2 as a solution?

(ii) Which of these differential equations are linear?

(iii) Which of the linear differential equations are homogeneous?

(iv) Write down the order of each of these differential equations.

(v) Try to find, by inspection, other solutions to each of these differential equations.

2. (a) Find a differential equation which has the family of curves y = Ax2 as solutions.

(b) If all the curves of one family f(x, y) = Cl intersect orthogonally all the curves of another family g(x, y) = ~, then the families are called orthogonal trajectories of each other. Find the orthogonal trajectories of the family of curves y = ce-~, c E 1R. Sketch a few members .of each family of curves on the same set of axes.

3. (i) For which value(s) of A is e,u a solution of the differential equation

2xy"(X) + (x + 2)y'(x) + y(x) = O?

(ii) Show that y = x2 - 4x + 8 is a particular solution of

2xy"(X) + (x + 2)y'(x) + y(x) 3x2.

(iii) 'Write down another two particular solutions of the differential equation in (ii). Explain how you arrived at your answers.

4. (i) Solve the differential equation

ye~1I + arctan y + (xe:tll + ~ + 1) ddy

= O. l+y x

(ii) Find the particular solution to the differential equation in (i) which satisfies y(O) = 1.

(iii) Show that F(x, y) = (ye:l:1I + arctan y, xe:l:1l + 1: y2 + 1) is a gTadient field, and find a

potential function J for' F.

1

5. One model for the spread of a rumour is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumour and the fraction who have not heard, the l'umour. In a course of 1000 students, 80 students have heard a rumour by 8 a.m. that the forthcoming class test has been cancelled. By noon, half the class had heard it. At what time will 90% of the students have heard the rumour?

6. A 2500 litre tank initially contains 500 litres of fresh water. Water containing 50% pollutants flows into the tank at the rate of 10 lit res per minute, and the well-stirred mixture leaves at the rate of 5 litres per minute.

(i) Find the concentration of pollutants in the tank at the moment it overflows. (Hint: Let A(t) be the amount of pollutants, in litres, in the tank at time t, and let V(t) be the volume of liquid in the tank at time t.)

(ii) If the tank 'initially contained 10% pollutants, find the concentration of pollutants in the tank at the moment it overflows.

2

----. . :>-.

p~ f;'A -""1 .... ..1 (7 V' ) (J - ,.dO

J( . b ~/ (J';:"~ s()

~~~ 1... 72. ....

6. (lJ 0/1-). r -- srA-- ,-. ""'Co ~+ ,s--(j-

12 ... ~- A ((;.) ~ ...r (/0"0 T'6) f c .

~oo+- 6 'V

MATHEMATICS 2080W

TUTORIAL 20

1. Which , if any, of the following arguments are correct:

(a) Let S = {x,x2,X3 }

Form the equation ax + /3x2 + 6x3 = O. (*) Put x = 1 in (*) to get a + ,8 + Ii = 0

9, 10, 11 SEPTEMBER 2008

This equation has a = 10, (3 = 2, 6 = - 12 as a solution, and since (*) has a non-zero solution, the set S is linearly dependent.

(b) Let S = {x, x 2 , x 3 }

Form the equa.tion ax + /3x 2 + b"x3 = O. (+) Put x = 1 in (+) to get a + ,6 + <5 = O. Put x = -1 in (+) to get -a + /3 - 0 = O. Put x = 2 in (+) to get 2a + 4/3 + 80 = O. This system of equations has a = (3 = c5 = 0 as the only solution.

However, since we have only investigated the values x = 1, x = -1, x = 2, we cannot conclude that S is a linearly independent set of functions.

2. Use the definition to decide whether or not the following sets of functions are linearly de­pendent on the indicated intervals:

(i) {cos x, sinx, sin 2x}, [0, (0)

(ii) {cos2 x, sin2 x, I}, (-00,00)

(iii) {2 + x, x - x2, 3x2 + X + 8} , (-00,00)

(iv) {x2 , xlxl}, [0, 00)'

(v) {x2 , xix!}, (-00,00)

3. Show that y(x) = x-I and Vex) = e-x are both solutions to the differential equation xyl/(x) + (x - l)y'(x) - Vex) = o. Find the general solution to this differential equation, and hence the general solution to the differential equation xY"(x) + (x - l)y'(x) - y(x) = 5. Which results have you used to obtain the general solution?

4. Write down the general real solution to each of the following differential equations:

(i) ylll(X) + 31/'(x ) - 4y(x) = 0

(ii) yl/(x) + 4y(x) = 0

(iii) y"(X) + 2y'(x) + 5y(x) = 0

(iv) y(4)(X) + 8y"(X) + 16y(x) = 0

(v) (D3 + 1) Y = 0

(vi) (D + 1)4y = 0

(vii) (D4 + l)y = O.

1

5. Let (*) denote the differential equation X2yll(X) - xy'(x) - 3y(x) = x3,

(i) Show that YI (x) = x3 and Y2(X) = .!. are both solutions of the homogeneous differential x . equation associated with (*).

(H) Write down the general solution of the homogeneous differential equation associated with (*) . State t he results you have used.

(iii) Use the method of variation of parameters to find a particular solution of (*) .

(iv) Verify that the particular solution found ;n (iii) does satisfy (*).

(v) Find another particular solution of (*) and ex;:>lain how you obtain it.

(vi) .Vrite down the general solution of (*). What result have you USE:d ?

(vIi) What fact gllarar.:.tees that the system of equations you obtained ~n (iii) bas a unique solution?

(viii) Find the solution of (*) that satisfies y(l) = 1, y'(l) = O.

6. You are given that Yl(X) = x-6 and Y2(X) = e-X -5 are both solutions of the linear differential equation T(y(x)) = 5. Find two solutions of the associated homogeneous differential equation T(y(x)) = 0, and another particular solution of T(y(x)) = 5.

2

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(ViV .kI(l)" J. L J +'.!.L +- 1,'; ~'L , (j /6 I'~ ~

t. IJd: (x- ')- (e-~-.r) '( .totk-Ife4A A>".N{ rtltex)= o.

v1-ftJ:=- ,;..!l:c-lj-(e-t..)J S c,Coc.,d.., ,l-e-t...A'o-.- 1( 7(fI Clt))=- 5" ~ .A.,t rr/;> (x --9 + [(:J.-'J-(e-csJj. v

MATE~MATICS 2080"'N

rpUTORIAL 21 :U1 , 17, 18 SEPTE!-.1BPR 2008

1. (i) Find the general real solution of the differential equat;ion

(D2 + l)y(x) = O.

(ii) Use variation of parameters to find a particular solution to the differential equation

(D2 + l)y(x) = secx.

(iii) Find another particular solution to the differential equation in (ii).

(iv) Solve the differential equation

(D2 + l)y(x) = csc x.

Check that your solution is correct!

2. (a) Find the general solution of (D3 - 8)y(t) = O.

(b) Find a particular solution in each of the following cases (use undetermined coefficients):

(i) (D3 - 8)y(t) = 4 + lOt2

(ii) (D3 - 8)y(t) = 3cos2t+ 4sin2t (iii) (D3 - 8)y(t) = 7tet

(iv) (D3 - 8)y(t) = 4e2t•

(c) Find the general real solution of (D3 - 8)y(t) = 7tet•

3. Find the general solution of the differential equation

D(D - l )y(x) = xez + 3x

(i) using undetermined coefficients,

(ii) using variations of parameters.

Compare your ~nsw'::r5 in (i) and (ii).

( 8 4. I..et A = \ ;

t' • 22 ) ~4 -;-i .

(a) (i) Are any of ( n · 02

) • ( D · 01

) • ( D eigenvoctoTh of A!

(ii) List the corresponding eigenvalues.

(h) (i) Decide whether or not 3 ( ~ ) ~ ( ~ ) is an eigenvector of A.

(ii) Is it true that any scalar multiple of an eigenvector is also an eigenvector? Verify youi'daim.

(c) (i) Decide whether or not ( ~ ) + ( ~ ) ( ! ) is an eigenvector of A.

(ii) In general, what can you say about the sum of two eigenvectors of a matrix? Prove your claim.

5. A mass m is attached to the lower end of a spring and allowed to reach equilibrium. Suppose that the mass is set in motion so that the differential equation governing the displacement y(t) of the mass, measured positively downwards from the point of equilibrium, is y"(t) + 2y(t) = 0, y(O) = 4, y'(O) = 2.

(i) Explain the physital meaning of the initial conditions y(O) = 4 and y'(O) = 2.

(ii) Determine the position of the mass at any time t.

(iii) What is the maximum displacement from the equilibrium position?

(iv) If the motion is damped and is governed by the differential equation

yl/(t) + 2y'(t) + 2y(t) = 0, y(O) = 4, y'(O) = 2,

find the function yo(t) that describes the motion of the mass. What happens to yo(t) as t -t oo?

(v) If the mass is driven by an external force so that the differential equation governing the motion of the mass is

yll(t) + 2y'(t) + 2y(t) = 2 sin t + 4 cos t,

find y(t) . What happens to yet) as t -t oo?

(vi) If the mass is driven by an external force so that the differential equation governing the motion of the mass is

yll(t) + 2y(t) = sin V2t,

fiu.i !J{t;. ,,vtat happens to yet) a.,S t -t oo'?

2

l1A-lV; ,rOi?{) \,/ - /4.N·..glUe/Lj To ;;rrol'U~'- 511- #{Jr, c;loc8 I.ft? '1 - AO>?"'-r .6 ~~) ~ If &IK. . -.--.--- .-----------.---.-.~-~-,~.......-

~'.) '11';<' x s. ... ..( -I- 6>1 "- ...R ..... / c"'" "'-/ ,(N!;.) -t -=- x s. :'..l"'< ~~ ..l~/t:~.y+- C04 oX.

f!v) -"1:> 4- CPy1L + D~·;,.(. - Jl CPt'X -f ..J-<'.-t .,..e.....;:..~ ,1 ( .4,,( {S-I< u /" " ~,(Q J -"1

0; A e ~ (;- t- e - t-('t CD; fT 6- +- C 'il.:"" Ii -.( ) A It C ($L ..R ·

(J ,/ ./

(6) (0 - j - E (. v Iv -:% ,J.~~ ~ ~I't uo (;- (;~./t c.-V (" 6- (I '/-f e J l-. .

(c.) (J:; Ae"c---f .e -t-· /,1~/IE-.. + C h~J.jI) - (~+I-)' e (;- A ~ cc-R. ( I.. ) ,1./ "

3 .~ (iJ /1J ~I':' x (Q~ 6) (ir)\ +- JL (ex +-0.1).

Q"cj 71;-~ / =- A Ct) ~ I -t-. IItl) eX.

fj~>-fA«-( A rvt..A:_ ~ -1/ IV' -4 -1-"'< e ~ -r- :f I.- e :x _ Y. -e ;). - i x '-- 1"- ·

It (q. J / ;2 ., " (- I · - (V ~..I-- ~cAr.-x/£. ~ "' .2 / Y h €- ~-<AJ-'/I L6 ~ (V..;, e- __ chr-.-cJ/R ~:::.,. ·

fpJ(j ~ a ... -t"-~c4--. & M,...-J"'-">~"-¥.."{~ - ~~-e-A-K';;,, l ' . ~ , lCj !) ~ 0-" .i? ~~ok. tv 7Z,.J,...u.... ,{,{..:.-3.e . P"'="- ~ ...... ~.,.~...;

.dlJ .ldoo;f t;;'.d. Ae<~'f . .L<~Y""'-) G,..J~ .3: i -Jj. U

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(i~) 11 a I ~ ..,. ~ J')' r -I- r... 4;; • .I'i- 6- •

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b - G- ~- -

(!'.J 11 Cf I " e - f,.. r A-Coo G -t-d .J./AO/) +.)- 4. ';.. t- ..-.> ..,;74.;;" f- Q.o G- -:;:, c;P> •

Cd) 'V {f I: ~CP3 {;- f- +- .14.. ~ (i t) +- ( _:,~ 6-;: fi) , )

1/0 (n 1/,.1.0

/f1 c51-<. ~ / -rJ ,pC 0/ -,.') ".z> a.:r G- ---=> Q;> -

MATHEMATICS 2080W

TUTORIAL :L~ . 23, 24, 25 September 2008

1. A non-zero vector x is an eigenvector of the matrix A if Ax = AX for some scalar ,\. Geomet­rically, this means that Ax lies on the same line through the origin on which x lies, that is, the linear map represented by A does not change the line of action of an eigenvector.

(a) Use the above geometric interpretation to find the eigenvalues and eigenvectors of each of the following linear maps:

(i) T:]R2 -t ]R2 is the linear map that projects points onto the line y = -x, (ii) S: lR3 -t ]R3 is the linear map that reflects points in the plane 2x - y + 3z = O.

(b) How many eigenvectors are there corresponding to each of the eigenvalues of S?

( c) How many linearly independent eigenvectors are there corresponding to each of the eigen­values of S?

(d) Write down the definition of an eigenspace of a linear map. Describe geometrically the eigenspaces of the linear map S.

( 7 -1 -2) 2. Let A = -1 7 2 .

-2 2 10

(i) Find det(A - AI) and hence the eigenvalues of A.

(ii) Compare the sum of the eigenvalues of A with the sum of the elements on the main diagonal of A.

tiii) Compare the product of the eigenvalues of A with the value of det A.

(iv) Use Gauss-reduction to find the eigenvectors corresponding to the eigenvalues found in (i).

(v) How many linearly independent eigenvectors are there corresponding to the repeated eigenvalue of A?

(

1 2 -2 4 -1 1

3. Let A = 2 -2 2

1 -1 1 ~ ).

-1

(i) Write down the value of d.et A, and explain how you obtained the result.

(ii) Is A invertible?

(iii) Given that A has -1 as an eigenvalue with algebraic multiplicity 2, find all the eigen­values of A.

(iv) Is the information you found in (iii) sufficient to conclude that A is NOT diagonalizable? Explain.

1

(3 1 2) 4. Let A = 2 2 2 .

. 213

(i) Find the characteristic equ~tion for A.

(ii) What are the eigenvalues of A? Check that the eigenvalues are correct.

(iii) Find the eigenvectors of A.

(iv) Check that the eigenvectors found in (iii) are correct.

(v) Write down the value of det A.

(vi) Is A diagonalizable? Why/why not?

(vii) Can you find matrices P and D such that P-1AP =;.·D? If so, what is D?

(viii) How many different matrices P and D can you find satisfying p-l AP = D? Explain.

(ix) Use your result from (vii) to solve the system of first-order linear differential equations

X~ 3Xl + X2 + 2X3

x~ 2Xl + 2X2 + 2X3

x~ 2Xl + X2 + 3X3

(x) Find the particular solution to the system in (ix) which also satisfies the conditions Xl(O) = 2, X2(0) = I, X3(O) = -10.

(xi) Modify your working in (ix) to solve the system

X~ 3Xl + X2 + 2X3 + 5 x~ 2Xl + 2X2 + 2X3

x~ 2Xl + X2 + 3X3

1 (3 -4) 5. Let C = 5" 4 3 .

(i). Find all the eigenvalues of C.

(U) Find the corresponding eigenvectors.

(iii) What does C represent geometrically? Did you expect C to have any real eigenvalues?

2

! - -; f- 8 .. ,6-_ {C-I Gje ~

.~ f-I:>t~ 6 f- ~ ~C e t­

. ~ 'f-t ~ (C~+- k e t--

;( .: - "- (,)/ "L&a; .I-:{", ­

t (~~I I e--(£/ ):/(),

..? J \ iiv) () .. n' c4J.,,-,,(~ ~-- ~~) ~O(:... CU<.-c.<>f {~ •

No~

MATHEMATICS 2080W

TUTORIAL 23 9, 10, 11 OCTOBER 2007

( 2 0 -1) L Let A = 0 2 -1 .

o 0 1

(i) Find the eigenvalues of A . Is this information sufficient to conclude that A is NOT diagonalizable? Explain.

(ii) Find the eigenvectors of A.

(iii) Describe geometrically the eigenspaces of A.

(iv) Find a cartesian equation for the eigenspace associated with the repeated eigenvalue.

(v) Is A diagonalizable? If so, find an invertible matrix P and a diagonal matrix D such that P-1AP = D.

(vi) Can you find an orthogonal matrix Q such that Q-1 AQ = D? Explain geometrically why this can/cannot be done.

2. (i) Identify the curve 12xy - 5y2 = 36 by reducing the equation to a standard form. Give the directions of the new axes.

(ii) Sketch the given curve and find the angle through which the x- and y-axes must be rotated to obtain the standard form.

(iii) Modify your working in (i) to reduce the equation

2 24 16 12xy - 5y + Ji3 x + vT3 Y = 32

to a standard form. Sketch its graph.

3. The curve C has cartesian equation x2 + 4xy + 4y2 + 4v15x + 3V5Y = 15.

(i) Identify C by reducing its equation to a standard form with respect to a new set of axes.

(ii) Sketch the graph of C with respect to the x- and y-axes, and show the directions of the new axes.

(iii) Through what angle must the x- and y-axes be rotated so as to reduce the equation of C to the standard form?

4. A surface S has cartesian equation

llx2 + lly2 + 8z2 - 2xy + 4xz + 4yz - 2V6x - 2V6y + 4V6z = 18.

(i) Write down a 3 x 3 symmetric matric A associated with the equation of the surface S.

(ti) Show that ( _ ~ ), ( -i) and ( ~ ) are eigenvectors of the matrix A.

Are these eigenvectors linearly independent? Why/why not?

(iii) Find a orthogonal matrix P such that p-l AP is a diagonal matrix.

(iv) Identify the surface S by reducing the equation to a standard form.

(v) Give the directions of your new axes and sketch the surface S.

(

2 1 _-L) 3 v'2 JIB 2 1 i..:-

5. Let A = ~3 v'2 ';;[8 . :3 0 Ji8

(i) Is A an orthogonal matrix?

(ii) Find det A.

( 3 - 2J2)'

(iii) Show that x = J21- 1 is an eigenvector of A. What is the corresponding eigen-

value? Did you expect this eigenvalue? Explain.

(iv) Calculate Ax where x = ( i )-(v) Compare the values of II Ax II and II x II. Did you expect this?

(vi) Let y = ( ! ). Find Ay.

(vii) Calculate the angle between x and y, and also the angle between Ax and Ay . Did you expect this?

(viii) What, if anything, can you say about the other two eigenvalues of A?

2

TUTORIAL 24

(

6 -3 -2\ 1. Let A = ~ -2 -6 3 I

, \-3 -2 -6)

MATHEMATICS 2080W

(i) Show that A is an orthogonal matrix .

(ii) Find det A.

16, 17, 18 OCTOBER 2007

(iii) Use the results from (i) and (ii) to determine what A represents geometrically.

(iv) Show that (y) is an eigenvector of A.

(v) What additional information can you add to your answer to (iii) as a result of (iv)?

(1) (-5) (vi) Note that x = ; is orthogonal to ~ . Find Ax and hence the cosine of the angle

between x and Ax. 1

(vii) Calculate "2(traceA - 1), and compare your answer with that found in (vi).

2. Let A be the matrix representing a reflection of}R3 in the plane 2x + 4y - z = O. Without constructing the matrix A, answer the following questions:

(i) What are the eigenvalues of A?

(ii) What are the corresponding eigenvectors of A?

(iii) Describe the eigenspaces of A.

(iv) What is the value of det A?

(v) Use the above information to construct the matrix A.

3. Construct a 3 x 3 matrix A representing a projection of]R3 onto the plane 2x + 4y - Z = 0

4. Do the following improper integrals converge or diverge?

100 1 (i) '2 dt,

1 t 100 1

(ii) 1. Vi dt, (iii) 100

e-tpn t dt

5. (i) Does the improper integral ~ dt converge? JOO -st

1 t

(ii) Does the improper integral ~ dt converge? 11 -st

o t

(iii) What does (i) and (ii) tell you about the function ~?

6. Find the Laplace transform of each of the following functions :

(i) t + 3t4 - 4 sin 2t

(ii) e2t cos 3t

(iii) lOe3t cos t sin t

(iv) (1 + e2t )2.

What property of the Laplace transform did you use in (i)?

2

I

1·'1~ ,J-OJ' () ~{- ~(~J1Qd Tb- /Zt.ro/4~4 L ;. ¥- O~ 14dcf

(ly ft(f)"' 6(f), A-(_::): /~(-J) ~(t)~/~(/) .

Q. ~ . (Lj

S-l

MATHEMATICS 2080W

TUTORIAL 25

1. Find C-l{F(8)} where F(s) is :

(i) (8 ~ 1)2 (ii) (8 ~ 1)3

982 - 4s - 4 (iv) 8 3 _ 82 _ 28

8+3 (v) (82 + 1)(82 + 38 + 2)

14, 15, 16 OCTOBER 2008

28 -7 (iii) (s _ 1)2 + 9

2. (i) Use the Laplace transform to convert the initial-value problem

(D2 + 4D + 4)y(t) = t2e-2t , yeO) = 3, y'(O) = I,

into an algebraic problem.

(ii) Simplify the algebraic problem in (i) and hence find the solution yet) to the given initial-value problem. .

(iii) Use the method of undetermined coefficients to solve the initial-value problem in (i).

(iv) Use variation of parameters to solve the initial-value problem in (i).

(v) Compare your answers in (li), (iii) and (iv).

3. Suppose J(t) and get) have Laplace transforms F(s) and G(s) respectively. Is it true that C{J(t)g(t)} = F(s)G(s)? Give reasons.

4. Find '

(i) C{tu(t - I)}

(ii) C{ u( t - 11"/2) cos t}

(iii) C{(t - 1)3et u(t - I)}

(iv) C{J(t)} where J(t) = { ~2

e-S e-3s

5. Let F(s) = -2- - -( 1)2 ' s +1 s+

(i) Find C-1{F(s)} = J(t) .

if O$t<2 if t 2:: 2

(ii) Express J(t) as a piecewise defined function.

(iii) Find the values of J(2), J{3} and J(4) .

1

6. We want to solve the initial-value problem

y"(t) + yet) = k(t), yeO) = y'(0) = 0,

{4 ifO<t<2 .

where k( t) = t if t ~ 2 usmg Laplace transforms. Proceed as follows:

(i) Sketch the graphs of get) = tu(t - 2), h(t) = (t - 2)u(t - 2) and k(t).

(ii) Express k(t) in terms of the unit step function u(t - a).

(iii) Find 1:{k(t)} using your expression in (ii) .

(iv) Find the partial fractions decomposition of each of

1

8(S2 + 1) and

1

82(82 + 1)"

(v) Use the above results to solve the given initial-value problem.

(vi) Check that your solution satisfies the conditions y(O) = 0 and y'(O) = O.

(vii) What is the value of y(2)'?

(viii) What is the value of y(4)?

(be) Use the Convolution theorem to find

1:-1 { I} S(S2 + 1)

and C-1

{ S2{ 8} + 1) } .

2

MATHEMATICS 2080W

EXERCISE SHEET 1

S refers to Calculus: Concepts and Contexts by Stewart, (3rd Edition).

1. S: §9.6 Exercises 15, 23.

2 . S: §11.1 Exercises 15, 17, 19, 21, 23.

3. Sketch the family of level curves and hence the graph of each of the following functions:

(i) z = In(x2 + y2)

(ii) z = arctan(x2 + y2 )

(iii) z = (x2 + y2)e- Cx2+y2) .

4. S; §11.1 Exercises 25, 31 - 36.

2008

5. Sketch the solid in the first octant that is bounded by the graphs of the given equations.

(i) x 2 + Z2 = 9, y = 2x, y = 0, z = 0,

(ii) 2x + y + z = 4, x = 0, y = 0, z = 0,

(iii) z = x 3, X = 4y2, 16y = x 2 , z = O.

6. Sketch the region in 1R3 bounded by the graphs of the given equations. 0) z = 0, Z = y, x2 = 1 - y, (ii) y = 2 - Z2, Y = z2, X + z = 4, x = O.

7. S: §9.7 Do a selection from Exercises 3-33.

8. S: §10.1 Exercises 1, 3, 5, 7, 9, 11, 17-22, 23, 31-:34.

9. Does the curve r(t) = (asintcos2t,asintsin2t,acost),a > 0 lie on a sphere? If so, find the radius of the sphere.

10. In each of the following cases find the intersection (if any) of the curve and surface:

(i) ret) = (2 + 3t, t, 1 + 2t), x + 3y - z + 7 = 0;

(ii) r(t) = (2 + 3t, t, 1 + 2t), x - y - z + 1 = 0;

1 (iii) ret) = (0, -, t), x2 + y + 2z = 4;

t

1 1 t2

(iv) r(t)=( -, I--,-), z= 2 '1; t t 2 l+x + y

1 1 t 2 1 1 (v) r (t ) = Ct, 1 - t ' 2 )' z = '2(x + y -1- ;;2 - 1).

11. Describe the curve of intersection of the surfaces and find parametric equations for the curve.

(i) z=8-x2 -2y2, z=x2+2y2, 'rr

(ii) z = )9 - x2 - y2, Z = )(x2 + y2) cot2 0:, 0 < 0: < 2'

12. S: §1O.2 Exercises 5, 17, 19, 21, 27, 28, 31, 32, 43, 47.

13. The curves rl(t) = (e t - 1, 2sin t , In(l + t)) and r2(t) = (1 + t , t 2 - 1, t3 + 1) intersect at the

origin. Find the angle of intersection.

14. S: §10.3 Exercises 1, 3, 15-17, 18, 21, 22,23,25,27,39,40, 52.

15. Find a cartesian equation for the osculating plane to r(t) (cos t, sin t , cos t) at the point 11'

where t = 4'

16. S: §10.4 Exercises 3, 8, 17.

SOME ANSWERS

8. (18) II , (20) I, (22) III.

9. The radius is a.

1 1 1 10. (i) (-4, -2, -3) , (ii) no intersection, (iii) (1 ,1 ,1) and ( ;;:;,3, -), (iv) (1,0, -2) '

. v3 3 lIe

(v) r(t) = (t' 1 - t' "2)'

11. (i) (2cost,J2sint,4), (ii) (3sino:cost,3sinasint,3coso:).

• o. 7 16 3 12. (28) (1,0,4),55, (32)( -, -, --).

13. ~ 2 '

3 5 'if

14. (16) j8 () v'8 ( ') I cos xl . . . (2 + 4t2)3/2' 18 3J3' 2_) (1 + sin2 X)3/2' (40) oscuiat1l1g plane 1S 3x ~ 3y + z = 1,

(52) 10x3 - 15x4 + 6x5•

15. x = z.

16. (8) At t = 0, Q = (1,0, I), ~ = (0, -2,0) , speed = J2.

2

MATHEMATICS 2080W

EXERCISE SHEET 2 2008

S refers to Calculus: Concepts and Contexts by Stewart, (3rd Edition).

1. S: §11.2 Exercises 1, 5, 7, 9, 11, 13, 15, 33, 35.

2. By considering what happens along each of the lines x = 0 and y = 0, show that the function

{~ sin(x2 + y)

f(x,y) = x+y 1

li (x,y)¥~,~

~ x=y=O

is not continuous at (0, 0).

3. S: §11.3 Exercises 9, 17, 21, 23, 25, 47, 51, 77, 79.

4. S: §1l.4 Exercises 1, 3, 9, 13.

{

X2(X - y) . 2 2 If (x, y) =1= (0, 0)

5. Letf(x,y) = x +y

o if x = y = O.

(i) Find J:c;(O, 0) and fu(O, 0).

(ii) Is f continuous at (0,0)'1

(iii) Is f differentiable at (O,O)? Give full reasons.

{

x2y . -4--2 If (x,y) =1= (0,0)

6. Let f(x,y) = x +y

o if x = y = O.

(i) What value does f approach as (x, y) tends to (0,0) along the x-axis?

(ii) What value does f approach as (x, y) tends to (0,0) along the y-axis?

(iii) What value does f approach as (x, y) tends to (0,0) along the parabola y = x2?

(iv) Is f continuous at (O,O)?

(v) Is f differentiable at (O,O)?

7. S: §11.5 Exercises 3, 7, 17, 23, 37, 44, 45.

8. The radius r and altitude h of a right circular cylinder are increasing at rates of 0, 01 cm/min and 0,02 em/min, respectively. Use a chain rule to find the rate at which the volume is increasing at the time when r = 4 em and h = 7 cm. At what rate is the total surface area changing at this time?

1

9. Use the definition of the directional derivative to find the directional derivative of the function in question 5 at the point (0,0) in the direction (i) (1,0), (ii) (0,1), (iii) (1,2), (iv) Do all the directional derivatives of f at (0,0) exist?

10. S: §11.6 Exercises 7, 9, 21, 23, 25, 27, 33, 35, 37, 41,47, 49.

11. Which of the following are true and which are false? For those that are true, provide a proof. For those that are false, give a counterexample.

(i) If fx(a, b) and fy(a, b) both exist, then f is differentiable at (a, b).

(ii) If fAa, b) and fy(a, b) both exist, then f is continuous at (a, b).

(iii) If /z(a, b) and fy{a, b) both exist, then all the directional derivatives exist at (a, b) .

(iv) If all the directional derivatives exist at (a, b), then t is differentiable at (a, b).

(v) If / is not continuous at (a, b), then fz(a, b) and fy(a, b) do not exist.

(vi) If all the directional derivatives exist at (a, b), then t has a tangent plane at (a, b).

(vii) If fz(a, b) and /y(a, b) both exist, then f has a tangent plane at (a, b).

(viii) If tx(a, b) = fy(a, b), then f is differentiable at (a, b) .

(ix) If fx(a, b) and fy(a, b) both exist but fx(a, b) =1= fy(a, b), then / is not differentiable at (a, b).

(x) If t is differentiable at (a, b), then all the directional derivatives of t at (a, b) have the same value.

(xi) If a curve C lies on a surface, then the osculating plane to C at the point P is the same as the tangent plane to the surface at P.

ANSWERS

5. (i) fx(O,O) = 1, fu(O, O) = 0, (ii) Yes, (iii) No.

6. (i) 0, (ii) 0, (iii) ~,(iV) No, (v) No.

8. 0, 8811' cm3/min; 0,4611' cm2 imino

9. (i), (ii) See answers to question 5; (iii) - 1~, (iv) Yes. 5v5

11. Get your answers and reasons checked!

2

MATHEMATICS 2080W

EXERCISE SHEET 3 2008

S refers to Calculus: Concepts and Contexts by Stewart, (3rd Edit ion).

1. S: §12.1 Exercises 13, 14, IS.

2. S: §12.2 Exercises 3, 11 , 13, 15, 17, 19, 20, 25 .

3. S: §12.3 Exercises 5, 10, 15, 17, 19, 33, 36, 38, 39, 41, 50.

4. S: §12.5 Exercises 1, 3, 9.

5. (i) S: §12.6 Exercises 1, 2,3.

(ii) Find the area of the surface z = J2Xij for 0 ::; x ::; a, 0 ::; y ::; b.

6. S: §13.6 Exercises 9, 12, 19, 27.

7. S: §12.7 Exercises 3, 5, 9, 17, 19, 25, 26, 27, 29, 31.

S. Which of the following transformations from ]R2 to ]R2 are linear transformations?

(i) f ( x ) = ( x + 1) (ii) f ( x ) = ( ~2) (iii) f ( x ) = ( 2x - y ). y y - 3 y sm y y x + 3y

9. Find the im(age10f ~he iin)e x(t) = (1,2,3) + t(2, -4, 1) under the linear map represented by

the matrix 4 0 1 . -1 1 2

10. Find the image of the plane 2x - 4y+ z = 0 under the linear maps represented by the following

matr(iceSi 2 3) (1 2 3) (i) 4 0 1 (ii) -3 10 1 .

-1 0 2 1 2 3

11. Let A = (i ~l ). Find the parallelogram to which A takes the square with vertices at

(0,0), (1,0), (1,1), (0,1). Find also its area (i) by using determinants, (ii) directly from the vertices of the parallelogram.

12. Let T be the transformation represented by the matrix (; !), and let R be the region

bounded by the lines y = 0 and x = 1 and the curve y = x2 . Find the area of the image of R under T.

13. (i) Show that the ellipse :: + r: = 1 can be given parametrically by x = a cos t, y = b sirli, o ::; t ::; 27r.

(ii) :i:d (thrmlc r the ellipse in (i) under the t ransfonnation represented by the matrix

a b /

(iii) Find det A.

(iv) Use the results of (ii) and (iii) to find the area inside the ellipse ~: +- r: = l.

(X) ( X2 + y2 + z2 \

14. (a) Let F y = xyz ) . F ind the derivative matrix of F and use the affine z l+x+ y+ z

("I) lOS)

approximation at ; to estimate F ( ;: ~ . Compare your estimate with the exact

value.

(b) Let F ( ~ ) = ( 3;x ~.; ) . Find the derivative matrix of F and use the affine approx-

imation at ( ~ ) to estimate F ( ~: ~ ) . How does this approximation compare with

the exact value?

15. (a) The region R in the xv-plane is bounded by the graphs of x + y = 6, x - y = 2 and y =0.

(i) Find the region R' in the uv-plane onto which R is mapped under the transformation

x = u + v, y = u - v.

(ii) Calculate ~~:: ~t and compare it with the ratio of the areas of Rand R' .

(b) Find the region into which the square with vertices at (0,0), (1,0), (0,1) and (1,1) is

( x) ( x2 _ y2 )

transformed by the mapping F y = xy .

Find the approximate area of the image under F of a circle of area 10-3 centred at (1,3).

(c) Find approximately the volume of the image under the mapping u = xyz, v = x + y + z , w = x2y, of a sphere of volume 10- 6 with centre at (1,2,0).

16. S: §12.4 Exercises 9, 11 , 13, 15, 28, 31.

17. (i) Use polar co-ordinates to evaluate J /(x+ y) dA where R is the region bounded by the

R

graph of x2 +- y2 - 2y = O.

(ii) Find the volume of the solid that lies inside the graphs of both x2 + y2 + Z2 = 16 and x2 + y2 - 4y = O.

t / y!2;_y2 (iii) Express Jo Jo dx dy as a repeated integral using polar co-ordinates. Do N OT

evaluate the double integral.

2

18. S: §12.8 Exercises 1, 3, 9, 15, 19, 27, 31, 33.

19. S: §12.9 Exercises 15, 19.

20. The transformation given by x = u2vw, y = uv2w, Z = uvw2 maps a region R onto a region

R'. Express J J J (xy~)3/4 dV in terms of the volume of R'. R

ANSWERS

5. (i) (2) 9v'30 1[, 6. (12) ~ (17V17 -1). 48

(ii) 2~ Vab(a + b).

8. Only (iii) is a linear transformation.

9. x = (14,7,7) + t( -3,9, -4).

10. (i) 6x - 5y - 8z == 0 (ii) x = ;\(1,1,1).

11. 10. 12. 2/3 .

13. (ii) x2 + y2 == 2, (iii) - 2/ ab, (iv) nab

14. (a) Estimate is 0, 9 . The exact value is 0,88 . (

2,8 ) ( 2~85 )

3,9 3,9 (b) The affine approximation is exact for a linear map.

15. (b) Get your answer checked 1

The approximate area of the image of the circle is 2 x 10-2 .

(c) 6 X 10-6 .

(28) 16. 16. 9

17. (i) 1r , (ii) 1~8 (31f - 4), (iii) get your answer checked I

20. 4(volume of R').

3

MATHEMATICS 2080W

EXERCISE SHEET 4 2008

S refers to Calculus: Concepts and Contexts by Stewart , (3rd Edition).

1. S: §13.2 Exercises 1, 3, 5, 7, 9, 19, 27, 30, 33, 35.

2. Find the area of one side of the curtain whose base is y = x2 , ° :::; X :::; 1, and whose height above the point (x, y) is xy.

3. S: §13.3 Exercises 5, 9, 17, 19, 21 , 31, 33.

4. S: §13.5 Exercises 3, 5, 10, 13, 15.

5. S: §13.4 Exercises 7, 8, 9, 11, 19.

6. Find t he area of the region bounded by the hypocycloid r(t) = (cos3 t , sin3 t), ° :::; t :::; 27r . In addition, sketch the hypocycloid.

7. S: §13.7 Exercises 1, 3, 5, 7, 10, 17, 19.

8. S: §13.8 Exercises 3, 5, 7, 13, 18.

1 + 25 V5 2. - ,---120

5. (8) 318 . , 5

31l" 8. (18) 2

6. 37r 8

7. ( 10) 91l".

Answers

1

MATHEMATICS 2080W

EXERCISE SHEET 6 2007

1. (a) Express (-3,0, -3) as a linear combination of (1,1,2) and (3,2,5). Can (-3,0, -3) be written as a linear combination of (1, I, 2), (3,2,5) and (4,3, 7)? In how many different ways is this possible?

(b) Is (1, I, 1) a linear combination of (1, 1,2) and (3,2, 5)?

(14) (213) 2. (a) If A == ~ 120 and B = 0 4 -1 find AB and BA.

(b) Find two 2 x 2 matrices A and B such that AB = ° but A =F 0 and B =F O.

(c) Give examples to show that the following statements are false:

(i) AC = BC, C =F O:::} A = B, (ii) (A + B)(A - B) = A2 - B2.

3. (a) Find the 2 x 2 matrix that represents a reflection of JR.2 in the line y = mx

(i) directly, (ii) using matrix multiplication.

(b) Write down the 3 x 3 matrix that represents a projection of lR3 onto the xz-plane.

4. (a) Can a homogeneous system of linear equations have exactly two solutions? Explain.

(b) Decide how many solutions there are to each of the following systems of equations. (i) x2 + y2 - 1 = 0 (ii) 6xy + 3x2 + 6x = 0

x - y + 1 = 0 3x2 - 3y2 + 1 = o.

5. Can you construct a matrix A so that the system of linear equations Ax = 0 is inconsistent? Explain.

6. Given that (3, -1,0) and (-7,1,2) are each solutions of Ax = b where A is a 3 x 3 matrix, find two non-zero solutions of Ax = o. Find also a third solution to Ax = b.

7. Find a condition on a, band c so that the system of linear equations

x - 2y + 3z a

2x - 4y + 6z b

-3x+6y-8z c

is consistent. What are the solutions in the case when a = -1, b = -2 and c = 17

8. Express the solutions of the equation Ax = 0 as a(li~ea~ cot)bination of vectors where A is

(i) (; ~ !3) (ii) ~ ~ ~

(1 -1 1 4) (1 -1 1 4) (iii) 2 2 -1 (iv) 2 -2 3 9

-1 4 1 -1 4 1

9. Verify that w = 2, x = 3, y = 4, z = 5 is a solution of the system of linear equations

w - x +y + 4z 23

2w - 2x + 3y + 9z 55

-w + x + 4y + z 22 .

Use your solution to question 8(iv) to write down the general solution to the above system.

10. In each of the following cases construct a system o(Iinear equations having the given' set as solutions.

(i) {A(-2, I, 1): A E lR}

(ii) {A(2, -1,3) + j.L(O, 1,4) : A, j.L E lR}

(iii) {(I, 4,5) -I- A(2, -1, 3) + j.L(O, 1,4) : A, j.L E JR} .

11. For which values of k does the system of linear equations

x + y + (k + 4)z

2x + y + (k + 8)z -3x - 3y + (k2 - 4k - 12)z

-1

-6

k2 + k + 3

have (i) a unique solution, (ii) infinitely many solutions, and (iii) no solution? List all the solutions (in terms of k) to this system.

12. (a) In each of the following cases, set up and solve a system of linear equations to balance the given chemical equation:

(i) Cu + HN03 -7 Cu(N03h + H20 -I- NO

(ii) Ga3(P04h + H3P04 -7 Ca(H2P04h-(b) Set up and solve the system of equations for the currents in the branches of the given

network. . C,. . 10 V ~l.. l y. V

3Jlf " 'I~l~~ 13. The following diagram shows a road network where all the streets are one-way. The flow

of traffic in and out of the network is measured in vehicles per hour, and is indicated on the diagram. Let Xl, X2, X3 and X4 denote the number of vehicles flowing along the various branches per hour.

~oo .)SD

tS'O (00 XI

:( It 12,.

to 0 ')0

~o

2

(i) Construct a system of linear equations in the unknowns Xll X2, X3 and X4 that describes the traffic flow in this road network.

(ii) Solve the system of equations you constructed in (i).

(iii) Use your solution to find the maximum and minimum values of X2'

14. (a) For which value(s) of k is the vector (1, -2, k) a linear combination of (3,0, -2) and (2, -1, -5)7

(b) Show that one of the vectors (1,4,5)' (1,3,2), (0,1,2) and (1,4,4) is a linear combination of the others. Does that imply that everyone of these vectors is a linear combination of the others?

(c) Express the polynomial 12x2 + 8x - 2 as a linear combination of the functions 1 + x, 1 - X2, X - 2x2 .

15. Which of the following subsets of R3 are linearly independent? Give reasons for your answers. (i) {(I, 2, 3), (1 , -3,4), (2, -2, -1), (-1, -5, 14)} (ii) {(2, 0, 1), (0,0,1), (1,0, I)} (iii) {(2,0,1), (0,0, 1), (0, 0, O)} (iv) {(2,0,1),(1,2,3),(2,4,6)} (v) {(2,5,-3),(4,10,6)} (vi) {(1,2,3),(4,-3,2),(1,1,1)}

16. For which value(s) of>- is the set A. = {(1,-1,1,0),(1,-2,1,4),(2,1,2,>-)} a linearly inde­pendent subset of ]R4? Where possible, express the third vector in A as a linear combination of the other two vectors in A.

17. (a) vVhich of the sets listed in question 15 are generating sets for ]R3?

(b) Which of the sets listed in question 15 are bases for ]R3?

18. Find two different bases for each of the following linear subspaces.

(i) {(x,y,z) E R3: x + 2y - 3z = O}

(ii) {(x, y, z, w) E R4 : X + 2y + w = 0, 2x - y - Z + w == O}

19. Any set on which operations of addition and scalar multiplication are defined, and which is closed under the formation of linear combinations is called a linear subspace. Decide, giving reasons, which of the following are linear subspaces.

(a) The set of all 2 x 2 matrices which are of the form (~ 7), where n is a non-negative

integer.

(b) The set of all polynomials of degree 2.

(c) The set of all polynomials with degree not exceeding 2.

(d) The set of all real-valued functions of a real variable whose graph passes through the point (0,1).

(e) The set of all solutions of the differential equation ~~ + Y cos t = t.

(f) The set of all solutions of the differential equation ~; -\- Y cos t = O.

3

(g) The set of all solutions of the matrix equation (~ ; ;) x = O.

(h) The set of all solutions of the matrix equation (~ ~ ;) x = ( ~1 ) .

ANSWERS

1. (a) (-3,0, -3) ;" 6(1,1,2) - 3(3,2,5); Yes, in infinitely many ways . (b) No.

( 2 17 -1) (31 24) 1 (1 _ m2 2m) (1 0 0)

2. (a) 10 45 5 , 12 38 . 3. (a) --2 2 2 _ 1 (b) ° 0 ° 16 16 22 1 + m m mOO 1

4. (a) No, (b) (i) Two, (ii) Four. 5. No.6. >-(1O, -2, -2).

7. b === 2a, x = (5,0, -2) + >-(2, 1, 0).

8. (i) x =·>-(3, -3, I), (ii) x = (0,0,0), (iii) x == >-( -1,2, -1,1),

(iv) x = >-(1,1,0,0) + J1.( -3,0, -1,1).

9. x=(2,3,4,5)+>-(I,I,0,0)+J1.(-3,O,-I,l).

10. Check your answers by solving your systems of linear equations.

11. (i) Unique solution if k #- 0, k #- 1. x = _1_(1 - 9k, 3k - k2 - 4, k + 1) k-l

(ii) Infinitely many solutions if k = O. x = (-5,4,0) + >-(-4, 0, 1). (iii) No solution if k == 1.

12. (a)(i) 3Cu + 8H N03 -+ 3Cu(N03h + 4H20 + 2NO.

(ii) Ca3(P04h + 4H3P04 -+ 3Ca(H2P04h. . 35. 38. 1

(b) ~1 = 9,'L2 = 9,23 = 3' 13. (Xl, X2, X3, X4) = (350 - a, a, 150 - a, a), 0 ::; a ::; 150.

14. (a) k = -S. (b) (1,4,5) is not a linear combination of the other three.

(c) 236

(1 + x) - 332 (1 - x2

) - ~(x - 2x2 ).

15. (i) Dependent, (ii) Dependent, (iii) Dependent, (iv) Dependent,

(v) Independent, (vi) Independent.

16. >- #- -12. If >- == -12, then (2,1,2, -12) = 5(1, ·-1, 1,0) - 3(1, -2, 1,4).

17. (a) (i) and (vi) are generating sets.

IS. Get a tutor to check your answer.

(b) Only (vi) is a basis for ]R3.

19. (8,) No, (b) No, (c) Yes, (d) No, (e) No, (f) Yes, (g) Yes, (h) No.

4

MATHEMATICS 2080W

EXERCISE SHEET 7

1. Let A = -1 1 2 . Find ( 2 3 4) \ 2 -3 5

(i) the minor determinant M12

(ii) the cofactor C12

(iii) det A •

1 -2 2 2 3 4 .) . I 2 1 -2 3 1 3 5 7

2. (1 Fmd 3 4 -8 1 and 2 3 6 7

3 -11 12 2 1 5 8 20

(ii) 'Without expanding, find the value of

1 1 1 1 a a+l a+2 x y z and b b + 1 b + 2

y+z x+z x+Y, c c+1 c+2

(iii) Let A be an n x n matrix such that A2 = J. Find the value of det A.

(iv) Suppose A is a 5 x 5 matrix for which detA = -3. What is the value of det(2A)?

3. SOlV

l

1 ~\ A 3 1 1

(i) 2 2 - .\ -1 = 0 , 5 -5 2 - A

a2 b2 c2

4. Factorise the determinant I be ae ab abe

1

4 - A

(ii) ~ 4

2-,\ 6

6 -5 1= O.

-3- A

(Hint: Try to extract common factors; don't just bash it out.)

x a b c

5. Solve the equation I a x b c 1= O. a b x c Q. b c X

x-3 x+2 x-I 6. Solve the equation I x + 2 x - 4 x = 0 .

x-I x+4 x-5

2007

7. Let A and B be 3 x 3 matrices with detA = -2 and detB = 3. Find det(AB), det(BT A), det(A3 ) and det(A-l) .

8. (i) Express det B in terms of det A, given that B = p-1 AP.

(ii) Let A be an invertible matrix such that A2 = A. Show that det A = 1.

1

( 2 1 4)

9. Let A = 1 2 -1 . 113

(i) Calculate det A and hence show that A is invertible.

(ii) Find the adjoint of A and hence A-I.

10. (i) Solve the given system by Cramer's Rule:

Xl - 2X2 - 3X3

Xl + X2 - X3

3Xl + 2X2

3

5

-4

(ii) Use Cramer's rule to determine for which values of k the system

kXl + X2 4

kXl + kX2 4

has a unique solution. For which value(s) of k is the system inconsistent? Explain.

1L :ea rrnT (nT lineMly d;:;rn ) , ( ! ) , ( i5 ) }

ANSWERS

1. (i) -9, (ii) 9, (iii) 53.

2. (i) 0, 16, (ii) 0, 0, (iii) ±1, (iv) -96.

3. (i) -3 or 4 (repeated), (ii) 1 Or 1 ± -133.

4. (a - b)(b - c)(c - a)(ab + ae + be) .

5. x=a, b, cor-(a+b+e).

6. ~ 3·

9. (i) 6,

1 7 -6 -6 -8 --. , , , 2·

(

7' 1 (ii) adj A = -4 2

-1 -1

10. (i) Xl = -4, X2 = 4, X3 = -5,

11. (i) No, (ii) Yes.

8. (i) det B = det A.

6 , so A-I = ~ adj A. -9 )

3 6

(ii) k i= 0, k # 1; k = O.

2

MATHEMATICS 2080W

EXERCISE SHEET 8 2008

1. (a) State whether the given differential equations are linear or nonlinear. Give the order of each equation.

(i) (1 - x)y" - 4xy' + 5y = cosx (ii) yy' + 2y = 1 + x2

(iii) x 3y(4) - x 2y" + 4xy' - 3y = O.

(b) Verify that the indicated function is a solution of the given differential equation. (i) 2y' + y= 0; y = e- x/2

(ii) :~ - 2y = e3x; y = e3x + 10e2x

b (iii) xy" + 2y' = O· y = a + -. , x

(c) For which real value(s) of m is y = xm a solution of the differential equation x 2y" _y = O?

(d) (i) Show that YI = x 2 and Y2 = x3 are both solutions of X2y" - 4xy' + 6y = O. Are aYl and {3Y2, a, {3 E JR., also solutions? Is YI + Y2 a solution?

~ ~12 (ii) Show that YI = 2x + 2 and Y2 = -2' are both solutions of y = xY' + -2-' Are aYI

and {3Y2, a, (3 E JR, also solutions? Is YI + Y2 a solution'?

2. In each of the following cases find a differential equation for the given family of curves.

(i) y = ce-x

(ii) cy2 + 4y = 2X2

(iii) y = aeX + {3e2x + oe3x .

3. Find the differential equation of the family of circles passing through the origin with centres on the x-axis.

4. Solve the following differential equations:

(i) 4y + yx2 - (2x + xy2)y' = 0

(ii) y'(y+l)2 = x 2ylnx

(iii) if sec2 x + cosec y = 0

(iv) sin x(e-Y + 1) = (1 + cosx)y', yeO) = O.

5. Find a singular solution for the differential equation y' = x.J'1=Yi. 6. (a) (i) B

Show that the functions y(x) = Ar+-, A,13 E JR., are solutions of the second-order, x

homogeneous, linear differential equation J}yll(X) + xY'(x) - y{x) = O.

(ii) 1 Verify that y(x) = 2ex (1- -) is a particular solution of

x x 2ylI (X) + xY'(x) - y(x) = 2x2ex •

1

(iii) Use the Fundamental theorem to write down a further three solutions of the differ­ential equation in (ii).

(b) (i) Write down G family of functions which are solutions of the differential equation (yll(X))2y(x) = O.

(ii) Show that y(x) = e2x is a particular solution of the differential equation (yll(X ))2y(x) = 16e6x .

(iii) Use the results in (i) and (ii) to show that the Fundamental theorem does not carry over to non-linear differential equations.

(c) (i) Verify that y(x) = x 2 + X + 3 and Yl,xj = 2X2 - X + 3 are both solutions of the differential equation X2y"(X) - 2xy'(;::;) + 2y(x) = 6.

(ii) 'Write down a non-zero solution to the linear differentia! equation X2y"(X) - 2xy' (x) + 2y(x) = O.

7. Solve the following differential equations:

(i) y' + y = e3x

(ii) x 2 y' + xy = 1

(iii) y' cos2 x sin x + y cos3 X = 1

(iv) (x + 2xy2 - 2y)y' + y = 0 (HINT: Write as ~: = ... )

y (v) y' = -, y(5) = 2.

V-x

8. Solve the following:

(i) 2y2x - 3 + (2yx2 + 4)y' = 0

(ii) y3 - y2 sin x - x + (3xy2 + 2y cos X)yl = 0

(iii) xy' = 2xex - y + 6x2

(iv) 4y+2x-5+(6y+4x-l)y'==0, y(-l) =2.

9. Find an integrating factor of the form xOyf3, a, {3 E JR, for the non-exact differential equation (y2 + xy)dx - x2dy = O.

10. (a) A large tank is filled with 500 litres of pure water. Brine containing 2 kg of salt per litre is pumped into the tank at the rate of 5 f./min. The well-mixed solution is pumped orit at the same rate. Find the number of kilograms of salt A(t) in the tank at any time.

(b) Solve the problem in (a) under the assumption that the solution is pumped out at a faster rate of 10 f./min. When is the tank empty?

(c) A. large tank is partially filled with 100 litres of fluid in which 10 kg of salt is dissolved.

Brine containing ~ kg of salt per litre is pumped into the tank at a rate of 6 f.jmin . The

well-mixed soluti;n is then pumped out at a slow'er rate of 4 ilmin. Find the D.umber of kilogTams of salt in the tank after 30 minutes.

2

l.(a)(i) linear, 2,

(d) (i) yes, yes,

(ii) nonlinear, I,

(ii) no, no.

ANSWERS

(iii) linear, 4. (c) ~±V5 2

2. (i) y' + y = 0, (ii) (x2 - y)y' == xV, (iii) ylll - 6y" + lly' --- 6y = O.

'~ 2xyy' - '1,2 - x 2 v. . - - & •

4 . (i) 8 In x + x2 - 41n y - y2 == C

(iii) 4 cos y == 2x + sin 2x + c

x 3 x3 y2 (ii) 3 1nx - 9 == 2 + 2y + In Iy l + c

(iv) (1 + cos x)(l + eY ) == 4.

5. y = 1. 6. Get your answer checked.

7. (i)

(iii)

(v)

8. (i)

(iii)

9. 1

xy2 '

y

y == sec x + c cosec x

x == ~ + ~ 2 y

x 2y2 - 3x + 4y == c

x y - 2xex + 2ex - 2x3 == C

(ii) xy == In x + c

(iv) yx == 1 + ce-y2

x2

(ii) xy3 + y 2 cosx - ~ = c 2

(iv) 4xy + x 2 - ,sx + 3y2 - Y = 8.

10.(a) 1000 - 1000e- 16o , (c) 64,38 kg.

3

l\;lATHElVIATICS 2080W

8XERCISE SHEET 9 2008

1. Use the definition to dete rmine whether the given functions are linearly independent or de­pendent on ( --;)(1 , 00):

(i) X , x2 , 4x - 3:Z;2

(ii) 2 + x, 2 + Ixl (iii) 1 + x, X, x2

.

2. Show by computing the Wronski an that the given functions a re linearly independent on the

indicated interval:

(i) x ~ , x 2; (0,,00) -

(ii) eX,e-x,e4x ; (-00,00)

(iii) 1, X , X2, _ . . ,xn ; (- 00,

(iv) eX , u''', x2ex ; (-00,00)

(v) cos x ,sinx,xcos :r,xsinx; (-00,00).

3. Verify that the given fUIlctions fc rm a ba::;is for the set of solutions of the differential equation on the indicated interval. Write down the general solution ,

(i) y" - y' - 12y = 0; e-<lx , e4x; (-c.o,<:::o)

(ii) x'2y ll -- 6xy' + 12y= 0; x3 , :r;4; (O,eo)

(iii) X3 ylll + 6X2 y" + 4xy' -- 4y = 0; :1:, x-2

) .:r - "ln x ; (0,00).

4. Find a second linearly independent solution y,J of each differential eq uation by putting yz (x) = U(X) Yl (x) and reducing the order:

(i) X2 y" - 7xy' + 16y = 0; VI = x-1

(ii) (1 + 2.x)y" + 4xy' - 4y = 0; Yl = e- 2x .

5. Find the general solution of the given differential equations:

(i-) 4y" + y' = 0

(ii) y" -- 36y = 0

(iii) y" + 9y = 0

(iv) y" _. 4,!/ + 5y = 0

(v) ylll - 4y" - 5y' = ()

(vi) ylll - 5y" + 3y' + 9y = 0

d5y dy (vii) -, - 16- = O.

dx" dx

6 Find the particular solution of the differential equation y'll + 12:;" + 36y' 0, s3.tis fying y(O) = 0, y' (O):::= 1, yl/(O) = --7.

7. Verify that the functions cos 2x, sin2x , sinxcosx and 1 - 2cos2 x arc solutions of the ho-. mogeneolls differential c4uatioll !/(x) + 4y (x) = 0, and that :<;(1:) = . 1 is a solut ioIJ of yl!(x) + 4y(x) = 4.

Find the general solution of the non-homogeneous di fferentia! equation.

8. (a) Show that the substitution 11, = In .1: reduces the difFerential equation

ax2 y"(:Y;) + bx:/(~:' ) + = 0, (J" b, c E lR

to a differential equation with constant coefficients.

(b) Use t he result from (a) to solve the following:

(i) X2y" + xy' + 4y = 0

(ii) X2y il + 5J.:y' + 4y = U.

9. Solve each di fferential equation by variat ion of parameters:

(i) y" + y = seCT

(ii) y" + y = sinx

(iii) y" + 2y' + y = e- x In :E

(iv) X2]/'_ .xy' + y = 4:1: In x, given that. T and x In x are solutions of the associated homoge­

neous differential equation .

10. So] ve the following differential eq uations by undetermined coefficients:

(i) y" + 3y' + 2:; = 6

(ii) y" -+ y' -. 6y = 2x

(iii) y" - 8y' + 20y = 100:1;2 - 26xex

(iv) y" - yl = -·3

(v) yll - y' + ly = 3 + d

(vi) yl' + 4y = 3 sin LX

(vii) y" + Y = 2x sin:;;

(viii) y" - 2y' + 5y = eX cos 2:;:

(ix) y" + 2y' + y = sinx -+ eos2x

(x) y'" - 6y" = 3 - cosx

(xi) yllf - y" - 4y' + 4y = 5 - e + e2x

(xii) y(4 ) - y" = 4x + 2xe - x .

2

ANSWERS

1. (i) Dependent,

4. (i) Y2=x41nx,

(ii) Independent, (iii) Independent.

5. (i) (iii) (v) ( vii)

(ii) Y2 = X.

x Cl + C2 e-"4

CI cos 3x + C2 sin 3x C1 + C2e-x + C3 e5x

C1 + C2e-2x + C3e2x + C4 cos 2x + C5 sin 2x

5 5 1 6. - - _e-6x + -xe - 6x ,

36 36 6

7. Get your answer checked.

8. (b) (i) CI cos(21n x) + C2 sin(21n ::c)

9. (i) Cl cos x + C2 sin x + x sin x + cos x In( cos x)

(.') . 1 11 C1 cos X + C2 SUi X - 2" x cos x

(iii) cle- x + C2xe- x + ~x2e-X ln x -- ~x2e-X 2 4

2 ( ) ') (iv) CIX + C2X In x + 3"x,ln x J

(ii) CI e - 6x + C2e6x

(iv) e2x (cl cosx + C2sinx) (vi ) cle- :7; + C2 e3x + C3xe3x

10, Check your answers by substituting into the differential equation.

3

MATHEMATICS 2080W

EXERCISE SHEET 10 2008

1. Use a geometric approach to find the eigenvectors and eigenvalues of

(i) the 2 x 2 matrix that represents a projection on the line y = x,

(ii) the 3 x 3 matrix that represents a reflection in the yz-plane.

2. Determine which of the indicated column vectors are eigenvectors of the giwm matrix A. Give the corresponding eigenvalue:

(i) A = (: i) ; X l = ( ~2 ) ,X2 = ( ~ ) ,X3 = ( ~2 )

( 1 - 2 2 ) ( 0 \ ( 4 ') ( - 1 \)

(ii) A = --;2 ~ ~2 ; Xl = i) , X2 = ~4 ,X3 = i . 3. Find the eigenvalues and eigenvectors of the given matrix:

(.) (-1 2) ( .. ) (-8 -1) ( ... ) (-1 2) 1 -7 8 11 16 0 III -5 1

(040) (123)

(iv) -1 -4 ° (v) 0 5 6 . o 0 -2 0 0 -7

4_ (a) Let X and y be eigenvectors of a square matrix A. Does it follow that x + y is an eigenvector of A?

(b) Suppose that x,y and x + yare eigenvectors of a square matrix A. What can you deduce?

(c) Let>. be an eigenvalue of the matrix A and let x be a corresponding eigenvector. In terms of >. and x, find an eigenvalue and eigenvector for each of A + I and kA, where k E 1Ft

( 17 2 -10)

5. (a) Let A = 2 8 -8 . Use the trace of A to find all the eigenvalues, given that 1 and -10 -8 14

10 are eigenvalues of A.

(

1 2 -2 4 -1 1

(b) Let B = 2 _ 2 2

1 -1 1

(i) Calculate det B. (ii) Is B invertible?

: ). -1

(iii) Find all the eigenvalues of B, given that -1 is an eigenvalue of multiplicity 2.

(iv) Is B diagonalizable? Give reasons.

6. Determine whether the given matrix A is diagonalizab!e. If so, find a matrix P that diagonalizes A and a diagonal matrix D such that p-l AP = D.

(.) ( 2 ! ) (ii) ( ~1 ; ) (in) ( ;_ 2 ~2 \

( 1 --1 n 3 (iv) \ ~ 1 ~l 1 3 8 ) -1 -0

(v) U 3 -1 ,

(vi) 0 2 D (vii) p: -10 7 ~9 ) 2 0 2

~ ) -1

-9 8 -9 0 0

-1 2

7. (a) Find a 2 x 2 matrix A that has eigenvalues 2 and 3 and corresponding eigenvectors ( ~ ) and

( ~ ). \

(b) F)n~ a)3 x ; sj~m~etric ffi(. at~ix)that has eigenvalues 1, 3, and 5 aud <.:orresponding tjgenvector~

~ ~1 , l ~1 ) and ~ .

(c) If A is an n x n diagonalizable matrix, then D = p-l AP, where D is a diagonal matrix. Show that ifm is a positive integer, then Am = PDmp- l .

( 6 -10) (d) Use (c) to find A10 given that A = 3 --5 .

8. The 2 x 2 matrix A has eigenvalues 1 and -1. Show that A 2 = I.

9. In each of the foHowing cases use diagonalization techniques to solve the system of linear first-order differential equations.

(i) x' 2Xl - X2 1

X2 12xl - 5X2 ,

(ii) x' -2Xl + X2 - 16e- 5t

, Xl(O) = 7,X2(O) = 3 1

X2 Xl - 2X2 - 16e-5t

X' 1 Xl - 3X2 + 3X3 (iii) x' 2 3Xl - 5X2 + 3X3 .

X, 3 6Xl - 6X2 + 4X3

( -3 2) 10. Let A = -8 5 .

(i) Find the eigenvalues and eigenvectors of A.

(ii) Is A diagonalizable? Explain.

(iii) Use the substitution x = (~ ~ 1 ) Y to solve the system of linear first-order differe~tial equations x' = Ax.

11. Find t he eigenvalues and corresponding eigenvectors of the matrix ( ; \ 0

eigenvectors form an orthogonal set arid comm~nt •. m t he reason for this.

2

2 0 \ 2 0) . U 1

Show that these

12. (a) Determine whether the given matrix is orthogonal:

(010) (0 ° 1) (i) 1 0 0 , (ii) -12/13 5/13 0 )

o 0 1 \ 5/13 12/13 0 ( 1 -1 1)

(iii) 1 - 1 -1 . 1 2 0

(b) In each case, construct an orthogonal matrix from the eigenvectors of the given symmetric matrix:

( 1 9 ) ( 1 0 1 \

(i) 9 1 ,(ii) \ ~ ~ ~ ) .

(c) Show that. if A and B are n x n orthogonal matrices, then AB is orthogonal.

13. (a) The eigenvalues o!the matrix A = ( ~ 22) 5 -4 are 0 and 9. Find an orthogonal matrix P

-4 5

(9 0 0)

such that pT AP = 0 0 0 . 009

( 2 1 1)

(b) Find an orthogonal matrix P such that pT i 2 ~ P is a diagonal matrix.

1 (VIO -3JIT 1) ( 2 ) ( 0 ) 14. Let A = r:.-:;-n VIO 0 -10, u = 1 and v = 1 . v 110 3VW vTI 3 2 1

(i) Is A an orthogonal matrix?

(ii) What is the angle between u and v?

(iii) Find the angle between Au and Av.

ANSWERS

1. (i) (1,1) is an eigenvector corresponding to the eigenvalue 1; (1, -1) is an eigenvector corresponding to O.

(ii) (1, 0, 0) is an eigenvector corresponding to the eigenvalue -1; any non-zero vector of the form (0, a, b) is an eigenvector corresponding to 1.

2. (i) X3, ). = -1. (ii) X2, A = 3; X3,). = 1.

3. (i) 6, ( ~ ); 1, ( ~) (ii) -4, ( ~4) (iii) 3i, ( 1 ~ 3i ) j -3i, ( 1 ~ 3i )

(iv)-2, ( ~I ), (n (v) I , (D; 5, 0); -7, ( ~~ ).

4. (a) No (why not?)

(b) x , y, x + y all belong to the same eigenspace.

(c) x is an eigenvector corresponding to A + 1; x is an eigenvector corresponding to k)" .

5. (a) T he other eigenvalue is 28.

3

(b) (i) 0, (ii) no, (iii) -1,-1, 0, 3, (iv) nO.

. ( -3 1) (1 0) 6. (1) P = 1 1 ) D = 0 5 (ii) Not diagonalizable.

C 2 0 ) COO) (iii) P = 0 2 1 , D = 0 4 0 1 1 -1 0 0 5

(I II ) C OG) (iv) P = 0 1 0 , D = 0 1 0 (v) Not diagonalizable \ - 1 1 1 0 0 2,F

/ 0 1+v'5 1- v'5) e 0 0 ) (vi) P = l 0 2 2 , D = \ 0 v'5 0

1 0 0 \ 0 0 - -/5 C3 -I -I I) COO 0) .. 0 1 00 0200 (vu) P = - 3 0 0 1 ' D = 0 0 1 0 .

1 0 0 0 0 0 -1

(4 -1) 7. (a) 2 1 ' (b) l 4 11 4 , ( 8 4 -1)

-1 4 8 (d) (6 -10)

3 -5

9. (i) Xl = ae- t + be- Zt , X2 = 3ae-t + 4be-2t

{ii} Xl = e-t + 2e-3t + 4e-5t~ X2 = e-t - 2e-3t + 4e- 5t

(iii) Xl = (a + b)e-2t + ce4t , X2 = be-2t + ce4t , X3 = -ae2t + 2ce4t .

10. (i) ( ; ) is an eigenvector corresponding to 1. (ii) No.

(iii) Xl = (b + 2a - lOat)et , X2 = (2b - a - 20at)et .

11. Get your answer checked.

12. (a) (i) orthogonal, (ii) orthogonal, (iii) not orthogonal.

1 3 (

6

13. (a) P = 3..15 0

(

J2 1 J2

(b) p= J6 J2

-V5 2V5 2..15

-V3 V3 o

14. Get your answer checked.

~4 ) is. possible solutioo.

1) (400) 1 givesD = 0 1 0 . -2 0 0 1

"

(b) Get your answers checked.

MA'I'HEMATICS 2080W

EXERCISE SHEET 11 2007

1. Reduce the equation x 2 + y2 - 4xy = 3 to a standard form. \Vhat sort of curve does it represent? Sketch the curve and find the angle through which the x and y axes must be rotated to obtain the standard form.

2. In each of the following cases, reduce the given equation to a standard form, identify the curve and give the directions of the new axes. Sketch also the graphs of the given equations in the xy-plane.

(i) 29x2 - 24xy + 36y2 = 180

(ii) x 2 + 3xy + y2 = 1

(iii) 75x2 - 360xy + 432y2 = 156x + 65y

(iv) 16x2 + 24xy + 9y2 + 4x + 3y = 0

(v) x 2 - 2xy + y2 + 3v'2x + 3V2y = 6.

3. Let R be the region within the circle x 2 + y2 = 4, and let T be the transformation represented by

h . ( v'3 2) t e rnatnx -2V3 0 .

(i) Find the area of T(R), the image of R under the transformation T.

(ii) Let C be the curve that bounds the region T(R) . Find the equation of C.

(iii) Use Lagrange multipliers to find the points on C which are nearest to and furthest from the origin.

(iv) Use diagonalization to convert the equation C to a standard form. Use this result to check your answer to (iii).

(v) Recallthat the area of an ellipse is nab. Use this to check your answer to (i).

4. In each of the following cases, reduce the given Sl1rf~r.!c to a standard form. idQn.t~fy n-U:l surface, and give the directions of thQ nkliW axes.

(i) x 2 + 2y2 + 2z2 + 2xy + 2xz = 1

(ii) 8x2 + 5y2 + 5z2 + 4xy + 4xz - 8yz - 3x + 6y + 6z = 0

(iii) lOx2 + 8y2 + 9z2 - 4yz - 4xz = 6

(iv) z2 + 4yz + 4xz - 2xy - 3!2x + 3!2Y + 8 = O.

5. Let A = ~ (:8~~6 =!~). 81 44 -23 64

(i) Show that A is an orthogonal matrix.

(ii) What does A represent geometrically?

(iii) From (ii), we know that A has a certain eigenvalue. What is it?

(iv) Find an eigenvector x corresponding to the eigenvalue in (iii). How does this add to the infor­

lllation you had in (ii)?

(v) Choose a vector y at right angles to x. By considering y and Ay, add further information to

your answer in (ii).

1

(vi)

( vii)

Obtain the same information that you have just found in (v) by using the trace of A.

What does the matrix B = ~ (618 ~~6 =~~) represent? 81 -44 23 -64

6. One of the matrices A = ; (~ ~ ~2) , B = ~ (~ !2 ~) represents a rotation in 3 2 -2 1 2 1 -2

}R3 . Which one? Find the axis of this rotation, and the angle through which the rotation takes place.

1 (6 -2 -3) 7. (a) Show that the matrix - -2 3 -6 represents a reflection in t he plane x + 2y + 3z = O.

7 -3 -6 -2

(b) Construct a matrix which represents a projection onto the plane x + 2y + 3z = O.

ANSWERS

1. Hyperbola; 3y2 - X2 = 3 with axes in the directions (1 , 1) and (-1,1) .

2. (i) Ellipse; ~2 + ~2 = 1 with axes in the directions (4, 3) and (-3, 4) .

(ii) Hyperbola; 5X2 - y2 = 2 with axes in the directions (1,1) and (-1,1). (iii) Parabola; 3y2 = X with axes in the directions (12,5) and (-5, 12). (iv) Pair of lines; Y = 0, 5Y + 1 = 0 with axes in the directions (3, -4) and (4,3). (v) Parabola; y2 = 3 (1 - X) with axes in the directions (1, 1) and (- 1, 1) .

3. (i) 16V37r (ii) 12X2 + 12XY + 7y2 = 192 2 '3 8

(iii) ± ~(3, 2) nearest; ± /1f}( -2,3) furthest. y13 y13

(iv) If ( ~ ) = vb (~ ~2) ( ~ ) we get ~~ + ~: = l.

4. (i) Elliptic cylinder; 2X2 + 3y2 = 1 with axes in the directions (0 , 1, -1), (1,1, 1) and (-2,1 , 1). (ii) Paraboloid; y2 + Z2 + X = 0 with axes in the directions (-1 , 2,2), (2,1,0) and (2, - - 4, 5). (iii) Ellipsoid; 2X2 + 3y2 + 4Z2 = 2 with axe.:; in the direr.tions (1,2,2), (2, -2, 1) and (2, 1, -2}.

(iv) Elliptic hyperboloid; (X - 3)2 + 3y2 - 3Z2 = 1 with axes in the directions (1, -1,0), (1, 1,2)

and (1, 1, -1).

5. (ii) A rotation. (iv) (1, -4,8). The rotation is about the line r = t(l, -4,8). (v) The rotation is through a right angle. (vii) B represents the rotation given by A followed by a reflection in the plane z = O.

6. B represents a rotation about the line r = t(2 , 1, I), through an angle of 7r radians.

7. (b) -t"4 (~~ ~~ =~). -3 -6 5

2

MATHEMATICS 2080W

EXERCISE SHEET 12 2007

1. Which of the following improper integrals converge?

. 100 dx .. 100

dx ... 100 -x (I) -1 2' (11) -I -, (Ill) e dx.

o + X 2 X nx 0

2. (a) Use the definition to find the Laplace transform of each of the following:

(i) J(t) = { ~1, ~ : : < 1 (ii) J{t) = {~~nt, ~ ~ ~ < 1r (iii) J(t) = te4t•

(b) Use the table of Laplace transforms to find £{f(t)} where

(i) J(t) = 2t4 (ii) J(t) = t2 + 6t - 3 (iii) J(t) = (t + 1)3

(iv) J(t) = 4t2 - 5sin3t (v) J(t) = sin 2t cos 2t (vi) J(t) == sin t cos 2t.

3. Use the result that £{el1t J(t)}== F(s - a) if £{J(t)}== F(s), to write down the Laplace transform of each of the following functions.

(i) t3e2t (ti) e3t sin 2t (iii) e-t cos 3t

4. Show that the function J(t) = ~ does not possess a Laplace transform.

5. Find the inverse Laplace transform of each of the following:

(.) 1 ( .. ) 1 48 ("') (.8 + 1)3 1 - 11 --- III --• $3 s2 85 54

111 (iv) - - - + --2

82 8 8-

48 (vii) 482 + 1

() 1 (.) 5 v 48 + 1 VI 8 2 + 49

( ... ) 1 (. ) 28 - 6 Vlll ~16 IX -r-g

5 - 8 +

( .) 8 ( .. ) 8

XI 82 + 28 _ 3 xu (8 - 2)(8 - 3)(8 - 6) 1

(x) 82 + 3s

(xiii) ~ 8

(xiv) (82 + 4}(s + 2) (xv)~, ~

6. (a) Find the inverse Laplace transform of

(i) (8: 2)3 (ii) ;2~ 8

(iii) 82 +45 +-

• 8 28-1 (IV) (S+1)2 (v) -~-, --,~.

(b) Find the Laplace transform of

(i) (t -l)u(t - 1) (ii) tu(t - 2) (iii) cos 2tu(t - 1r) (iv) (t -1)3et-lu (t -1).

(c) Find the inverse Laplace transfonn of

. e-2s •• e-7rS ••• e-S

(I) -3 (n) -2- (lll) -(--)' 8 s+1 88+1

7. Write each of the following functions in terms of the unit step function and hence obtain its Laplace transfonn.

n J(t) = {2, 0 ~ t < 3 1 -2, t;:::: 3

(ii) J(t) = { 0, 0 ~ t < 1 t2 , t;:::: 1

{I -s} 8. Sketch the graph of the function J(t) = £-1 :52 - e

s2 .

9. (a) Find the Laplace transform of each of the following:

(iii) J(t) = { t, () ~ t < 2 0, t;:::: 2

(i) lot Tet- r dT (ii) 1 * t3 (iii) . t2 * t 4 (iv) e-t * et cos t.

(b ) Use the result that £-1 { F (8) G (5)} = 1* 9 to find the inverse Laplace transform of the following:

1 (i) 8(s+1)

(ii) . _ .1. s (iii) (82 + 4)2'

10. Use the Laplace transfonn to solve the following initial-value problems:

(i) 1/' - 6t/ + 9v = tj v(O) = 0, y'(O) = 1 (ii) V'" - 4V' + 4y = t 3e2t

; v(O) = 0, y'(O) = 0

(iii) V" + V = sint; V(O) = 1, y'(O) = -1 (iv) 2y'" + 3y" - 3y' - 2y = e- t ; y(O) = 0, y'(O) == 0, y"(O) = 1

(v) V(4) - y == 1; y(O) = 1, y'(O) = 0, yl/(O) = -1, y'''(0) = 0

(vi) y' + V = J(t) where J(t) = { ~: ~ ~ ~,<:(O) = 0

(vii) y' + 2y = J(t) where J(t) = { ~, ~ ~ ~,< ~(O) = 0

{

0, 0 ~ t < 1r (viii) V" + y = J(t) where I(t) = 1, 11" ~ t < 21r, v(O) == 0, y'(O) = 1

0, t? 21r fix) y" + 2y' + V = OJ yl(O) = 2, V(l) = 2.

11. Use Laplace transforms to solve the given systems:

. x'(t) = x - 2y (1) y'(t) = 5x _ y j x(O) = -1, y(O) = 2

.. x"(t) + V"(t) = t2 I

(11) x"(t) _ y"(t) = 4t ; x{O) = 8, x (0) == 0, v(O) = 0, v'(O) == o.

1r 1. (i) converges to 2'

2.(a) (i) ~e-s - ! 8 8

48 (b) (i) 5

5

8 15 (iv) ~ - 52 +9

A.NSWERS

(ii) diverges, (iii) converges to 1.

1 + e-S7r

(iii) (ii)~ 1

(s - 4)2 2 6 _ ~

(ii) :sa + if- 5

2 (v) 82 + 16

( ... ) 6 6 3 1 111 -+-+-+-

54 83 82 8

(vi) ~(82: 9 - s2 ~ 1)'

6 3. (i) (5 _ 2)4'

2 (ii) (8 _ 3)2 + 4'

5+1 (iii) (s + 1)2 + 9

2

t2

t - 2t4 3 t3 5. (i)

2 (ii) (iii) 1 + 3t + _t2 +-

2 6

(iv) t - 1 + e2t (v) 1 _ 1

(vi) 5 .

4e 4 "7sm7t

(vii) t

(viii) ~ (e4t _ e- 4t ) (ix) 2 cos 3t - 2 sin 3t cos 2" 8

(x) ! _ ~e-3t 3 3

(xi) ~e-3t + !et

4 4 (xii) ~e2t _ e3t + ~e6t

2 2

( .. . ) t 1. XUl 4 - 8 sm2t (xiv) - ~e-2t + ~ cos 2t + ~ sin 2t ( ) 1 . 1. 2 xv "3 sm t - 6" sm t.

6.(a) (i) ~t2e-2t 2

(ii) e3t sin t (iii) e-2t cos t - 2e-2t sin t

(iv) e- t - te-t 3 (v) 5 - t - 5e- t - 4te- t - _t2e- t

2

e- S

(b) (i) ~ e- 2s

(ii) ~_~_ 2e-2s

82 + .­s

-'/fS se (iii) 82 + 4

6e- s

(iv) (s _ 1)4

1 (c) (i) 2" (t - 2)2u(t - 2) (ii) - sin tu(t - 11") (iii) u(t - 1) - e-(t-l)u(t - 1).

7. (i) 2 4

2 - 4u{t - 3); - - _e-3s .• 2 2e-& 2e-s e-s

(11) t u(t - 1); - + - + -s s 8 3 8 2 8

(iii) t _ tu(t _ 2); ~ _ e-2s

_ 2e-2s

8 2 8 2 -8-·

1 9. (a) (i) 82(8 _ 1) (ii) ~ (iii) ~~

85 88

-t (b) (i) 1 - e (ii) _~e-t + ~e2t 3 3

(iv) , 8 -1

( ... ) t . 2 ill 4 sm t.

10. (i) t 2 2e3t 10te3t

"9 + 27 - 27 + -9-t5e2t

(ii) 20

8 t e-2t 5et e-t (iv) --e-2 + -+-+-

9 9 18 2 ()

1 tIt 3 v - e + - e- + - cost - 1 442

1 t -2t 1 1 - 2(t- i) (vii) -- + - + _e __ -u(t _ 1) - -(t - l)u(t - 1) + _e --u(t -1)

4 2 4 4 2 4

(viii) sint + (1- cos(t - 1I")]u(t - 11") - (1 - cos{t - 211"}Ju(t - 211")

(ix) (e + l)te-t + (e -1)e-t .

11. (i) x = - 'cos3t - ~ sin3t, y = 2 cos 3t - ~ sin3t

.. t3 t4 t3 t4

(11) x = 8 +"3 + 41' y = -"3 + 41·

3

( ... ) 1. t Ul cos t - 2" sm t - 2" cos t

(vi) [5 - 5e- (t-l)]u(t - 1)

MAM2080W - MATHEMATICS 2080W

Table of Laplace Transforms

j(t) £ {J(t)} = F(s)

eat 1

s-a

tn n! sn+l

8 cos at

82 + a2

sin at a

82 + a2

tcosat 82 - a2

(S2 + a2)2

tsin at 2as

(82 + a2)2

eat j(t) F(s - a)

u(t - a)j(t - a) e-asF(s)

j(n)(t) sn F(8) - sn-l j(O) - sn-21'(0) - . . . - sj(n-2) (0) - j(n-l)(o)

---- ------ - ------ -- -

MATHEMATICS 2080W

. ":)J:1BET 1 2006

S refers to Calculus: Concepts and Contexts by Stewart, (2nd Edition).

1. S: §9.6 Exercises 15, 23.

2. S: §11.1 Exercises 15, 17, 19, 21, 23.

3. Sketch the family of level curves and hence the graph of each of the following functions :

(i) z = In(x2 + y2 )

(ii) z = arctan(:r 2 + y2)

(iii) z = (x2 + y2)C ·_( x2+y2).

4. S: § 1l.1 Exercises 25, 31 - 36.

5. Sketch the solid in the first octant that is bounded by the graphs of the given equations .

(i) x2 + z2 = 9, Y = 2x, y = 0, Z = 0,

(ii) 2x + y + z = 4, x = 0, y = 0, Z = 0,

(iii) z = x 3, X = 11y2, 16y = x 2

, z = O.

6. Sketch the region in jR3 bounded by the graphs of the given equations. (i) z= O, z=y, x 2 =1-- y, (ii) y c-= 2-- z2 , Y=Z2, x+ z =4, x=O.

7. S: §9.7 Do a selection from Exercises 3-31 .

8. S: §10.1 Exercises 1, 3, 5-10, 11, 14, 15, 17, 19, 27-30.

9. Does the curve r (t) = (asintcos2t,asintsin2t,acost),a > 0 lie on a sphere? If so, find the radius of the sphere.

10. I n each of the following cases find the intersection (if any) of the curve and surface :

(i) ret) = (2 + 3t, t, 1 + 2t), x + 3y - z + 7 = 0;

(ii) r(t) = (2 + 3t , t, 1 + 2t), x - y - z + 1 = 0;

(iii) r(t) = (0,}, t), x2 + y + 2z := 4;

. 1 1 t2 1 (iv) r(t) = (t' 1 - i ' "2)' z = 1+ x2 -t--?;;

. 1 1 t2 1 1 (v) r(t)=(t, 1 - i'2) ' z=2(x+ y +;} --I).

11. Describe the curve of intersection of the surfaces and find parametric equations for the curve .

(i) z=8-·-x2 -2y2, z =x2+2y2,

(ii) z = J9- X 2 ':"" y'l. , z = ~2 + 1pfrot2Q , 0 < ex < ~.

12. S: §J.O.2 Exercises 5, 17, 19, 21 , 27, 28. 31 , 43, 47.

13. The curves 1') (t ) = (ei -- 1,2 sin t, InO + t)) amI r2(t) ::-.:. (1 + t , t2 -- 1, t3 + 1) intersect a t the

origin. Find the angle of intersection .

14. S: §10.3 Exercises 1,3, 15-17, 19,21,22,23,25,27, 37, ;18.

15. Find a cartesian equation for the osculating plane to r(t) 7f

(cos t, sin t, cost) at t he point

where I. = 4-'

16. S: §10.4 Exercises 3, 7, 17.

SOME ANSWERS

8. (6) II , (8) I, (10) III .

9. The rad ius is a.

1 1 10. (i) ( - 4,--2, -3), (1i) no intersedion, (iii) (1,1 , 1) and ( (:., 3, 3")'

y3 ,. , 1. 1 t 2

I v) r(tl = ( -- . 1 - - .- ). \ / l ' t 1 2

11. (i) (2cost , V2sint,4) , (ii) (3sinQcost,3sinasint,3cosa).

12. (28) (1,0,4),55°.

7f

13. 2

1 (iv) (I, 0 , 2) '

, .j8, I cos ~c I ' ." . .l4. (16) -(' --2)3/2' (22)~. '--~'-,~, (38) osculatmg plane IS 3x - .3y + z = l.

2 + 4t J (1 -I- SlIl X)·1/

15. x = z .

2

MATHEMATICS 2080VV

EXERCISE SHEET 2 2006

S refers to Calculus: Concepts and Contexts by Stewart, (2nd Edition).

1. S: §11.2 Exercises 1,5,7,9, 11, 13,15,33,35.

2. By considering what happens along each of the lines x = ° and y = 0, show that the function

{

sin(x2 + y) f(x,y ) = ~fY-

is not continuous at (0,0).

3. S: §1l.3 Exercises 9, 17, 2 ] , 23, 25, 47 , 49 , 73 , 75.

4. S: §lL4 Exercise!::> 1 , 3, 9, 13.

f X2(X - y)

r _ 2 5. uet J (x, y) ~ 1 x: y'

if (.1:, y) i (0 , 0)

(i) Find fx(O , O) and fy(O, 0).

(ii) Is f continuous at (O,O)?

if x = y =-= o.

if (x, y) t (0,0)

if x = y = 0

(iii) Is f difrerentiable at (O , O)? Give full reasons,

{

x'2y

( x4 + y2 6. Let h x, y) = 0

if (x, y) ::J (0 , 0)

if J; = Y = O.

(i ) What value does f approach as (x , y) tends to (0,0) along the x-axis?

(ii) What value does f approach as (x, y) tends to (0, 0) along the y-axis?

(iii) What value does f approach as y) Lends to (0,0) along t he parabola y = ];2,?

(iv) Is f cont inuous at (O,O)?

(v) Is J differentiable at (0, O)?

7. S: §l L'i Exercises 3, 7, 17, 21, :i5, 42 , 43 .

8. Th radius r and alti tude h of a right circular cylinder are increasing at rates of 0, 01 em/min alld 0,02 em/min , respectively Use a chain ru le to find t he r ate a t which the volume is increasing at the time when T = 4 em and h == 7 em. At what rate is the total surface area

changing at. this time?

9. Use the definition of the directional derivative to find the direct iona.l derivative of the function in question 5 at the point (0,0) in the directiun (i) (1,0), (ii) (0,1), (iii) (1,2), (iv) Do ill the directional derivatives of f at (0,0) exist?

10. S: §11.6 Exercises 7, 9 ,21,23,25,27,33, 35, 37 , 41 , 47,49.

11. Which of the followi ng are t rue and which are false? For those that are true, provide a proof. For those t ha t are false, give a counterexample.

(i) If fx (a, b) and fy(a , b) bot h exist, then f is differentiable at (a , b) .

(ii) If f x(a , b) and fy(a, b) both exist , then f is continuous at (a, b).

(iii) If fx (a, b) and fy(a, b) both exist , then all the directional derivatives exist at. (a , b).

(iv) If all the directional derivatives exist at (a, b), then f is differentiable at (a, b).

(v) If f is not continuous at (a, b), then fx(a, b) and fy(a, b) do not exist.

(vi) If all the directional derivatives exist. at (a ,b), then J has a tangent plane at (a ,b) .

(vii ) If Jx(a, b) and fy (a, b) both exist, t hen f has a tangent plane at (a, b).

(v iii ) If fAa, b) = f y(a, 0), then f is differentiable at. (a, b).

(ix) If f x(a, b) and fy(a, b) bot.h ex.ist but fx (a, b) t fy(a, b), then f is not differentiable at (a, b).

(x) If f is differentiable at (a, b), then all th e directional deri vati yes of f at (a, 0) have the same va lue .

(xi) If a curve C lies OIl a surface, then the osculating plane to C at the point P is the same a..'-l the tangent plane to the surface at P.

(xi i) If fxy(a , b) = fyx(a, 0) , then f is different iable at (a , b).

ANSWERS

5. (i) fx(O ,O) = 1, fy(O, 0) = 0, (ii) Yes, (iii) No.

1 6. (i ) 0, (ii) 0, (i ii ) 2' (iv) No, (v ) No.

8. 0, 881f cm3/rnin ; 0,461f cm2/ min .

1 9. 0), (ii) See answers to qll est iol1 5; (i ii) --~ , (iv ) Yes.

oy 5

1]. Get your answers a nd reasons checked'

2

MATHEMATICS 2080W

EXERCISE SHEET 3 2006

S refers to Calculus: Concepts and Contexts by Stewart, (2nd Edition).

1. S: §12.1 Exercises 13, 14, 18.

2. S: §12.2 Exercises 3, 9, 11, 13, 15, 17, 18,23.

3. S: §12.3 Exercises 5, 10, 15, 17, 19, 29, 32, 34, 35, 37, 46 .

4. S: §12.5 Exercises 1, 3, 7.

5. (i) S: §12.6 Exercises 1, 2, 3.

(ii) Find the area of the surface z = .J2XY for 0 ~ x ~ a, 0 ~ y ~ b.

6. S: §13.6 Exercises 9, 11, 19, 27.

7. S: §12.7 Exercises 3, 5, 9, 15, 17,23, 24, 25, 27, 29.

8. Which of the following transformations from ]R2 to ]R2 are linear transformations?

(i) f ( x ) = ( x + 1) (ii) f ( x ) = ( ~2 ) (iii) f ( x ) = ( 2x - Y ). Y Y -- 3 y sm y y x + 3y

9. Find the iID(agc 1 of ~he ;in)e x( t) = (1,2,3) + t(2, -4, 1) under the linear map represented by

the matrix 4 0 1 . - -1 2

10. Find the image of the plane 2x - 4y+ z = 0 under the linear maps represented by the following

matr(icesi 2 3) (i) 4 0 1

-1 0 2 (] 2 3)

(ii) -3 10 1 1 2 3

11. Let A. = (~ -;1). Find the parallelogram to which A. takes the square with vertices at

(0,0), (1,0), (1,1), (0,1) . Find also its area (i) by using determinants, (ii) directly from the vertices of the parallelogram.

12. Let T be the transformation represented by the matrix (~ ~), and let R be the region

bounded by the lines y = 0 and x = 1 and the curve y = x2. Find the area of the image of R

UDder T .

2 2

13 (i) Show t.hat the ellipse :'2" + lLb2 = 1 can be given parametrically by x = a cos t, y = b sin t, a o ~ t ~ 2Jr.

(ii) Find the image of the ellipse in (i) under the transformation represented by the matrix

A~ (i Yd (iii) Find det A.

2 2

(iv) Use the results of (ij) and (iii) to find the area inside the ellipse :2 + ~ = 1.

(

I X ) (X2 + y2 -+- z2 ) 14. (a) Let F y = xyz . Find the derivative matrix of F and use the affine

z l+x+y+ z

appcoximation at ( l ) to estimate F ( :: ~ ) . Comp.,e your estimate with the exact

value.

(b) Let F ( x) (3"X -- 4y ). Find the derivative matrix of F and use the affine approx-y a+y

.. ( 2 ) . F ( 1, 9) H d h' . t' . h ImatlOn at 3 to estunate 2, 9' ow oes t IS approxuna IOn compare Wit

the exact value?

15. (a) The region R in the xv-plane is bounded by the graphs of x + y = 6, x - y = 2 and y = O.

(i) Find the region R' in the uv-plane onto which R is mapped under the transformation x = U + v, y = u - v.

(ii) Calculate ~i:: ~. and compare it with the ratio of the areas of Rand R'.

(b) Find the region into which the square with vertices at (0,0), (1,0), (0,1) and (1,1) is

( x) (X2 _ y2 ) transformed by the mapping F . :.= ' . y xy

Find the approximate area of the image under F of a circle of area 10-3 centred at (1,3).

(c) Find approximately the volume of the image under the mapping u = xyz, v = x + Y -I- z, 'W = x2y, of a sphere of volume 10-6 with centre at (1,2,0).

16. S: §12.4 Exercises 9, 11, 13, 11, 15, 31.

17. (i) Use polar co-ordinates to evaluate I / (x + y) dA where R is t he region bounded by the

R

graph of x 2 + y2 - 2y = O.

(ii) Find the volume of the solid that lies inside the graphs of both x 2 + y2 + z2 = 16 and x2 + y2 - 4y = O.

t l v'2Y--;2 (iii) Express r.. dx dy as a repeated integral using polar co-ordinates. Do NOT

.0 0 evaluate the dou ble integral.

,)

L.

18. S: §12.8 Exercises 1, 3, 9, 13, 17, 25 , 29, 30.

19. S: §12.9 Exercises 15, 19.

20. The transformation given by x = u2vw, y = uv2w , Z = uvw2 maps a region R onto a region

RI. Express J J J (Xy!)3i4 dV in terms of the volume of RI . R

ANSWERS

5, (i) (2) 9v'30 1T.

(ii) 2~Vab(a+b). 8. Only (iii) is a linear transforn1ation .

9, x = (14,7,7) + t( -3,9, -4).

10. (i) 6x - 5y - 8z = 0 (ii) x=).(1 , l , l ).

11. 10. 12. 2/3.

13. (ii) x 2 + y2 = 2, (iii) -2/ab, (iv) flab

(

2, 8 ) ( 2; 85 ) 14. (a) Estimate is 0,9 . The exact value is 0,88 '

3,9 3, 9 (b) The affine approxiInation is exact for a linear map .

15. (b) Get your answer checked! The approxin1ate area of the image of the circle is 2 x 10- 2

.

(c) 6 X 10-6.

8 1T 16. (14) 3 - 2'

.. 128 17. (i) 1T, (11) 9(311" - 4), (iii) get your answer checked!

1 1T185/ 2

18. (30) (1 - v!2) 10 20. 4(volume of It).

lVIATHEMATICS 2080\V

EXERCISE SHEET 4 2006

S refers to Calculus: Concepts and Contexts by Stewart, (2nd Edition).

1. S: §13.2 Exercises 1, 3, 5, 7, 9,17,2.5 ) 28) 31, 33.

2. Find the area of one side of the curtain whose base is y = x 2, 0 :s: x :s: 1, and whose height

above the point (x, y) is x y.

3. S: §13.3 Exercises 5, 9, 17, 19, 21, 31, 33.

4. S: §13.5 Exercises 1, 5, 10, 13, 15.

5. S: §13.4 Exercises 7, 8, 9, 11 , 19 (Sketch the hypocycloid.) .

6. S: §13.7 Exercises 1, 3, 5, 7~ 9, 17, 19.

7. S: §13.8 Exercises 5, 7, 13, 18.

1 + 25V5 2.

120

5. (8) 318 . 5

37r 7. (18) 2

Answers

1

MATHElVIATICS 2080\V

EXERCISE SHEET 6 2006

1. (a) Express (-3,0, - 3) as a linear combination of (1 , 1,2) and (:3,2,5). Can (-3, 0, -3) be writtJ'n as a linear combination of (I , 1, 2), (3 , 2, 5) an , ~ (4, :3, 7)7 In how many different way:" I;' this possible?

(b) Is (1 , J, 1) a linear combination of (1 , 1, 2) and (3, 2, 5)?

(14) ( 213\ 2. (a) If A = ~ 120 , and B = 0 4 _.J ) fi nd AB and 13A

(b) Find t wo 2 x 2 matrices A and B such that 1113 = 0 but A i= 0 and B f:- O.

(c) Give examples to show that the following statement.s are false :

(i) AC = BC, £' i= 0 '* A = B, (ii) (A + B)(A - B) = A2 - B2.

3. (a) Find the 2 x 2 matri.>:: that represents a reflection of lR.2 in the line y = mx

(i) direc tly,

(ii ) using matrix multiplication .

(b) Write down the 3 x 3 matrix that, represents a pr0ject ion of 1~3 onto the xz-plane.

4. (a) Can a homogeneous system of linear equations have exact,ly two solut ions? Explain.

(b) Decide how many solutions t here are t o each of t he following systems of equations. (i) x2 + y2 - 1 = 0 (ii) 6xy + 31-,2 + 61': = 0

1': -- Y + 1 = 0 3x2 - ,3y2 + 1 = O.

5. Can you construct a matrix A so that the system of linear equations Ax = 0 is inconsistent? Explain .

6. Given that (3 , -1,0) and (-7, 1,2) are each solutions of Ax = b where A is a 3 x 3 matrix, find two non-zero soluti.ons of Ax = O. Find also a third solution to Ax = b.

7. Find a condition on a, band c so that the system of linear equations

x - 2y + 3z a

2x - 4y + 6z b

- 3x + 6y - 8z c

is consistent.

What are the solutions in the case when a = -1, b = - 2 and c = 1?

8. Express the solutions of the equation Ax = 0 as a(lin2ea~ COln)bination of vectors where A is

(ii) 1 2 0 04 2

(

1 -1 1 (iv) 2 -2 3

- 1 1 4

(iii) ( ~ - 1 -1 4", ) 2 -1

4 i D (

1 2 3 ) (i) 2 -3

9. VerifY that 1JJ = 2, x= 3, y = 4, Z = 5 is a solution of the system of linear equations

1JJ - X + 'J + 4z 23

21JJ -- 2x + 39 + 9z 55

-11) + i + 4y + z 22 .

Use your solution to question 8(iv) to W ' ; ,e down the general solution to t he above system.

10. In each of the following cases construct ,', s · ' em of linear equations having the given set as solutions.

(i) {,\( -2 , I , 1) : ;, E IR}

(ii) {A(2, -1,3) + j.t(O, 1,4) : ;" j.t E lR}

(iii) {(1 , 4, 5) + ;'(2, -1 , 3) + j.t(O, 1,4) : ;" fJ E lR.}.

11. For which values of k does t he system of linear equations

1': -+ y + (k + 4) ,,:

2x + y + (k + 8)z

-3x - :3y + (k2 - 4k - 12)z

-1

-6

k2 + k + 3

have 0) a unique solution, (ii) infinitely many so lutions, and (iii) no solution? List all the solutions (in terms of k) to th is system.

12. (a) In each of t he following cases, set up and solve a system of linear equations to balance t he given chemical equation:

(i) C'u + HN03 -+ CUU'103h + H20 + NO

(ii) CU:3(POl\h + H3 P04 -+ Ca(H2P04 )z.

(b) Set up and solve the system of equations for the currents in the branches of the given

i, IOV, I~a .t..1 II < . ~ "3

3Jl~ 6..I'L 5.JL

network.

13. The following diagram shows a road network where all the streets are one-way. The flow of traffic in and out of the network is measured in vehicles per hour, and is indicated on the diagram. Let Xl , X2, X3 and X4 denote the number of vehicles flowing along the various branches per hour. f'%oo ~ 2.S0 .

15'0 (00

]£,

x.,3

5'0 ' 0 :::

.fOO

')

(i) Construct a system of linear equationti in the unknowns Xl, :1,'2 , :[;;3 and X4 that describes the traffic flow ill this road netwo: k.

(ii) Solve the system of equations you constructed in (1).

(iii) Use your solution to find t.he maximum and minimum value~ of :1:2 .

14. (a) For which value(s) of k is t he vector (I , -2, k) it linear combination of (3, 0, - 2) and (2, - I , -5)?

(b) Show that one of the vectors (: , . 'i ), (1,3,2), (0,1 , 2) and (1,4,4) is a linear combination of the others. Does that imply tha t every one of these vectors is a linear combination of

the others?

(c) Express the polynomial 12x2 + 8x - 2 as a linear combination of t he functions 1 + x, 1 - x 2 , x·- 2x·2 .

15. Which of the following subsets of]R3 are linearly independent? Give reasons for your answers. (i) {(1 , 2,3),(1 ,-3,4) ,(2,-2,- I ),( --1,·-5, 14)} (ii) {(2,0,1) ,(0,0, 1),(I,O, 1)} (iii) {(2, 0,1) , (0,0,1) , (0, 0, O)} (iv) {(2, 0,1), (1, 2,3), (2,4, 6)} (v) {(2, 5,-3),(4, 10,6)} (vi) {(1,2,,3),(4, ·-3,2),(1, 1, 1)}

16. For which value(s) of 1\ is the set A. = {(1 , -1, I , 0) , (1, -2, 1,4), (2 , 1,2, A)} a linearly inde­pendent subset of ]R4 ? \Vhere possible, express the third vector in A. as a linear combination of the other two vect.ors in A.

17. (a) Which of the sets listed in qnestion 15 are generating sets for R J?

(b) Which of the sets listed in question 15 are bases for ]R3?

18. Find two different bases for each of the following linear subspaces.

(i) {(x,y,z) E]R3 : x + 2y - 3z = O}

(ii) {(x , y , z, w) E ]R4 : x + 2y + 10 = 0, 2x - Y - z + w = O}

19. Any set on which operations of addition and scalar m ultiplication are defined, and which is closed under the formation of linear combinations is called a linear subspace. Decide, giving reasons, which of the following are linear subspaces .

(a) The set of all 2 x 2 matrices which are of the form (~ ~), where n is a non-negative

integer.

(b) The set of all polynomials of degree 2.

(c) The set of all polynomials with degree not exceeding 2. I' ~

(d) The set of all real-valued fun ctions of a real variable whose graph passes through the point (0 , I).

(e) The set of all solutions of the differential equation dy + Y cos t = t. dt

(f) The set of all so lutions of the differentia l equatio:l illY + 71 cos t = 0. ct

~~

(

1 ? (g) The set of all solu t ions of the matrix equation 4; 3 ) 7 . x = O.

( \ I' f . - . (1 2 h) The set of all so utlOns 0 t h \~ matrIX equatiOn 4 5 3 ) ( - 1 \ 7 x = . 6)'

ANSvVERS

, ) (-3, 0, - 3) = 6( 1,1,2) - 3(3,2, 5); Yes, in infini tely many ways. (b ) l'~

( 2 17 -1) (~) ( 2' I 31 24 1 1 - m 2m 2. ~a) 10 45 5 , 12 38 . 3. (a) -. - . -2 ,) . 2 _ ' . )

16 16 22 1 + Tn ~m m , ( 1 0 0)

(b) ° ° 0 .

4. (a) No, (b) (i) Two, (ii) Four. 5. No.6. A(10, - 2, -2).

7. b = 2a, x = (5, 0, -2) + A(2, 1,0).

8. (i) x =/\(3, -3, 1), (ii) X= (0 , 0,0) , (iii) x=A(-1 , 2, -1 , 1) ,

(iv) X= A(l,l, O,O) +{l(-3,O,-1 , 1).

9. x = (2,3,4,5 ) + .-\(1,1,0,0) + {l( -3,0, -1,1).

10. Check yo ur answers by solving your systems of linea~J equations.

11. (i) Unique sol ution if k =/: 0, k 1= L x = -k 1 -(1 - 9k,3k - k2 -- 4, k + 1) - 1

o ° 1 /

(ii) Infinitely many solu t ions if k = 0: X = (-5,4, 0) + A( - 4, 0,1). (iii) No solution if k = 1.

12. (a)(i) 3Cu + 8HN03 --+ 3Cu(N03h + 4H10 + 2NO.

(ii) COdP04h + 4H3P04 --+ 3Ca(H2POJh. . 35 . 38 . 1

(b) t] = 9' ~2 = 9' Z3 = 3' 13. (Xl, X2, X3, X4) = (350 - ct , ct, 150 - ct, ct ), ° S; a S; 150.

14. (a) k = - 8. (b) (1,4 , 5) is not. a linear combination of the other three.

( 26 ( ) 32 ( 2 2 2) c) '3 1 + X - '3 1 - x ) - 3 (x - 2x .

15. (i) Dependent , (ii) Dependent, (iii) Dependent, (iv) Dependent ,

(v) Independent, (vi) Independent.

16. A 1= -12 . If) = -12, then (2,1,2 , - 12) = 5(1, -1 , 1,0) - 3(1, -2, 1,4) .

17. (a) (i) and (vi) are generating sets. (b) Only (vi) is a basis for JR.3 .

18. Get a tutor to check your answer.

19. (a) No, (b) No, (c) Yes , (d) No, (e) No, (f) Yes, (g) Yes, (h) No.

MATHErvlATICS 2080W EXERCISE SHEET 7

1. Let A = (~1 ~ ~). Find 2 -3 5

2.

(i) the minor deterrninant M12

(ii) the cofactor C12

(iii) det A .

11 -2

(i) Find ~ ! 3 -11

2 1 1 2 3 4

-2 3 and 1 3 5 7 -8 1 2 3 6 7 12 2 1 5 8 20

(ii) \Vithout expanding, find the value of 1 1 1 a a+1 a+2 x y z and b b+ l b+2

y+z x+z x+y c c+l r:.: + 2

(iii) Let A be an n x n matrix such that A2 = I. Find the value of det A.

(iv) Suppose A is a 5 x 5 matrix for which det A = -3. What is the value of det(2A)?

3. Solve for A I-A 3 1 4 -A 4 6

(i) 2 2 -- A -1 = 0 ) (ii) 2 2-A -5 =--= 0 . 5 -5 2 - )., 7 6 -3 - A

a2 b2 c2

4. Factorise the determinant be ac ab abc

(Hint: Try to extract common factors; don't just bash it out.)

x a b c I 5. Solve the equation

a x b c = O . a b x c a b c x

x-3 x-+-2 x-I 6. Solve the equation x + 2 x - 4 x = 0 .

x-I x+4 :r -5

2006

7. Let A and B be 3 x 3 matrices with det A = -2 and det B = 3. Find det(AB ), det(BTA) , det(A3

) and det(A- 1).

8. (i) Express det B in terms of det A, given that B = p-l AP.

(ii) Let A be an invertible matrix such that A 2 = A. Sho\v that det A = 1.

1

9. Let A = (f ~ ~l ).

(i) Calculate det A and hence show that A is invertible.

(ii) Find the adjoint of A and hence A -I.

10. (i) Solve th e given system by Cramer 's Rule:

Xl - 2X2 - 3X3 3

~r l -+ ·'];2 - X3 5

3X l -+ 2X2 -4

(ii) Use Cramer's rule to determine for which values of k the system

kXl -+ X2 4

kXl -+ kX2 4

has a unique solution. For which value(s) of k is the system inconsistent? Explain.

ANSWERS

1. (i) -9, (ii) 9, (iii) 53.

2. (i) 0, 16, (ii) 0, 0,

3. (i) -3 or 4 (repeated),

(iii) ±1, (iv) -96.

(ii) 1 or 1 ± v'33.

4. (a - b)(b - c)(c - a)(ab -+ ac -+ be).

5. x==a, b, cor --(a-+b+c).

2 6. 3'

9. (i) 6,

1 7 -6 -6 -8 --. , , , 2'

(

71 (ii) adj A == -4 2

-1 -1

8. (i) det B == det A.

-9 ) ~ , so A -1 = ~ adj A.

10. (i) Xl == -4, X2 == 4, X3 == - 5, (ii) k # 0, k # 1; k == O.

11. (i) No, (ii) Yes.

2

MATHEMATICS 2080VV

EXERCISE SHEET 8 2006

1. (a) State whether the given differential equations are linear or nonlinear. Give the order of each equation.

(i) (1 - X)y" - 4:.cy' + 5y = cos x

(i i) yy' + 2y = 1 + x2

(iii) X37/ (4) - X2y" + 4xy' - 3y = O.

(b) Verify that the indicated function is a solution of the given differential equation.

(i) 27/' + y = 0; y = e-- x/2

(ii) ~¥. -- 2'U = e3x . y = e3x + 10e2x I ax '- ,

b (iii) l:Y" + 2y' = 0; y = a + -.

x

(c) For which value(s) of m is y = xm il solution of the differentiai equation X 2 y" - Y = O?

(d) (i) Show that Yl = x2 ilnd Y2 = x3 are both solutions of X 2y" - 4xy' + 6y = O. Are OiYJ and ,BY2 , 0:, ,13 E I~, also solutions? Is YI +)j2 a solution?

x2 (yl)2 (ii) Show that YI == 2x + 2 and Y2 = -"2 are both solutions of y = :cy' -+ -~ 2' . Are aYI

and JiY2, Oi , {3 E JR:. , also solutions? Is YI + Y2 a solution?

2. In each of the following cases find a differential equation for the given family of curves.

(1) y = ce- x

(ii) cy2 + 4y = 2X2

(iii) y=.o aex + !3e2x + 6e3x .

3. Find the differential equation of the family of circles passing through the origin with centres on the x-axis .

4. Solve the following differential equations:

(i) 4y -+- yx2 - (2x + xy2)y! = 0

(ii) y' (y + 1)2 = x2y In x

(iii) y' sec2 x + cosec y = 0

(iv) sin + 1) =: (1 + cOS:C)y', y(O) = O.

S. Find a singular solution for the differentiai equation y' = XVI _. 1/2•

R Show that the functions y(x) = A.x+ ~, A, B E JR, are solutions of the second-order,

homogeneous, linear different.ial equation X2y"(X) + xy!(x) - y(x) = O.

6. (a) (i)

( ii) Verify that y(x) = 2ex (1 -- ~) is a particular solution of X

x'2. Y"(:r;) + X7/'(x) - y(x) = 2x2ex.

(iii) Use the Fundamental theorem to write down a further three solutions of the differ­ential equation in (ii).

(b) (i) Write down a family of functions which are solutions of the differential equation (y"(X))2y(X) = O.

(ii) Show that y(x) = e2x is a particular solution of the ditIerential equation (y"(~r;))2y(x) = 16eox .

(iii) Use the results in (i) and (ii) to show that the Fundamental theorem does not carry over to non-linear differential equations.

(c) (i) Verify that y(x) = x2 + J; + 3 and y(x) = 2.1:2 - X + 3 are both solutions of the differential equation x2y"(x) - 2xy' (X) + 2y(x) = 6.

(ii) Write down a non-zero solution to t he lincar differential equation x2yl/(x) - 2:ty'(x) + 2y(x) = O.

7. SDlve the following differential equations:

(i) yl + y = c3x

(ii) X2y' + xy = 1

(iii) y' cos2 x sin x + y cos3 X = 1

(iv) (x + 2xy2 - 2y)y' + y = 0 (HiNT: \Vrite as riT 0.: . • . )

dli

(v) y' = 2_, y(5) = 2. y -x

8. Solve the following:

(i) 2y2x - 3 + (2yx2 + 4)y' = I] Oi) y;~ .- y2 sin :r - x + (3xy2 + 2ycOS :C)Y' = 0

(iii) x'll = 2xex - y + 6x2

(iv) 4y + 2:1; -- 5 + (6y + 4x -. l)y' = 0, !ie-i) ,= 2..

9. Find an integrating factor of the form 17('1/, a, {J E JR:., for the non-exact differential equation (y2 + xy)dx - x2dy = O.

10. (a) A large tank is filled with 500 litres of pure water . Brine containing 2 kg of salt per litre is pumped into the tank at the rate of 5 Clmin . The well-mixed solution is pumped out at the same rate . Find the number of kilograms of salt A(t) in the tank at any time.

(b) Solve the problem in (a) under the assumption that the solution is pumped out at a faster rate of 10 ejmin. When is the tank empty?

(e) A large tank is partially filled with 100 litres of fluid in which 10 kg of salt is dissolved.

Brine containing ~ kg of salt per litre is pumped into the tank at a rate of 6 P./min. The

well-mixed solution is then pumped out at a slower rate of 4 f./min. Find the number of kilograms of salt in the tank after 30 minutes ,

2

l.(a)(i) linear, 2,

(d) (i) yes, yes,

(ii) nonlinear, 1)

(ii) no , no.

ANSWERS

(iii) linear, 4. (c) I -+. r­----'- Vi)

2

2. (i) y' + y = 0, (ii) (x 2 -- y)y' = xv , (iii) y'" -- 6Z/' + 11y' - 6y = O.

3. 2xyy' = y2 - .1:2 .

4. (i) 8In x + x2 - 4ln y- y2 = c

(iii) 4 cos y = 2x + sin 2.1: + c

.. ' x 3 x3 y 2 (11) --- In :r - - = - -+ 2t'j + In Iyl + c 3 9 2 . . (iv) (1 + cos.1:) (1 + eY ) = 4.

5. y = 1. 6. Get your answer checked.

7. (i) y

(iii) y = sec x + c cosec x y 8

(v) x = 2 + Y

8. (i) x 2y2 - 3x + 4y = c

(iii) xy - 2xeX + 2ex -- 2x3 = C

(ii) xy = In x + c .)

(iv) yx = 1 + ce- Y-

x 2

(ii) xy3 + y2 cos X - 2 = c

(iv) 4xy + x 2 - - 5x: + 3y2 - Y = 8,

1 9. xy2 '

t

lO.(a) 1000 -lOOOe- Ioo , (c) 64, :38 kg .

3

l\;lATHElVIATICS 2080W

8XERCISE SHEET 9 200fj

1. Use the definition to dete rmine whether the given functions are linearly independent or de­

pendent on (--00 , 00):

(i) X, x 2 , 4x - 3:Z;2

(ii) 2 + x, 2 + Ixl (iii) 1 + x, X, x2

.

2. Show by computing the Wronskian that the given functions are linearly independent on the

indicated interval:

(i) x ~ , x 2; (0,,00) -

(ii) eX,e-x,e4x ; (-00,00)

(iii) I , X, x 2, _ .. ,xn ; (-00,

(iv) eX, u''', x2ex ; (-00,00)

(v) cosx,sinx,xcos:r,xsinx; (-00,00).

3. Verify that the given fUIlctions form a ba::;is for the set of solutions of the differential equation on the indicated interval. Write down the general solution,

(i) y" - y' - 12y = 0; e-<lx , e4x; (-c.o,<:::o)

(ii) x'2 y ll -- 6xy' + 12y = 0; X3,:[4; (O,eo)

(iii) X3 ylll + 6X2 y" + 4xy' -- 4y = 0; :1:, x-2

) .:r - "ln x; (0,00).

4. Find a second linearly independent solution y,J of each differential equation by putting yz(x) = U(X)Yl (x) and reducing the order:

(i) X2 y" - 7xy' + 16y = 0; YI = x-1

(ii) (1 + 2.x)y" + 4xy' - 4y = 0; Yl = e- 2x .

5. Find the general solution of the given differential equations:

(i-) 4y" + y' = 0

(ii) y" -- 36y = 0

(iii) y" + 9y = 0

(iv) y" _. 4,!/ + 5y = 0

(v) ylll - 4y" - 5y' = ()

(vi) ylll - 5y" + 3y' + 9y = 0

d5y dy (vii) -, - 16- = O.

dx" dx

6 Find the particular solution of the differential equation y'll + 12:;" + 36y' 0, s3.tisfying y(O) = 0, y'(O):::= 1, yl/(O) = --7.

7. Verify that the functions cos2x, sin2x , sinxcosx and 1 - 2cos2 x arc solutions of the ho-. mogeneolls differential c4uatioll !/(x) + 4y (x) = 0, and that :<;(1:) = . 1 is a solutioIJ of yl!(x) + 4y(x) = 4.

Find the general solution of the non-homogeneous differentia! equation.

8. (a) Show that the substitution 11, = In .1: reduces the difFerential equation

ax2 y"(:Y;) + bx:/(~:' ) + = 0, (J" b, c E lR

to a differential equation with constant coefficients.

(b) Use the result from (a) to solve the following:

(i) X2y" + XVi + 4y = 0

(ii) X2yil + 5::r;y' + 4y = U.

9. Solve each differential equation by variation of parameters:

(i) y" + y = seCT

(ii) y" + y = sinx

(iii) y" + 2y' + y = e- x In:E

(iv) X2]/' _ .xy' + y = 4:1: In x, given that. T and x In x are solutions of the associated homoge­

neous differential equation .

10. So] ve the following differential equations by undetermined coefficients:

(i) y" + 3y' + 2:; = 6

(ii) y" -+ y' -. 6y = 2x

(iii) y" - 8y' + 20y = 100:1;2 - 26xex

(iv) y" - yl = -·3

(v) yll - y' + ly = 3 + d

(vi) yl' + 4y = 3 sin LX

(vii) y" + Y = 2x sin:;;

(viii) y" - 2y' + 5y = eX cos 2:;;

(ix) y" + 2y' + y = sinx -+ eos2x

(x) y'" - 6y" = 3 - cosx

(xi) yllf - y" - 4y' + 4y = 5 - e + e2x

(xii) y(4) - y" = 4x + 2xe- x .

2

ANSWERS

1. (i) Dependent,

4. (i) Y2=x41nx,

(ii) Independent, (iii) Independent.

5. (i) (iii) (v) ( vii)

(ii) Y2 = X.

x Cl + C2 e-"4

CI cos 3x + C2 sin 3x C1 + C2e-x + C3 e5x

C1 + C2e-2x + C3e2x + C4 cos 2x + C5 sin 2x

5 5 1 6. - - _e-6x + -xe - 6x ,

36 36 6

7. Get your answer checked.

8. (b) (i) CI cos(21n x) + C2 sin(21n ::c)

9. (i) Cl cos x + C2 sin x + x sin x + cos x In( cos x)

(.') . 1 11 C1 cos X + C2 SUi X - 2" x cos x

(iii) cle- x + C2xe- x + ~x2e-X ln x -- ~x2e-X 2 4

2 ( ) ') (iv) CIX + C2X In x + 3"x,ln x J

(ii) CI e - 6x + C2e6x

(iv) e2x (cl cosx + C2sinx) (vi ) cle- :7; + C2 e3x + C3xe3x

10, Check your answers by substituting into the differential equation.

3

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS MATHEMATICS 2080W

CLASS TEST 1 - MONDAY, 27 IVIARCH 2006

Time: 1 ~ hours Full Marks: 50

Notes: (i) Only approved calculators may be used. (ii) Full answers are expected.

1. (a) Find parametric equations for the curve of intersection of the surfaces z = 12 - x 2 - 2y2 and z = 2X2 + y2.

(b) Does the curve T..(t) = (v't, t , -4t) intersect the surface z = 2X2 + y27 If so, at which point(s ) .

(c) Find a cartesian equation for the tangent plane to z = 2X2 + y2 at the point where x = - 2 and y = -1.

[12]

2. Let C be the curve r(t) = (t, 2t, i), t > 0.

(i) Find two surfaces on which C lies and sketch theln and C on the same set of axes.

(ii) Obtain a vector equation for the tangent line to C at the point corresponding to t = 1.

(iii) Calculate the curvature of C at the point corresponding to t = 1.

(iv) Find a cartesian equation for the osculating plane to C at the point (1, 2, 1).

(v) Express the length of the portion of C frorn (2, 4, ~) to (3, 6, ~) as a definite integral. Do not evaluate the definite integral.

[16)

3. Express the region within the sphere x2 + y2 + (z - 1)2 =:-.: 1 and above the plane z = 1 in terms of spherical co-ordinates. [5]

{

X3

4. Let f(x, y) = x2

: y2

if ( x, y) = (0, 0)

(i) Is f continuous at (0 , 0)7 Verify your clairn.

(ii) Find fx(O, 0) and fy(O,O). Show your working.

(iii) Calculate fx(1, 2).

(iv) Is f differentiable at (0, O)? Give full answers.

(v) Find a vector equation of the tangent line to the curve of intersection of z = f(x, y) and the plane y = 0, at the point where x = ° and y = o.

[15]

5. Is the following statement true or false. Verify your claim. "If gx ( a , b) and gy ( a, b) both exist and are equal, then g is differentiable at ( a , b)." [3]

/lI}H ;108C5'v-/ - SO'-uTiC)lcJS' /D a~ ~T/- c2~#4,12c~ c2oo6' I

IIC/.) ~ k....r-. C5 fl;; ~ ... C<-S ~ f!~ "01 ~ ::(.) -~ (f >--;- .;L£ '---+ {j c....J .. ~e. x-J..."l-ff ~ 4.. /2; ~ "'t- ..&I.';-c./ 4-<-/~~ ~ -f2oL'..de ~ cg~f~cr/l~ q/ft>~~J'.

4- .x::.2- C04 -C "11 == .:J ,J-. ~ f /Z.IA>.-?;? ceo 2..(- -+- 4. kA-Lt 0:5 t5 S-C:;;;-: / V . c? /

(.6) h 6~~ 2r ~1.d-: -4< c- ; .;-t- r- (;- J- as> 6{6+~) :0 o.

7Zwr 6-;0 or- 6::.-6. ~~ 6-'4°1

de> /~~(tf~~aI'7~

-- (0) 0) 0). C7Z:.....;; /f;;O,uLY ,oCX;r-() A-·T~'ct-.tj

(C) 6,tL,'j) =- *-.x. J btl ('~'v)= .1"(/ . ~ .. J.J-.1./ -/); -.g/ 5-yl-~ -I)$- -~ . ~-<:-~ ~-- ~ ~ «-d~: J =- 9 - $"&; ~ ) -.1. (1/ -N).

~ C£ilL.ff!.

~i= ~. . c.. ,j.(tj 7Z..-~ .... ~'..u """- .-d:; ~~

(J :::-.,2;1.... ~~ ~ ~:::. ~

1V.A-...t4t- x ")o) .j <io) J- > ~ .

(ii) A' (i- ) :: (/ ~ - 7~) r ) / ) - ) - J c. 4'"- I=-(. ~ ~ -/ •

ttcc:lo.'-i~J.o..... t' ~"S-.. ~~,,;/ 3:-0'): (~)'/)-I->'{i,~-/) (ik) -:: %) =- (0) 0, -/i). S'o ~ '(I) ~ (~0.1.,;I).

~ '(,) 1< A' '(I);:' (11:1 -I) >Co (0 0.;2.)';:' ('4:- --' 0). ....." /,,) ) ) --:>

C~roY-~ 1<; 1(1f./-~6)/.= ~ /(,1, 2;_)/3 ~ r6

eV ) .-?1~ ~ G!>SC.....(~tr ~'-<- <Ur «~/)';" ~'(i) y ~"{I); (y/ -~ (J) .

t:;"-~----- o.g ~C~?! r" ~ 4: (X-I) -,;}-t(j - oJ) = O.

~.' J'''~'-L C.q "" r-"'~ C-~ I ~ tPs:cJ ~ ~6 rl-f... .u L4 ,rf"K ~ -SZ'-c>I... C .A-~S./ "-s. . fJ ;:. ..2..t. ·

~, J ~

(II) ,(~cd "- j /-:'({;)/~(-:: IP-f.:;'~cf~~(;--=-.f~cA . J. I .2. 0 - to .2 .2

3 . ~ ~. 0 ~ I ~ 1-) 0 £ 8 6. .,.11 J C~f :S:. ~ ~ .2 CDIO y . "i ~ ~""- ~--r ~::J ~ / ~/.:;/ I or..ol ~ ~ ! >lJ..-I-~~ ij1- "'.1.3 ..;. r""" h[J ?.2.,; ).;P~ I

4,.(iJ &- ex) V ::;i/~o) ~_~~ O~! :r-J

/ =- /:L/ J~"-/ ~ !~/ I :r4.:t I ~ z.,. /f 7

S',,'ACA_I.X-! -~ 0 <...r "'~(f iJott /.J!.-t ~ ~ .,£'-. -/r~'J) =-0 . c~,~ ~(t:J.)(J)

r( A. Co<....- f ('OJ 6) ==- 0) -I- ~ CC>1-f Y y ~ c<,ov.1' er..::t- (()~ (;)) •

e0 (x (0 ... 0) 0:. ...-e.~ ·tf.(..A,o) - ;fIo~ 0) <:-~"--. ~_oo) ..,e. -z:> 0 ~ ..£ -'> 0 :.A. - =-- /.

'5y (0)0) ::-~:.- lro,, ~) - (f~d) ~~;."... ~o =- o. ~ --. 0 ."l ;d. -!> 0 ./L

(i'tj F~ / V ¥ (o) d) I <- (.I .. "I)::' J' ~ C:r l-~ '1 ') - ;(..l • .2 A.. _ :1..9

.(- s:x.;; I-/ ():L if cVf.-cl)~ - ~ '--t-rJ/

e"v) {~~~fe- ~e~,. 6J ..(' (~ J/!{o..+-.i, ,->.,..,i.) -tfIQ.;O) - ..I<tf.cCc.,kJ)-~{y~,Io)-.> l:> ~

M L-I-A-:J, ~ ~ ~ 0.

I ~$ ,

(1') ~s : ~ "'-I-l2. - 0 _. 12. I - A . 0 _ _ ~ ~ l.

vU~+-~ >- - { u L~';;:/ ~ ...e,.. ..l J d,," k~ ~ -0 - -4. -1 .-b, t:> O«}.,L ~ ~ #>

1i.A.J ~/-'

-I ...a ~ oe. 'fI..e- ...p<, aJ/"" o.>t- (' o. 0 ).

(II) (~( o~ 6) =1. S'o ~ t-ct~~ A'--L h o-t.-~#~ .

(II 0) I). J401~ JO~· ___ 't C-Cd'(" ~d-~~ ~ ~ ( ~)= (o~ OJ 0)7 'Au/ OJ ~-

I 0 A 6-1< .

~ "- <J:J- ( 0 ) 0);:;- ? y (OJ 0) -'" 0 I b.-Y ~ ~ /If../t e.o--I, ~~ 0c.X Co ... 0 ~ So ; ..i.? AA-or ~tI ~ ... ¥-l·~ (., ",.#- C o.A)

1L ~~~.-..r ~ Ft4t:-S"E.

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS MATHEMATICS 2080W

CLASS TEST 1 - TUESDAY, 27 MARCH 2007

Time: 1 i hours Full Marks: 50

Notes: (i) Only approved calculators may be used, (ii) Full answers are expected,

1. Let rl(t) = (t, t2, t2) and r2(s) = (cos s, 1 - sin2 s, cos2 s), s, t E IR.

(i) Obtain two surfaces on which rl (t) lies, and sketch these two surfaces and rl (t) on the same set of axes.

(ii) At which point(s) do rl(t) and r2(s) intersect?

(iii) Find, if any, the points of intersection of rl (t) and the surface z = 1 - 2X2 +- 2y.

(iv) Find the curvature of rl (t) at the point (-3, 9, 9). At which point is the curvature greatest? Explain.

(v) Find a cartesian equation for the osculating plane to rl(t) at the point (-3,9,9) .

[19]

2. Find liIn h(x, y) where h(x, y) = { Y (x,y)-+(l,O) x

if y # 0 l'f . Give full reasons,

y == 0 [3]

3. The function g(x, y) satisfies the conditions

g(3, -2) = 11 , gx(3, -2) = 4 and gy(3 , -2) = -9.

Find a vector equation for the tangent line to the curve of intersection of z = g(x, y) with the plane x = 3 at the point where :,; = 3 a.nd y = -2. [3]

4. Let f(x,y ) = {Yx2

if y # 0 if y = 0

(i) Use the definition to find fx(O, 0), jy(O, 0) and fy(l, 0).

(ii) Do all the directional derivatives of f exist at (0 , O)? Explain .

(iii) Is f differentiable at (0, OJ? Prove your claim.

(12]

5. Find parametric equations for the curve of intersection of the surfaces z = J x2 +- y2 and x2 +-(y-l)2+ z2= 1. [6)

6. (i) Express the region lying within X2+y2 = 4 and X2+y2+Z2 = 9 in cylindrical co-ordinates.

(ii) Express the region inside x 2 + (y - 1)2 + Z2 = 1, and above the plane z = 0, in spherical co-ordinates.

(9]

1

, /rc.N!J ~ 'J ~~ C~C.

-r; 6) L'-e t" • ~1! ~ ~ .1.. o.-</? :::- (J) 6tcJ -I £ Cod S ~ /, ~dG7 ~ f~t't- ~. J-f. c9 ( CVJ .d M ""- ~/' arZ'=-ctj d fl.).) -

(i'') G L =- I - :(6j). -I- .J f: 1.- 6) t;: L I . ~.:...-;{ 01"!-- [.. -I, ~ 1/ /).

'~ I(~ ):: (l1;l C/ :J- 6) if , -y 1&) ;: (D

J J; ;.. )

('. II (. ) 00 :~_ I It) X./: 'It)::: 0 -.2- ~

- -' )) .

1«6-) ~ ~_,Jg _ , /%J' k.t-J) =-

t+- ft:t)1f;...

I< ({., J..<? 0.. ~ ('? v. ; U<(A....... {ct Ii /-.,1",,-

(V) k~~- ~ (1:,) ~; ~ M~ C--'-U/V( ~ r~ /

;K~~ .

J.. ~C7KO ~ p ~ == I frJ-tO) ~ Af/.-y} "= d -~ 0 t« tJ -~ o .

at7 Af4 feed (7""0 ! .-K(,/:;)= L -) I CV3 x-- /,

Sl ~~ L{:t,"/) ~,!-X>, ./V'I'ST,

l _:( ~J) -) (I~ (J)

6.. (l-j - J "1- ".- '- ::': Jy:!' J q. ---r >-­

o ~A--- £ A

£\ ~ -o ~ (..7 _~II

(il) x~ ::JI--_.;V+-J ~ ~ ~J f' ~ - "l ~f.A..-t, e ~ 9c. 0 ~ p ~ ,) ,.J....-'-I/J..,. ..... e

If

o ~f~ L--

o 1!:..e~· I/.

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS MATHEMATICS 2080W

CLASS TEST 1 - MONDAY, 31 MARCH 2008

Time: 1 ~ hOllrs FUll Marks: 50

Notes: (i) Only approved calculators may be used. (ii) Full answers are expected,

1. Obtain parametric equations for the curve of intersection of the surfaces z = x 2 + y2 and z = 2x + 4y - 1.

[5]

2. Express the region that lies ~ the hemisphere z = J 4a2 - x 2 - y2 and outside the cylinder x 2 + Ii = a2

, a> 0 in spherical co-ordinates. --

3. The curve C is given by

r(t) = (3 + cos t, 2 sin t, cos2 t), 0 :S t :S 21f.

(i) Find two surfaces on which C lies.

(ii) Sketch C and the surfaces you found in (i) on the same set of axes.

(iii) At which point(s) does C intersect the surface 5z = (x - 3)2 + y2 - 4?

[5]

(iv) Find a cartesian equation for the tangent plane to the surface 5z = (x - 3)2 + y2 - 4 at

the point (1,5,5),

[3,3,5,3J

4, Let C be the curve defined by ret) = (t 2 , et-

l, In t), t > 0, and let P be the point (1,1,0).

(i) Give a vector equation for the tangent line to C at P.

(ii) Find the curvature of C at p,

(iii) Give a cartesian equation for the osculating plane to C at P.

r: _ {y2 if y of x u. Let f(x, y) - 'f . X 1 Y = x

(i) Use the definition of the partial derivatives to calculate fx(1, 1) and j~(l , 1) .

(ii) Do all the directional derivatives of f exist at the point (1,1) ? Give reasons.

(iii) Is f differentiable at the pOi~lt (1, 1)? Give fill! reasons.

1

[3,5,2]

[4,4 ,4]

{

x2y . I -- if fX~ ' 0 0 6. Let fllx,y) = X4 +y2 \ ,y) =I ( , )

o if x = y = 0

Does lim g(x, y) exist? Explain. (x ,y) ...... (0,0)

[4]

1'0<-< ~ Jf~ 1:.4..,...... 7)

~~- -~C ;. ~;~/~~" j ~ .. A I <-~~~ r~

e,y l (?~'/~c~ '~CL.-W-L- S-Cc,:;.)C . ", CctJ.)f-l-4 h;':~/-tf .f'17

L --;; 4- (c. ~ ;.. t- - ~ ,,-.L {- ) fl4'" ' - Lj c-;.) c (j6:J t ~ - / ~ ~ 6- ._;;. o-r ~ ~

?o-<-~h 1; ~r..-ke:JIl'o-... q..-e (i/,}..-6) ad (~-.,1/ 0),

(IV) /--<--{- ;:- 6, 'f / 6 ) = (: -}/ -I- :/- 4. -sJ ~ /' r " F{x/ 1/ <5 );- ( .,1 (.x -J~ dflj -s). _ g r-(~ s: r );: ( - ~ /0) - -)

7:;",) ~;v(-ra~; - 't< (..x -I) -f /o(j --S' ) --'S{j "1~r )-== 0

4 , (.y .", 'H) c:- (J-~ e ~ -; t ) . ? L-J C = I ,

~-

;;;"(j ~~~. ,~{f.,J "" (f, ~ oJ -I- A(~ 1/ 1)./ A 6-j~ . (it) ::: ''(f:) '" (.!)..> e 6 -I - :.--)). ,4" i~ ): f.r) I _/)

J ..-- ", / '

4--(tJ X 4--''{tJ;;- {-~ tt.. () --- ~ / .//

L~'l.Tc0~ /<' ~ I (-J) 4-/ C)~(' ~ ._r~c:. !(t')" ~ 1)/] (/7

tv 6lsci<...(~ 2; jK~ ,' - .;.. (j -I) 1- 'f (j -/) f- 0(5 - 0) -==- 0

L-=--.J X - ~ (I + I;.. 0 ~

S'.(y -{({,I):: A'~ «(I-f.-{r}-i(~ /J.,. ~"- .!.J._1 .= 0 11 ~~O ~ ~~O ~

(y{f, I): ~~ .(/;'II'..{I-({(/) ""A:..... tI' AiJ-I:::I , v ~-)U A. A 't.. -7 0 .,i

(it; ~( (41) .~ ~~ -(tf -/- A fA,; 1+ A V;J) -- t!.('~r/ I A -Ju t r .

~ I..,.e..~ ~ -t I\U~):l._1 ~ ~ q ~ lI, 1= t.t t1-

~ -"0 - '" )- J... P

h '-' l.,:.. n iI/ ,-1 ...~/ ~-'O l' - - - ~I 'cr; ",="'''' /

Ac u",~,/ d~ 04~cylr'~cv( ~:,~ Jbl-I s 't cd"- C/~ I)

(,J ~(f!f -:'!. I-ul) -f [1,1/- .A r: {i,tj - ~6/i,1! "'j(:-f.-'J~/ - 0- .21 .1i'.l. r~J-+~2.- ~ Lf-h-).-' }

l.Lg-I.-o~~A. -4~1, -" .r;-~~ J

~ i, ~ _~ -t~~) !S<-Qs6- ~ --4 D Q,a . .J2 -;) oj. , 40-/ ~ ;ooT' 1) li-A.. r- I

~jI~~~(~ a)/' (i,l)~

OJ< ': ,7-<.-: aL;~d-,'Cfl".l' 5LL60J'~~ A:"n fz:,d C~s ( ~ f-~ ,'0-- *{' 6 ~ (£.%,7 )~.uL jWo-~f' 'u ,~/) '1' ';:::-/) J __ .JL .

(. r /'" ) c S' " J ~

CI.4-( '/ 0) 0)/ (t> / I) .;J) cY~ vi L ~ I) / .rf<-Y" " ~ .It~., ~-t'cJ~ ~ t;

CO'"' ~ L·o>~a.~~-I.0 lUor~tlh~t/~(e c:u'- C /, /),

Department of Mathematics & Applied Mathematic

MATHEMATICS 280W

CLASS TEST 2 MONDAY, 7 MAY 2001 Time: 1 t hours Full Marks: 5C

Marks Available: 50

Notes: 0) Only approved calculators may be used.

(ii) Full answers are expected. Marks will be deducted for incomplete

solutions.

, 1. (a) Find and classify the stationary points of the function f( x, y) = ' x4 + y4 - 2x2 + 4y.

(b) Classify the stationary point (0,0) of the function g(x, y) = x2y + xy2 - x 3. [11]

2. Use Lagrange Multipliers to find the points on the curve x 2y , 6 that are nearest

to and furthest from the origin. [8J

[5]

4. Find the volume of the ~ll~ . _that lies in the first octant and is bounded by the

surfaces z = 4 - x 2, Y = x and y = 5. [5]

5. The repeated integral

represents the volume of a solid region R. Sketch and describe the region R. [5]

6.(a) Find the mass of that portion of the surface x = 4 - y2 which lies in the first octant,

between z = 0 and z = 4, if the density is given by p(x, y, z) =y.

(~) Calculate the mass of the region that lies between the surfaces z = 0 and z = x 2 + y2

and between the cylinders x 2 + y2 = 4, x 2 + y2 = 9, if the density is given by 1

p(x , y,z) == 2 + 2 [10] X Y

x 3

7. Let f(x , y) == - + 2xy. Y

(i) Find the Taylor expansion of f (x, y), up to the second degree terms, about the point

(2, 1) .

(ii ) Obtain the tangent plane to the surface f(x,y) == 12 at the point (2,1). (6]

A-H d-5'o'vJ - SOl- uli 0 I\J I ro C L"f\.:t /k--r7 .J - 1-,J1A-Y cJ-OO /

I. r ... ) {"L = 4X...!-lf-x.o 1 (o,-I}Ot-<'" (:t-'.~)o-e~ ~~.c...~I/t1T:';t .

./ -= 4- J -f'f': 0 S U U rt' U .A d C IJ"--+e G..-~:-..

(H -- I.) ... J. - ~ (0 I -I) ~ 0 Ij, *''' "'-_«' ""'-~l'" {«,.'1'::=- 0 (r,,-I) r 0 I.J. -" 4~JMe'A'A4I.A-"'. fyy: ~u l.-

e 6) ~:L -- .JfJ+iJ ~-J~~ ) '/]'1:' .rJ-frJ.(J. N~-L J.~JO)4)~O .a..(Jy ~o, oJ.

{x ~ : 02 U - 6 ~ ~ ~"1 -- .;1-J(.. +-01 ~> ~ry ~ ~-t- . .1 U , .4 • 16 ... A--c .,.I:! 0 ~ (0 CI ~ I 41- ~ f...)V-~ ~~«0+t v-4. •

~ / J _

q (OJ O)"::'CJ. ~(€Jt:.); cJ~ 1 :>o~C~O . - So ('O.Jo)~~Ja~d~l'r: f/ ' (J , . ..:::: 0 fOl- £. 0 I

,. r ,

(6)

(:l (j "I)~ /~ ./ ~ I} ~ -~ / (.;1 I)~ IwJ ./ ~ I)::: -ID · ./ ~.,,),;: I, ) 'y, I f 'f: Y '* J r z V ., '" ty y

16 ,'1) =: 1.2 + .';,.(x. -~) -It.(!/ -I) ~ ~~ (~(x-1.) .. -,106 .... )(,J --')"" 16t;, -//).

("; n;. ~(;5 ->I-~~ 10 f.f'F, 'f ) ;; t.t.. or;- ('.).,') ~

~\ :!! I.). +- Iv (:t -~ ) - It- (d -() .

(~L '\ °6 rl~E'S'T ./ I~,D... , ~ . .

Thne: 1+ hours ...

scJutians-.

1 L'·, ~t · , ,f-,") • ., . - ~ =: u::;:'" ,·",1n(J : .~ ~ • (. •• ". ) "' . I ... · ~ ;f: 4 " ;$. v; - '," T ~ / ...

4. ".1 "J .

F>;~ll.l.a k: I I ;}~ <:o~ (yt.) d!J d;.: . .In ,:>::.,

that.

[l1.J

f51

P ,'l' .O.

') .:.

.;,..- -

('ll dzdydz. /1J / .. :: ... -"_ .. '-) ---.-

- ·V.l 0- :t:· - y."t, [10]

(ii) The <IGll.ble int~~gnd l/ f( ~i~, }!)dA ITprcsen.ts t.he::·Y<~:Y:W.i~:5)f~·th:(i ; ~,:~lt(1 O'<;t:~X the Jv

/"., ,lH}

I" Il

region R~ 1y.!ng bd:'.:~~€~cn ,;. :::; 0 arid :t :.:;; f(;c :y)..

) \l ({"X,p J) = (~~ ("-;; J, I, '-7-) } vf{s, ~ 0) ;: (D) I, 3).

r~~jYto,,-<- ~ 0 (1:-J) -fl(!i -~) +-J(2 -0) =- o . v' .. v (..-

(iC) .A41a-,)(.,v<U.u-L f ~d-'/~~~Uc. ~ 1'V({S;~O) i =- . J',

("') ~~ A.A..)QA..r ~ 13' /0 6 c) =:L Ct~ q '-+ .6 ~ L L-~ /. rc../ / /' ("J )

c===-) ~ -f 3 c.. :::- J a- ....,(' &!l' '- -I- ~ L. 'f- t:- 1- .:::... /

Ej b = .;1- 3c. Qo~ q'" ~ /c - ~-t-fi Vc - ~--..R; L 0 . . l ' lo~ . ID~

fo Ok'L I'~ 'bk. ~ '"",-""'; ( 0 . .:z -J a ~) ,) - /0 / /0 .

;!. ./ _ I_ .J J ~ ~ o t - ~ x. -V -l! .::. 0 - - - ~

~ ;: -J:t'/ - ..tXJ ~o --_.<ffi .

2J Zr-. 7i<J-l' ~ 't'J ~'~ &JJ;oeJ>: fx~ = /:;".1. J". ' .

® ~J )( -=- c> cry (J .=... 0 or /J:-

(9 o~ (0) -)/ (f -1).

{xJ ; -31 ~-,)J f'lY': - {, x(j - .)..J( .'

Ctt-(i ,-l} ) ((xJ)"'-{(v~)(?Y)= -f cu~ {oJ -t}; ~)('1)J..-~)<)rlrr):: I)

aJ- (0, d) ) {(xv ~ ~vfYj/= . 0) ,10,,",,0 C:O-~O-. .

N'ct-~ .z4r- '""" -">L O//'''' ~ (' o~ d) a>\. t! =- () ) f ;> 0 fjsw--.1. ~ 0)

01'O"c.-l~ c,?o) ~ 1/=.1.:.; 1:=-x...J.L..e> (;?- x....:c»/ ,Jd~ ~= , IO'\.-~ (0) d) ~ '\. /S-ooU>lk /~~ ~.~

5".

b.

I

. / ""

4 :, I

~\C>(~~ ~e>l..rf ,£;~

- ~::;;7J..~o "'~1'7aA.,4 (~ 0- 1.. 4 ~~_ *-. S .

' - . --~--- - ~. If (" 1(1 11 'J- (J . 0 - , ., ' , J. ... _. :-11.1 '''~ - - "4 ____ . _ ,- .

. 4 _. - ~ / .-' . .. ... t.I - 4-'J.-

~:::-1l (f'"

"cc:r -,h. <\:~) f-C~ '1}:- -/ ~ 4.&:.c...a.. - -XLf-,.L.:=. /.

1[-fo(4-.::.. -Ii· 6...d //~<"k -/-71. ;e )

(ttl) hx-t~-e.' 'V :f(J(o.y,) A ~ ~ ~ ..d~""f; t'~ ~r. 9Y C/.." , - \. -' ''if.- , oj ' - >:)/.

/ V / iJ-( -" Vi - '~ ' , ' ~" . ~ .,

. , . • (';""L, 'fl/:J.; -1/ ~ ~ ,M~ ,-b ~ 4-u.rOC _~ =- p4'''I<,

Department of Mathelnatics and Applied Mathelnatics

MATHEMATICS 280W

CLASS TEST 2 - MONDAY, 3 MAY 2004

Tilne: 1 ~ hours Full Marks: 50

Notes: (i) Only approved calculators may be used. Oi) Full answers are expected. tvlarks will be deducted for incomplete solutions.

1. Find the points, if any, on the surface z = 4x2 + y2 at which the tangent plane is parallel to the

p lane x + 2 Y + z = 6. l.) ;

2. Locate and classify the stationary points of f(x, y) = X4 + y3 + 32x - 27y.

~3. Use Lagrange multipliers to find the points on the paraboloid z = x 2 + y2 that are nearest to and ' furthest from the point (0, O. ~). r 9 ~

8 2

4. Evaluate J J 1 4 dx dy. l+x ow

;"). Let S be that part of the surface z = y2 which lies over the triangular region having 'vertices at

( 0, 0) , (3, 0) an d (3, 3). 1

Calculate the mass of S lfthe density is given by p (x,y,z) = ----;:=== J1 + 4y2

1 v'1-z2 2-y

6. Let I = J J J Z d.l' dy dz. o 0 y

Sketch the region of integration for the integral I , and express I as a sum of repeated integrals of

the form J J J z dz dy dx. l '.

7. Let T : R,2 -+ R,2 be defined by T ( ~ )

g(;L', y) = x - 2y.

( !(1:, y) )

(', ' ) where f( x, y)

9 ,r~ Y

(1) Is T a linear rnap? COY\.-J'11,\I\ 0th''4 ~ G~~.Jif((\{j (01 (V1- .

(i1) Find the Taylor expZlllSioll~ up to the second degree tenns, for f( x, y) about the point (1. :2'1.

(iii) Find the approxin13te ,H(,(\ of the inlage under T of a s111all circle, centered at (1, 2 ). of (',rea

10-2 crn 2 .

T } ft ' ., 7" l . ' I 'J) . T ( 1, 1 ) (iv) l ise t le allnc apprO\J 111 cl tl0l1 to at t.ne pOlllL ( '- , to estlInate 1,9'

Il-t -

J. -{x {i" J}::: 4- .:J...3f' 3 .)=-0 a.=J .;( ~ - ~

,(y r,C'1) ,= J(j.}-.Jr.=o Q~ (J; ~..?', ~ ~U"-? ~cn~~ (-~ 13)

(X: (1,:;); /() X/ ~;; lx'J)~~" 1r J'(:('~); 68 . g - A C ::= - f;; :x. ,"1 = 5 - 3 (. If- cJt" t~, 3) "....to

. U Z S' 6, If. aJr C-.}.,1 --3 J (-J, 3) ..i;. 4. ~ /UAIA( 'UA.CLU. (470) ; (-J. -.1) ;, c;, Se-~(I!' /oc;. t

.:{

o(.r =- ..L -fA (if-.f f-) 1 r- ~o

._ ".L.£.... / '"7 -tf- .

~~~~-I---~~ :. ,

1.(~ Ttl) ~ (~{:(-r'f') . .L~,< T(g)=(Vt(~ r~ U-, jJor <\ A '--c.-r /147'-

(It) t 6,'1) =- 3 ;cJ-"j ) t; (1:'J)~ xl- ~() -tJl/x ''1)'' 6;t if ) (xv b >'} ) '" 3.f ~ (,. ,,6>,:/)~ -:l

do f{I,J)d:; .;2"r ('{X-')'-3(y-2)+}{t.16--t}~£fx-/)!,y-.2.) -.U

r

U-.2/'J· (/t'l) T' ~ ?J '" /3 x.) (J x..-.J -<11-) T i/;):; (~ ~)

IJ l I .- ;;).-_ _ } ( Q I

ol~ ref); - '1 - .1 -.<. <

0",,-0<.. .J6 ~''''S e-: /-1!" /D .:: '1 )(. /l) c-~.

tv) T(/:~) ~ r(j) -r T(:)f-~:)=(_i)I-(6 ~~C;~/) == ~?j r(:;1) :- C-~ ~),

.J

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS

MATHElVIATICS 280W

CLASS TEST 2 - MONDAY, 9 MAY 2005

Tin1e: 1 ~ hours Full Marks : 50

Notes: (i) Only approved calculators may be used.

(ii) Full answers are expected.

1. Find and classify the stationary points of the function f( x, y)'~ ;c2y + xy2.

[7J

2. C'se Lagrange IVlultipliers to find the points on the curve of intersection of the surfaces x 2 + y2 = 16 and 2y + z = -2, that are nearest to and furthest ' from; the origin. Explain why there is a maximum and a minimum, value in 'this case.

[9 J

4 2

3 Evaluate J f , 1 - dxdy. j jg - x 3

[6J o.,;y

4 The surface S (':onsists of that part of x 2 + y2 = 4 which, lies in ,the first octant." between z = 0 and z = 3, Find the mass of S if the density IS given by

p(x, y~ z) = x 2yz.

~ 7)

2 V4-x2 y'S-x2 _ y 2

5. Sketch and describe the region whose volume is given by J J J dzdydx.

-2 0 x:t+y2_2

[5 J

.:r2

6', 0 btai ~-l the Taylor expansion, up to the second degree terms, for f( x, iJ; ::? ---:-) ?f

abou t the point (2,1).

I. Let T ~ 2 -r R 2 be defined by T ( ~ ) "" ( : : 3~ ) .

(i ) Find a cartesian equa.tion for the in1age under T of the line y = 1 + 2~r. (ii) Is T a linear transforrnation? Explain.

[ 6]

;61 L J

S. Let F: }R3 ~ ]R3 be defined by F ( ~ J\ = ( x +x:~ Z2 ) .

z x + yz

Use the affine approximation to F at the point ( ~ ) to approximate F ( ~:; ) 2, 1

(5J

3

(x x()(, y} = oZ.;;! /J ~ A-c ::=- 0

D /~ 1/ =- x. :

4- /fees> ~ /jr(x, Y,J)d S "-Jf ~ :V}; oIS s s U

~ £{(4-J~ J'l,} ) l~oLi-ft#t)1- d4

= J IJ~ rJ- ~-Y'-1? o 0

j .

X:=~-J 1-

Xj =- 0

g , / 0) / -* .;J- u

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS

MATHEMATICS 2080\V

CLASS TEST 2 Tuesday, 9 MAY 2006

Time: 1 ~ hours

Notes: (i) Only approved calculators may be used. (ii) Full answers are expected.

1. Let g(x, y) = 5 - x + eX y2 and let P be the point (0, -2).

(i) Find the directional derivative of g at P in the direction (5, -1).

Full Marks: 50

(ii) In which direction is the directional derivative of g at P a n1inimum? What is this minimum value?

[5]

2 2x

2. Express J J e~? dy dx as a repeated integral, integrating first with respect to x and then

o 2x-4

with respect to y. Do NOT evaluate the repeated integral. [6]

3. A solid is enclosed by the surfaces 2z = 2 - y, y = 2 - x 2, Y = ° and z = 0.

(i) Sketch the solid and express the volume as a repeated integral of the fonn

J J J Idx dz dy.

(ii) Calculate the volume of the solid.

[7]

4. Find the surface area of the portion of the paraboloid z = h(l - x2 - y2), h > 0, lying between z = 0 and z = h. [6]

5. Use a suitable transformation to evaluate

where R is the region in the first quadrant bounded by the curves xy = 1, xy = 3, y = x 3 and y = 2x3 . [7]

6. Express the mass of the solid lying within the sphere x2 + y2 + (z - 1) 2 = 3 and below the plane z = 0 as a repeated integral using spherical co-ordinates, if the density function is p(x, y , z) = y'x2 + y2 + (z - 1)2. Do NOT evaluate your repeated integral. [8]

1

7. The transformation T : ]R2 -t ]R2 is defined by T ( ~ )

(i) Is T a linear transformation? Give reasons.

(ii) Find the image under T of the portion of the line y = :C, x :::; -1

(iii) Use the affine approximation to T at ( ~ ) to find an approximate value for T ( ~: ~~ ). (iv) Find the approximate area of the irnage under T of a small circle with centre (2, -1),

radius r.

[12]

2

Ii\.

:2 1'I11f/ R' i

"Jo () I ~~ ~

b . X ~i-J ).. r () -J ~ 3~) 1.~J )..-t-J'--= :l~.J J­

~ ~ (J.-:;. 2 +~~C-DOr

$"'0 j> '" :Zc""l :t f If. C G'J Y T 3 ;; Cot) I ;! J='/ :~, ;J..

/1~ss '1 ~~ ~ 111(r, L( & d-I/ := II/J ~ y 'l '9 ;>--cill

~ iT 1/ C'VJ;I 1- J c:.c:?f + ~ I< ~ J J j ) /1~-.2?C';t'';-1 'I<-M~ ott ~I d&

o 17f~ 0

(~c/k ~ COIfJ? - )c;xr',t-i' ~ 0)

. ., ( ' "r • ::J

[it; .-4.",:r- '1: 50 T (:x.1 ) ~ (~ ':- v l-) ~ (,';). . .v-;.. x. . .,. J . / . .Jf- ~ " ':':/~

~e 1J;'~) X .~ - I M .~ "Ir~L ck,;t4, /crt-+,'~ r ~ ~>-<--A.:=- 0) ,;~ ?o/.)~;,cc 1...--- z< -/~ x.;t~.J.) ';/0 .

IT -' ~ / .----,;:.

/71~~~

(p~: .,w,-,- CaAjL~ i,~~:-cr O<.~ ~~ ~It . ~ ft ~ ~~ 11 "'1, X &. -I /.M~ / M; pc.,. X lL~o.r. )

~',) r 1(; ) ~(~L -;:) ~ (.2 ~-) aY;de /~'~ (,I)

r (J 0:[1 ) :; T () . (7 '(/J]0' 0/ ) ; (;J I-{' -.lt%

: )

.• I + '-O>~~)J-~o~.oo{):i;~:;. · ~~) QHo..~'()J~ .;. iT~~ IITe-VI/ ;; T/ /~ 1/; -; 1/

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS MATHEMATICS 2080W

CLASS TEST 2 - MONDAY, 14 MAY 2007

Time: 1 ~ hours Full Marks: 50

Notes: (i) Only approved calculators may be used. (ii) Full answers are expected.

1. Let I = r1 jv'4-X2 r2

+y2

Idz dy dx. ~o vf:3x ~o

(i) Sketch the region whose volume is given by I.

(ii) Express I as a repeated integral using cylindrical co-ordinates.

(iii) Express I as a repeated integral, integrating first with respect to x, then z, and lastly y.

[12]

2. Express the mass of the solid lying within x 2 + (y - 1)2 + z2 = 1 as a triple integral using spherical co-ordinates, if the density is given by f(x, y, z) = Z2.

Do NOT evaluate your integral. (6]

3. Let F(x, y , z) = (z, x2 y , 3) represent the velocity field of a fluid. Let S be that part of the surface Z2 = 4x lying in t he first octant , bounded by x: = 0, y = x and x + y = 2. Find the flux of F through S in the upwards direction. [8]

4. Use the transformation u = ax, v = by, with suitable values of a and b, to calculate JR J 1 dA, where R is the region lying within the curve 4x2 + 9y2 = 36. [5]

5. Find the work done in moving a particle along the lower half of the circle (x - 3) 2 + (y - 1) 2 == 1

from the point (2, 1) to the point (4, 1) , through the force field F(X 1 Y) = (2xy, x2 + e y2

). [5]

6. Obtain the area of that part of the cylilidQr y = 4 - x 2 lying in the first octant, between z == 0 and z = x. c.,u"tlEl1.. [5]

7. Let F : ]R2 ---+ ]R2 be defined by F ( ~ ) = ( X:x-1 __ 3~~ ). Use the affine approximation to F

h . (-2 ) fi -d . IfF ( -1, 9 ) at t e pOInt 1 to n an approxunate va ue 0 - 1, 2 . [4]

8. Find the norrnal of the plane onto which A = maps the plane x = 3. (

022 _Ill 4°3) (5)

1

I. (~)

o 0 o

II 11 . ~ ~.:~~~8

:0 / I I /'cO:j. /"-~'''I 01/ 0I;! c-/L9 o 0 (:;>

y Ii\

0-

-Ul-

(~,I) (It-, f ) c..

2 3 v Jl

C'; :1-:>- (;­

J :;-4-6 t­

D t6:!:.l

DEPARTMENT OF MATHElVIATICS AND APPLIED MATHEMATICS MATHEMATICS 2080W

CLASS TEST 2 - TUESDAY, 12 May 2008

Time: 1 i hours Full Marks: 50

Notes: (i) Only approved calc ulators m ay be used. (ii) Full answers are expected.

1. Let T: ]R2 -+ ]R2 be defined by T ( x ) = ( x2

-2 Y2 \ . 'Y

1 y-x )

(i) Is 'T a lineae u antiform",tion ': Explain

(ii) Use the aftlne approximat ion to T at the point ( ~ ) to find the approximate value of

T ( 1,1 \ 0, 9 ).

(iii) Find the approximate area or the image under T of a circle center ( ~ ), having

radius to. (iv) Find the images under T of the curves y = x2 and y = o.

[2,4,3,4]

2. Let R be the region in the second quadrant bounded by x2 + y2 = 4, y = -x and the y-axis.

(i) Express the area of n. as a repeated integral, integrating first with respect to y and then with retipect to x.

(ii) Express the area of R as a repeated integral, integrating first with respect to x and then with respect to y.

1 (iii) Evaluate I I , dA using polar co-ordinates.

R J2 +x2 + y2

[2 ,4,5]

3. Calculate the surface area of that part of the surface z = y'X cut off by the surfaces y = 0 ) x = 4 and y2 = x. [7J

4. Use spherical co-ordinates to evaluate J J J(x2 + y2 - z2)dV

R

where R is t he region lying within t he sphere ;1; 2 +- y2 + Z2 = 4 wit h y 2 O. [7]

5. The region R is bounded by the graphs of .c +- 3p . .:: 0, X -t-- 3y = 2, 4:r - y = 0 and 4:t -- )) :.-::. 5. Use a. sui table transformat ion to evaluate J )'(5,l; + 2y)dA.

R

[7]

6. Let EJ~c, y) = (x2 + y, Y + :z: ). Find the work done in moving a particle alol1g the curve C t hrough the force field E. where

(i) C is the circle x2 + y2 = I , and

(ii) C is the upper half of the circle x2 + y'2 = 1 traversed iIl an anti-clockwise direction.

[6]

2

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MAM 2080w

Paper 1 revision

A. Coordinate systems:

1. Polar 2. Rectangular 3. Spherical 4. Cylindrical

B. Functions of 2 or 3 variables

1. Level curves 2. Limits 3. Continuity (squeeze theorem) 4. Partial derivatives

a) Directional derivatives b) Gradient c) Tangent plane d) Slope of tangent line

5. Linear approximation 6. Differentiability

C. Vector functions & Space curves

1. Limits (continuity) 2. Derivatives 3. integrals 4. Arc length 5. Curvature 6. Normal vector/plane 7. Osculating plane 8. Osculating circle 9. Chain rule

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UNIVERSITY OF CAPE TOWN

University Examinations - May/June 2004

Department of Mathematics and Applied Mathematics

MAM280W - Mathematics 280W (Paper I)

Time: 2 t hours Full Marks: 100

Marks available: 100

This paper consists of 2 pages. Only approved calculators may be used. Full answers are expected. !vlarks will be deducted for incomplete solutions. ,---------------------------------------------------------------------------

1. The curve C is given by r(t) = (In t, 21n t, (In t)2), t > O.

(i) Find two surfaces on which C lies, and use them to sketch the curve C.

(ii) Express the length of that part of C which lies between the points (0,0,0) and (2,4,4) as a definite integral. Do not evaluate your integral.

(iii) Find the curvature of C at the point (0,0,0).

(iv) Find a cartesian equation for the osculating plane to C at the point (0,0,0).

. . { x + 2y 2. Let f(x, y) = x2 + y2

if x = 0 or y = ° otherwise.

(i) Does lim f(x, y) exist? (x,y)-+(O,l)

(ii) Is f differentiable at (0, 1 )?

(iii) Does fx(O, 1) exist?

(iv) Is f differentiable at (O,O)?

3. Let f(x, y) = x3y2 + ¥.., and let P be the point (1,2). x

(i) Obtain the Taylor expansion of f, up to the second degree terms, about the point P.

[14}

[10]

(ii) Find a cartesian equation for the tangent plane to the surface z = f(x, y) at the point P.

(iii) Find all directions in which the directional derivative of f at the point P is equal to 11.

[13]

1

4. Find and classify the stationary points of f( x, y) == 2x3 .-; x 2y + y2. [8]

5. Use Lagrange Multipliers to find the maximum and minimum values of x 2y subject to the constraint y = x 2

- 8. [8]

6. Let F : ]R2 -+ ]R2 be defined by F ( ~ )

(i) Is F a linear transformation?

(ii) Find the approximate area of the irnage~ under F, of a circle centered at (1,1) with radius 1 10 cm.

(iii) Use the two-variable Newton's Method, with Xl = ( ~ ), to find a second approximation

X2 to the equation F ( ~ ) = o.

3 7'l -v9-x2

7. Let I = J J (9 - x 2 - y2) dy dx.

o x

(i) Sketch the solid which has the integral I as its volume.

[9]

(ii) Express I as a sum of repeated integrals, integrating first with respect to x and then with respect to y.

(iii) Use polar co-ordinates to evaluate I.

8. Use spherical co-ordinates to find the volume of the solid lying within the sphere x2 + y2 + (z - 1)2 == 1 and below the cone V3z = Jx 2 + y2.

[11 ]

(7]

9. Calculate the surface area of that part of the surface z = 12 - 3y2 lying in the first octa.nt, bounded by the planes z = 0, x == 0 and y = x. [6]

10. Let S be that part of the sphere x 2 + y2 + (z - 4)2 = 9 which lies above the plane z = 2, and let F(x, y, z) == (-y, x, 1). Use the Divergence theorem to calculate the flux of F passing through S. [7)

11. Use Stokes' theorem to evaluate

J(y2 Z 2 + z) dx + 2xyz2dy + 2xlz dz

c

where C is the curve r( t) == (cos t; sin t, sin t), 0 ~ t ~ 27T'.

2

[71

UNIVERSITY OF CAPE TO\VN

University Examinations-May/June 2006

DEPARTMENT OF MATHElVIATICS A.ND .APPLIED MATHElVIA:TICS

IvlAl'vI2080\V - J\l1.A.THEMA.TICS 2080\V

,.ime: 2 hours

1his paper consists of 2 pages .Vnly approved calculators rnay bev.sed ttl answers are e.rr;pected. ivf ark.s will be deduct~d for -incorrtplete solutiorlts:

, . . {/x+y 1. Let f(x, y) = x 2

if Y =1= 0 if y = 0

Full l\1'arks: 67 IVlarks Available: 70

. .. - .' -_._ . . ..•. _---

(i) Find /x(O, 0) and /y (0 1 0) if they exist, (--;ive reasons for your clainls.

(ii ) Do all the directional derivatives of f exist at (0: O)?

(iii) Find the directional derivati'V·e of f at (0: 0) in the direction (2: 1).

(iv) Is f differentiable at (0, O)? Explain.

f { x' + 71 if Y =f.. 0 (v) If g\x,Y) = x2 +CJc if y = 0 '

C E JR. , give with reasons, a valLie of r. such that ii~ . 9 (:;. y) does not exist. (;r.·Y ;'--)-\,IJ ,O)

2. The curve C is defined by r (t) = (s in2 t , cos2 t , cos tJ: t E JR,

(i) Is C a planar curve? E.'<:plain .

(ii) Find two surfaces on which C lies] and hence sketch the cu rve C

(1'1'.1"/ 0 J btain a vector equation of the t angent line \:.0 G at the point (L 0 ; (i').

(iv) Calculate the curvature of C at the point (LO, '0),

". 1'4~ /Y . ~. Express f(x, y) d1.; dy asa rep ented integra,l, integrati ng firST \'\-'it h · espcct to 7) aud . 2 li. .

~~hen \vith res~ 'ect to x. . .

1

4. Let S1 be the paraboloid z -= x2 + (y + 1)2, 52 the plane z= 5 + 2y, and let

F(x, y, z) = (y, -x, 3z).

(i) Find the projection of the solid region enclosed by S1 and S2, onto the xy-plane.

(ii) Obtain parametric equations for the curve of intersection of 51 and 8 2 ,

(iii) Calculate the volume of the region R enclosed by 51 and 52.

(iv) Use the Divergence theorem to obtain the flux of F, through the portion of Sl that lies below 52, in the downwards direction.

(v) Find curl F. Is F a conservative force fieid?

(vi) Find the work done in moving a particle t hrough the field F ) aiong the curve of inter­. section of S1 and 52 in an anti-clockwise direction .

5. Find the surface area of that part of the surface x = 1 - y that lies in the first octant, bet'ween z = 0 and z = 1 + y2 + x3 . . [j]

6. Find and classify the stationary points of f (x, 11) = xy + ~ x2 . [4]

7. Use the two-variable Newton's tvlethod, with Xl = ( ~ . ), to obtain a second approximation

X2 to the soh~tion of the systern of equations x 3 - - ~/' :=: 1, xy2 =: 2. [6]

2

UNIVERSITY OF CAPE TOWN

University Examinations-June 2007

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS

MATHEMATICS 2080W - MAM2080W - Paper 1

Time: 2~ hours Full Marks: 100 Marks Available: 103

If you would like to access your examination script(s) please see the noticeboard in the lVIathematics Building for application details or alternatively go to http://www.mth.uct.ac.za

This paper consists of 2 pages. Only approved calculators may be used. Full answers are expected. Marks will be deducted for incomplete solutions.

----------------------~~------

1. Let C be the curve r(t) = (t, t2, t4), t E lR, and let P be the point corresponding to t = !. (i) Does the point Q (2, 4, 8) lie on C? Explain.

(ii) Obtain a vector equation of the tangent line to C at the point P.

(iii) Find the point(s) of intersection of C with the surface x 2 y + ~ = 3.

(iv) Find a cartesian equation for the tangent plane to the surface in (iii) at the point (2, 1, -~).

(v) Obtain a cartesian equation for the osculating plane to C at the point P:

[2, 3, 4, 4, 4]

{ 2 - x - 2y if y f. 1

2. Let f(x, y) = ° if y = 1 .

(i) Use the definition to calculate ix(O, 1) and j~(O, 1).

(ii) Show that all the directional derivatives exist at (0, 1).

(iii) Decide whether or not f is differentiable at (0, 1).

(iv) Find the direction(s) in which the directional derivative of f at (3, 5) has value 1.

[2, 4, 4, 5]

3. Find parametric equations for the curve of intersection of the surfaces 2z + x 2 + y2 == 5 and Z2 = 3 + x2 - 2y2. [5]

4. Locate and classify the stationary points of f(x, y) == X4 + 5xy3 + 3x2y2. [7]

5. Use Lagrange multipliers to find the points on x + y2 == ° that are nearest to and furthest from the point (0, 3). Explain why these are the nearest and furthest points. [7]

6. Evaluate r1 r2

x 2 sin(y4)dy dx. Jo J2X

[5J

1

7. Calculate the surface area of that part of z = y2, lying above the triangular region in the first quadrant which is bounded by the lines y = x, y = 1 and x = O. [6]

8. Let S be the sphere with equation x2 + y2 + (z - 2)2 = 4 and let F(x, y , .z) == (2x, y, x2 + z) be the velocity field of a fluid.

(i) Express S in spherical co-ordinates.

(ii) Find the volume of the solid lying within the sphere S and inside the cone z = J3(X2 + y2).

(iii) Use the Divergence theorem to calculate the flux of F' through the upper half of the sphere S in the upwards direction.

_y3 xy2 9. Let P(x, y) = (2 2)2 and Q(x, y) = '( 2 2)2' x+y . x+y

(i) Show that Py(x, y) = Qx(x, V) .

(ii) Evaluate L P(x, y) dx + Q(x, y) dy where C is the ellipse 9x2 + 4y2 = 36.

10. Let F(x, y, z) = (5z + ex2, x, yz ).

(i) Find curlF.

[4,5,11]

[2, 4]

(ii) Use Stokes' theorem to find the work done in moving a particle through the force field F (x, y, z), along the curve of intersection of the surfaces x2 + y2 = 6 and x + 2y + 3z = 1, in a clockwise direction when viewed from above.

[2, 7]

11. Use the two variable Newton method to find a second approximation ;£2 to the solution of the

system of equations y3 - X = 0, x2 - 2y = 0, starting with ::r:l = ( \ 5

) . [6]

2

UNIVERSITY OF CAPE TOWN

University Examinations-May / June 2008

DEPARTMENT OF MATHEMATICS AND APPLIED MATI-IEMATICS

MATHEMATICS 2080W - MAM2080W - Paper 1

'rime: 2~ hours Full Marks: 100 Marks Available: 102

If you would like to access your examination script(s) please see the noticeboard in the Mathematics Building for application details or alternatively go to http://www.mth .uct.ac.za

This paper consists of 2 pages. Only approved calculators may be used. !,ull answers are expected. Marks will be deducted fc:r incomplete soltdions.

1. Let C be the curve r.(t) = (2t2, -1, 4t - 4), t E IR, and let S be the surface x 2 - 4 - 4yz = 0.

(1) Obtain a cartesian equation for the tangent plane to S at 1,he point P(2, -1,0).

(E) How many points of intersection are there of C and S?

(iii ) At what angle do C and S intersect at the point P(2, -l ,U).

2. A particle moves along the curve C with its position at time t given by

'[ ( t )::::: (cos t, sin t, t 2 ), t 2: 0.

[3, 5, 3]

(i) \tYrite down a vector equation for the tangent line to C) at the point corresponding to t - 'IT - 2'

(ii) Find the curvature of C at the point where t = ~.

(iii) Find a cartesian equation for the osculating plane to C at. t = ~.

(iv) Express the distance that the particle travels in moving :?tlong C, from the point (1,0,0) to the point (-1,0, 1f2), as a definite integral. Do not evaluate the integral.

. In(x2 + y2) 3. Let f (x, y) = 2 2'

X +y

(i) Find all the stationary points of .f. Do not classify them.

(ii) Obtain the level curve on which the point (1,0) lies.

(iii) Use the above information to give a rough sketch of z =-= f (x, y) .

1

[3 ,5 ,2,3]

[4,2,3]

4 1 '1 t" . . d fi d b ( ) {6 - 2x - 3y if y f. 2 " . le unctIon 9 IS e ne y 9 x, Y = 'f 2 . Use the definition of differ-

x 1 y = entiability to decide whether 9 is differentiable at the point (0,2). [6]

5. Find and classify all the stationary points of f (x, y) = 2X2 + 3xy + y3 . [7]

6. Use Lagrange Multipliers to find the maximum and minimum values of (x - 1)2 - 3y2 , subject to the constraint x 2 + 4y2 = 36. Justify your claims. [8]

7. U'se the two variable Newton method to find a second approxirnation ~2 to the solution of the

system of equations x 2 + 2y2 = 4 , x2 - Y - 1 = 0 starting with ±l = ( \5

). [6]

8. Express the area of the region enclosed by the parabola y2 = x and the line y = x - 2, as a repeated integral, integrating

(i) first with respect to y, and then with respect to x,

(ii) fi rst with respect to x, and then with respect to y .

Do not evaluate these integrals.

[7]

9. Let 81 be the surface x 2 + y2 + Z2 = 25, and let 8 2 be the surface z = 3. C is the curve of intersection of 81 and 82 .

(i) Obt ain a third surface on which the curve C lies.

(U) Give a set of parametric equations for C.

(iii) vVrit e down a triple integral, using spherical co-ordinates, which gives the volume of the solid lying within 8 1 and above 32 . Do not evaluate your integral.

(iv) Calculate the mass of that part of 8 1 that lies above 82 , if t he density function is given by p(x, y, z) = z.

[1,3,4,6]

10. Let F(x, y, z) = (2z, y, x) be the velocity field of a fluid, and let 8 be t he curved surface Z = J x 2 + y2, lying below z = 2. Use the Divergence theoreln t o find the flux of F through S. [8J

11. Use the transformation u = y - x 2, V = Y + x 2

, to evaluate J J x (y - x 2 )dA, R

where R is the region in the first quadrant , bounded by y = x 2, Y .= x 2 + 2, y = 2 - x 2 and

y = 3 - x 2 . [7]

12. Lt~t F(x , y, z) = (z, y, x).

(i) F ind curl F.

(ii) Calculate the work done in moving a particle through t he field E along that part of the curve r(t) = (cost,sin2t, (7)3) from the point (0,0,8) t o tI1e point (-1,0,1).

[2,4]

2

Department of Mathematics & Applied Mathematics

MATHEMATICS 280W

CLASS TEST 3 - Monday, 4 June 2001

1 Time: 1- hours

2

Notes: (i) Only approved calculators may be used.

Full Marks: 50

Marks Available: 51

(ii) Full answers are expected. Marks will be deducted for incomplete solutior.

1. Let T : ~3 ~ ~3 be defined by T G) = G ~ n· (i) Find the image of a normal to the plane 2x + y - z = 0 under the mapping T.

(ii) Find and describe the image of the plane 2x + y - z = 0 under T. [5]

2. Let F: ~2 ~ ~2 be defined by F(:) = C4X:

yy4).

(i) Is F a linear map? Explain.

(ii) Use the affine approximation to F at the point to find an approximate (20)

(1,9) value of F '). 0, ....

[6)

3. Find the mass of a piece of wire in the shape of the parabola y = x 2 , ° ~ x ~ 1, if

the density is given by p(x, y) == xy. [5]

4. Calculate the work done in moving a particle along the curve r{t) == (cos3 t,.sin3 t),

7r 7r -2 ~ t:::; 2' through the force field F(x,y) == (x..y2 + arctanx,y2 + x2y). [4)

? T.O

5. Use a sui table transformation to evaluate

/ / ysin(x2 y)dA R

where R is the region in the first quadrant bounded by the graphs of y = x, y = 2x ,

[6]

6. Find the mass of the solid lying between the plane z = 0 and the paraboloid

z = x2 + y2 and within the cylinder x2 + y2 = 2y, if the density is given by

p(x,y,z) = Jx 2 +y2. [8]

7. Use the Divergence theorem to find the volume of fluid flowing through the upper

h_9:l.L of the hemisphere x 2 + y2 + z2 = 4, z ~ 0, in the upwards direction, if the

velocity of the fluid is given by F(x,y,z) = (x 3 ,y3,l +z3). [10]

8. Use Stokes' theorem to evaluate / F.dr. where C is the curve of intersection of the

C

surfaces x 2 + y2 = 4, x + y + 2z = 4, positively orientated, and

[7]

. • • -....) , _" , ~ __ f;/ u~ d-C1O ,I . I. (tj

J . (tj

( ii)

I

.3. """-rS s J~("X.'1)o(.r ::. j t.l/' j,L1(!6r •. H Co.

"t,

~r~-4'-· c, ~ d4:r/Q,t f1

, ,.. 'I "'0 'III J'de..,. 7r (o~ -I) ~ (0,1)

Department of Mathematics and Applied Mathematics

NIATHEMATICS 280W

CLASS TEST 3 - TUESDAY, 19 AUGUST 2003

Time: 1 ~ hours Full Marks: 50

Notes: (i) Only approved calculators rnaJr be used. (ii) Full answers are expected. IVfarks ~vil1 be deducted for incomplete

solutions.

1 For ,vhich value(s) of A does the system of linear equations represented by the augmented matrix

2.

.. 1 0

( 0 A-I o 0

1 ) A(A - 1) (,\ - 1)('\ - 2)

have _ (i) no solution, (ii) a unique solution, (iii) infinitely many solu­tions? Find all solutions \vhenever possible. [7J

r ('i Let X = 1 1

\ 1

~ ) , 2

y== -1 4 1 o I

1 0 1 2

1 3 1

-1

~l ),

1 /

and

1 ) 1 ~ I ~. 1) J

(i) Is X a linearly independent subset of ]R1? Explain.

(ii) Find a cartgSiaD eqJ1a.tiQp for the linear subspace of 1I{1 generated by Y.

(iii) Find the dimensioIl of, and give a basis for the linear subspace of mr 'generated by Y.

(iv) Is X a generating set for the linear subspace of JR4 generated by Y? Explain 0 [11 J

3. Which of the following sets are closed under scalar multiplication, and which are closed under addition? Are either of the given sets linear subspaces? Give full reasons.

(i) {x E R,5 : 2Xl - X2 + X4 + X5 = 0, Xl - X3 + Xs . l}

(ii) {( ~ !) : a, b, e, d E R, ad - be > o}. {7]

4. Which of the follo,\ving statemeIlts are true and \vhich are false? Prove those that are true aIld provide an example to sho\v those that are false.

(i) Let A be a given 3 x 3 matrix such that the homogeneous system Ax = 0 has infinitely many solutions. Then Ax = Q has infinitely many solutions for all Q E R3.

(ii) If B is a linearly dependent subset of R3 , then B cannot generate JR3 .

(iii) If det A == .0, then the system ~4.x = 12 has no solution for all Q E ]R3.

(iv) If A is a 3 x 3 matrix having det A = 10, then det (2A) = 20. - ,

(v) If the square matrix B is obtained from A by repla~ing the first row of A by adding the second row to three times the first ro\v, t hen det A = det B. [10J

5. Let A =

,1 0 A -3 \ 1 1 -1 - A 3J\ ,

AD 2 -1 )· a 1 0 4 0

(i) Solve the equation del A = O.

(ii) Does _4- 1 exist when ,.\ == 4 ? Explain.

(iii) Find the element in the fourth TO'V and first column of A -1 in the caBe where A == o. rlol

l. .J

6. Consider the differential equation

x2 y!l(x) + xY' (x) + y(x) == 5 .

(i) Show that y (x ) = sin (In x )is a solution of the associated homo­geneous differential equation.

(ii) Given that y (x ) == 5 is a particular solution t o the given differential eauation .. find t \VO furt her solutions. r51

.... ~ ... .,J.

s- (i.~j /; I -i~/\ ~:i::/ / : -~~\ !} J i: ~ = 1 I:/~ ,:l~).>' =/~J X I c I 0 ~ jJ 0 II-,/- 4--1~ -/ I-I-'/- 1t-.lA 0 J+~).. 1-3~

:~ ! .P.-~~ -INA 1:= I,-JA )j:;'-r\ 1 -II = (I_J~IJ.3.j.J./\-~'!- o/::0- 3 "A )/3.-AV;+~) I H }. , -1 (\ l' "./ /' +d-}.. I / \. . ) /' r~" I \.' A ~ ~ I

S~'~ ~ d.u:- 4-:=-0 ~ ~;:- - / A;::-..J /l ::a- ¥'3 . ./ )

( "j Y4!s. ~- A =9- ~11 =to S--¢ A--1 ~/(-~. / ) .J

(/~ }JJ A.::- o. 7Z~ ~~:3,

('~{'r :: (-j J; ~ -Df!: ~. $'0 01J = -L. C:f-:= ~. I ~ I ;;., dd If 3

;. "i; ~:::-~~(~:t) ~/=- J. 6J6(~:L) ~/'- -~ ~(.-6,x) - --1 AA-'--(~) , -;:j J V X- ) (j - 1-z- :f-.:l..

~ X'tf+X -.:t'+:1 =. -God(--&~) -~'JA~). + Ccd(~X.)-f~~~.i) .=--0..;

,.~~;. 11 JJ~'f-e>-~) ~ ~ /J~~ 1_d ~5~QM d~. '.:, ~-0<"<--~ ~_-e. . ....;; k-"-€o.../'} IJfj Lk r,-<-""/=-..~~ ~

4, ,( X) '=:' S-+ ~~ (~:t) Ckrc{ 4 C:t).::= S- -J-.3 ~( J?;.../ ) ~ 6Jt J Va-

AfvL,u~\' Cv-J 1 ~-: /J't.~ - ~C5lJ ~~_( ~e .

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS MATHEMATICS 280W

CLASS TEST 3 - THURSDAY, 25 AUGUST 2005

Time: 1 hours Full Marks: 50

Notes: (i) Only approved ca.lculators may be used. (ii) Full answers are expected.

L Let X = { (. il ) , ( ]1 ) ,( 1J ,( ;2 ) , ( n } (i) Is X a linear subspace of Itt~?Fxplain .

(ii) Is X a basis for R"? Eh--plain.

(iii) tor each value of A, find a cartesian equation for the linear subspace of}R4 generated by the set X.

(iv) Find a subset Y of X that is a basis for ]R" in the case where). = -3.

(v) Express each of the element ( s) of X that are not in Y, as a linear combination of the elements of Y (in the case A = -3).

[13]

2. The following diagram shows a road network where all the streets are one-way. The Bow of traffic in and out of the network is measured in vehicles per hour, and is indicated on the diagram. Let Xl, X2 and X3 denote the number of vehicles flowing along the various branches per hour.

(i) Construct a. system of linear equations in the unknowns Xl, x2 and Xl that describes the traffic flow in this road network.

(ii) Solve the system you constructed in (i) and deduce the maximum and minimum values of x~.

[7]

1

3. 1£ A is a. 4 x 4 matrix with detA = 3, write down the value of det( - 2.43).

o 3 - A 2 -2

( 1 -A 7 1 -1)

4. Let B = 1 0 1 - A -1 .

1 0 -1 - A

(i) Solve the equation det B = O.

(ii) Use Cramer's Rule to find the value of X3 if

B ( :: ) = ( ~2 ) x;} -2 X4 -4

in the case where A = 2.

( 1 3 0)

5. Let A = 2 1 1 . -1 0 4

(i) Find the elements in the second row of the adjoint of A.

(ii) Find the elements in the second row of the matrix product (adj rl)A.

(iii) Deduce from (ii) the value of det A .

6. Solve the initial-value problem

( x) dy IT in y + 2 - Y cos x == sin x - I - - -, y( - ) = 1. y dx 2

7. Consider the differential equa.tion

t3Y"'(t) + 2t2yl/(t) + tyl(t) - y(t) = t ....... (*)

(i) Show that Yl(t) = t tin t is a solution to (*).

(ii) Is Yo(t) = t a solution of the associated homogeneous differential equation? Explain.

(iii) Is the set of all solutions to (*) a linear subspace? Explain.

(iv) Find two further solutioIlB to (*). Which results have you used?

2

[3]

[71

[6]

[6]

[91

<.J Not-,"" --L.~Q.-.. ,.-'~yo,~~ _ 0 I- x o-r- ~ /J'1J-e-/O'j...V<.AA~ ~-co-(g,r ~~'Jf-r.;:rt c..O--; ~ .I A,,(o-Y-' ~c--t..~crv-, •

. ~ ',K ~ ~.;~~?Ic oC.j<J..e.,c4A.-r (sAH'cS1Or:1 ...<---' £ Y-)J .du.r .. ~ A /J A / "

c.:.:~-",,-~ o---c ,~ 0.... "'~.J-e..,.'J

f x I -+ IOu - -1.). + +, S~ (;''J / I -I

~ I ~~] ~(~ 0 '/ ~OO) .j 1.. ) :: ~ J.s~ ( ~ i I ;).)..s i ':I.J ~

! :,t. I + .{J :::- Joe 0 i ). uO 0 0 o 0 ;{

('.A} __ "':' (- _ ( :") ...-1 3//) _- ( _ 'i)' 7, __ / , __ ,' (A, ,J) __ : / (~~ 4)) 1: /6 i<..;2 ~ - 'f. 3',) . /( -~v 7 ' . _ ~ \ / 'C ( ' .

/1-'" ?2 , 0 3-~

/ J 0 I J 0

So ~.

- (.

. \:J

I - J / / / _ ~ . "7 ;2 '-rl /::: b :i-A

1-'" -/ I J 0

I ·-1- I' J, 0

)0 ctc-l/.4 :­

"J (~i 4) lJ = (-~ ~.~ lj' . . ;( .:.,

( "J /t~<-e ({~CA~'I}/A = ol.d'4

(ii ) ..A1 () ;..- t , .j

lie. ) / U i) ....... " f'

1 I , ~00:4 )

\

-./ C-j { i.c, ) ..:

' .. . -..

~ .. { ~""".g, . ~ . ":-- ,. ,: .-.:. -.,'f~ ~ - v

__ ....- f ..

. _____ ' t . ....... _ " I - _

- ;2 1:

.- \ -I --)\ _I _ ~ I (.}

o .- .,...

~ ~~ /- )':}- l~ 0 0 0

-/ ) -~

-/

-'"

o ~ .. , C.: c: -C, ..A-..... G<-..- 3 .,..

C i:J.. )

o -+ 0 + 6·/ '- t ;:::.. D

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS

MATHEMATICS 2080W

CLASS TEST 3 Tuesday, 5 SEPTEMBER 2006

Time: 1 hour Full Marks: 40

Notes: (i) Only approved calculators may be used. (ii) Full answers are expected.

1. Given that A is an n x n matrix, express each of the following in terms of detA:

(i) det( -2A)

(ii) det(A-l)

(iii) det(adj A ).

[6]

2. Which of the following are closed under scalar multiplication? For those that are linear subspaces give a basis and the dimension.

(i) The set of all vectors (x, y , z , tv) E 1R4 such that y = 0 and x = Z = w;

(ii) The set of all solutions to the differential equation 1J (x)y"{x) - x + y(x) = o.

3. The vectors ( ~ ) and ( i ) are both solutions of a matrix equation Ax = b, bE 11.',

b #0.

(i) Find two solutions of Ax = o. (ii) Give as many solutions of Ax = b as possible.

4. Prove or disprove the following statements:

(i) Let A be an n x n matrix, b E ]Rn, b # O. If Ax = 0 has a non-zero solution then Ax = b has infinitely many solutions.

[6]

[4]

(ii) If X is a linearly dependent subset of ]Rn, then each element of X can be written as a linear combination of the other elements of X.

[4]

1

5 Let X = { ( -!). ( -D' ( -D' ( -D } (i) Find a cartesian equation for the linear subspace of]R4 generated by X .

(ii) Is X a linearly independent subset of ]R4? Explain.

(lli) Does ( ~~) belong to the linear subspace generated by X? Explain.

(iv) Write down a subset of X that is a basis for the linear subspace generated by X .

6. Let B = ( : 2-,\

1 2-,\ -1

(i) Calculate detB.

1 2 - /\

1 -1

2 ~ 1\ ) 1 .

1

(ii) For which value(s) of >. is detE = O?

(Xl) 1

0)

(iii) If B ~: = l ~ , use Cramer's rule to find x, in the case where A = O.

7. Find all solutions to the differential equation

y eX = (1 _ 2y _ eX) dy dx '

2

[8]

[9]

[5]

M/-}/1 d-O ~o 'y ... (- SO£a-·/7(j~ TD eL~ ~T-~ -:-s./oy~~'

1'(0 01«- (-J4) = (-~) ~dlf / (y oId41;:. "'/;'+' (;i<) J;;'<t- A- ("'..1;',4. ) ~ ( old-~ .:z;.,. ) (o.t d- A) old-( acI; 'A ); ~~ ~ ¥t-d--J. ! ~ o-IJ-(a,lI ,4):: {!d-A)M~~ ' I I

J,(iJ 'P< ~ -"J X: loL ( f)) ,LGJ?/ Fcr-au.;",; If(,f))= y(,j/r;X. £ X kI ~~v.L. _~ S'ccv(QrA4'C.eeL-~v/t/'co~·'~ •

[X k ~ ~(' I~.' . ~/Q1:--\ x ~«<- ~ ;" .d:~X' ~~'O<.<r C£ ,A ¥~J o 0 ,-I 0

D I () 0 0 6) a ~ lor x..;, I ( !) f a-< oe.::...)('" I.

(it) No{ e~ (f el) ~.x ~ ~ ~ ........ J 6...Y' (J ex J ;. ,J. ~ .M Alo 7:

[~l (~V{~(f)K- ~f&J:= 41(J'-~ j.~# #0]

3(9 A ((j) -(fJ) = /1(1)- A-(J)-- ~ -! -~J 50 fJ)~ ~ A~' CoA- tb .4_I,~.2. S'o .,;:, ( V c.r (£). .

(jiJ ~ -- (j) f- p( (/}) utG-K.

,*,~) ?<the: (: ;//);;-(V --a h~~~~ b..Y

~ ;I;); (;J k~( ~?i'~'~' tY r~e,' ~ x:- ((!.) CtJ/

X / '" ~, (.. ~) (I) I ..J, . ..i:a ~ • 1/. ~ f't..CL V;:L () 0./ /.) tJ;t ..6{ere- ~ /1'1.. .0 ..scc.-l..OA.... ot.

~.dJ (:j; .t (00) .

~: .6 ~/J~ '5~$~~~ 1 <6, ,((j)r(J}t-(I)+ (j);.(f).

(_i J- -1 -1/;),~ (1 -/ -} f /~:::) "'\,; G~ ~ ~ ?!l:~~~k) ~ If -b l. Y ~ , (> (;; / ~ d)~ 0 0 0 0 ~ ~ ~~ .

'r o

,,/ I I I .2-7'1' ' \/ ~ , L..l ~ I I ~'- - f' I f ,;1-,' I 0'

I 1_ \. ::; U f ," A .... /-f? f' I I D I '-. }. .

&--'!- -I _I I J'-\ o?--I r () 0 J .. -~

(~ -,-' vA -')(S~"')/ J '-~ "-1 c\ !).l(3 \ )1' I _,I :l-A/ /t f 0 0 I :; 1\ -I 3 ·- ry 0 '0 , I I ;;:.. o - I 0, _ () -/ 0 I I 0 0 I C, -I -I " -I

/~-y2yl~~) / ,-~ -: / / ' ~1A'-I)4j-,,)/~~ ~I , ,' /:-(A-,}J.(3-~.),. C t ~ _I -I "-I l'/ (..J 0 - j A -.1, / '-

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS J\;IATHEMATICS 2080W

CLASS TEST 3 - TUESDAY, <1 SEPTEMBER 2007

Time: 1 ~ hours :Bull l\1arks: 50

Notes: (i) Only appro1/ed calcula tors rnay be used. (ii) Full answers aTe expected.

f( l) (11 (1\ (1\ ( 0 \) _ 1 2 3151 oil 1. Let A-I ; 3 ' 3)' I 9 I: I 0 'f' l 3 \4} 6 \7) \0 / (i) Is A a linear subspace of ]R4? Explain.

(ii) Find a cartesian equation for the linear subspace X of JR.'1 generated by A .

(iii) Is A a basis for the linear subspace X generated by A? Explain .

(iv) Give a subset B or A that is a basis for the linear subspace X generated by /1-

( v) What is the dimension of X?

(vi) Extend the subset B that you found in (t v) to a basis for ]R4.

2. Let A = ( .• ~ x -;) 0 \

x 3 3) - 1- x 0 .3 :c .

x 3 3

(i) Calculate der.A.

(ii) For which value(s) of x, if any, is detA = O?

(iii) For which value(s) of x, if any, is A an invertible matrix?

(iv) For the case x = 4, find det2A, detA2 and detA -1 .

(v) Find the element in the first row and fourth column of A-1 in t he case 'where x = 4.

[1.3]

[13]

3. For wh ich vaIue(s) of k does the system of linear equations represented by the augmented matrix

(10 k I 1 ) o k -16 3 - k o 0 k2 - 16 k - <1

have (i) no solution, (ii) infinitely many solutions, and (iii) a unique solution? Give reasons for your claims [4]

4. Express 3 -:r + 4x2 as a linear combinn.tion of the fUIlctions 1 + :r, 2 -- :r2 and 3:1: + 2 );2 . [4]

5. You are given that Yl (x) = 2 + x + 2:r3 and Y2(X) = x + 3x In x + 2);3 are both solutions to a linear, third order, non-homogeneous differential equation.

(i) Give three solutions to the associated homogeneous differential eqll;),tion .

Oi) How many soluti ons are there to the non-homogeneous differential (~q uatjon?

(iii) Vlrite down as many solmion~' as you can to the non-hl)illogeneous differenti al equation.

6. Consider the differential equation

ylny+y./e+v+(x+ =:0

(i) Show that (t) is not an exact differential equation.

(ii) Show that ¢(x, y) = tis an integrating factor for (t).

(iii) Find the solution to (t) which sa.tisfies y(l) = 2.

2

[8]

[9]

/1/9-/1 ~o8'o 'V.; - S"'o?-c.iT;'o....s~ Tn C?~:5 ~T/J - 4- ~, ~~

/. cy "yO '- :2 (J)"' (V f. A I 50 A ..,;, Jl.>o. v c..-t'cI>-<.,.( ~~c4/-s=-/.v-..-. <d.f..

( :<; r o-r --L ---t. x,) I ~ JJ ) J S' ~ /L~ S'~ ~. .,t~ /- cr:./> r .,J..<.oI2..14c

~ (i)~/G~') ~ f({) ~/(i) + E(t) >(iZ) ?

r/ i j l ; I:J),...; D' / ,2' ; ()p(;)l~:rl) ·~a/~ / ,;)' 3 ¥- 6. 7- 0 I ;. . J. ~"-lJ - :1:., 0 () ~.l

5-- 3 ¥ 0 ~' Cf ~ j 1../ 0 0 I

N Ii ~( f i ! ~f~;: -~XI )

(~C> 0 0 0 ~:'f -+ ~ -t,.l.2, -1~1 G..:r..e~,q",- 'i K~' <'V-o. f ~ <-e X M ,,1../')' -+~ - 4--0. . -J Jo1 = 0 .

(ity /l....i:, Li~ . Yep ('c-~~ 0 1 .Ae ~ ~-e;:-. ~. ~~ . , .- .1;

~vJ~~:;Z 1(0: (t),y(~{;A~~~=~~:~ ~ .. 70LMA d <.t X - d O<-¥ ,;;., <><.. 6 ~ fr )<..

(iI) o-(,'I(,L X ;: 3 .

(vj ! (I)I({),(f),(Uj - ~ 00&( C::::'1~AA>..£~~ JUO ,-.

~ a.-I..,. r1J ,;2.)' I' .f- ~'J - L,.1~ - J' -r, ::. 0 .

;.(~j!,~ ; }-J fl ;/-/1. i ;~ i/''''jJ ; ~-3 {J= L I 3 J / I·x. I [) 3 :1- I l> 3/

3,(,!) .K ~o : ( .~ oO_/~/; ) / .40.4£0 4 '~;""'" .

(:, 0 -/6/--t/ .,U..q."1.. ;?J ,~""'S (0 6 0 ( -!l ) A-6 .4'U» 4~ ~

(~tj /(:::: *' :' (I (.) it-j')' 'L ' ". ./ / ~~. () 'f -/6 f_/ 1....Ie ~7 ,;"ul'.~ ~l ,4} O '-t..c- . av-P.

o 0 0 0 (j

(ii'J ~ 1"'- <l. ~.Ct.--- .:.r f<.::t 0) /<. * .;f 4:- . ( 3 /' I ' o-eI9-....... .-e, VIA.. J/! .. ---6-) .

(§; .[). t: 6.~ CO>f1.~' .. k:J + () e ). + ( -f ( ; -I- -e~ :/1-=0 ~ ( ~(Jf- -<t ~:t +- t}:: d 1- e:.f :=:. ~L (! + -e x) / .M> .D~ A400.P¥oc~ .

9~ ,"," , .~ ~""'- MO

~ :;:::1) :;:;;;l- -k.,4 :

DEPARTMENT OF MATHEl\!IATICS A ND A P P LIED MATHEMATICS MAM2080\V - CLASS TEST 3 - MONDAY, 25 August, 2008 MAM2084S - CLASS TEST 1 - MONDAY, 25 AUGUST 2008

T ime: 1 ~ hours Full Marks: 50

Notes: (i) Only approved calculators may be used. (ii) Full ans wers are ex pected.

l. 'F'or which values of k does the system of linear equations

(

l kk-l\ ( k \ Ok k ! x = k - l j Ok 1 / \ 0 /

have (i) no solution, (ii) a unique solution, or (iii) infinitely many solutions9

Find all solutions, in terms of k, whenever possible.

2. Let A = { ( )1 ) , ( ~: ) , ( ~ ) , ( ~~ ) }, B = { ( ~ ) , ( ~ ) }

c= { U ). U ). UJ }, and let X be the linear subspace of ]R4 generated by the set A.

(i) Obtain a cartesian equation for X.

(ii) Find a subset D of A that is a basis for X . Give reasons for your claims.

(iii) Extend the set D found in (ii) to a basis for Ri.

(iv) Decide whether or not Band C are bases for X. Give full reasons.

[8]

[14J

3. Which of the following statements are true and which are false? Prove those that are true and provide a counter-example for those that are false.

(i) Every subset of R3 which does not contain the zero vector is linearly independent.

(ii) The sum of two solutions to a homogeneous system of linear equations is again a solution.

(iii) T he set of a.ll solutions to the differential equation y' (t) + y(t) subspa.ce,

et2 forms a linear

[6)

(/ ~

4. Let A= 0 i 1 - 2

~ ~ !) 3 02 . 50 7

o 4

o

o

(i) Evaluate det A. Show your working.

(ii) Use Cramer 's Rule to find the valUe of x·) sat isfy:np; Ax =

(iii) V':rite down th,;; ";J.i .. ws of det (2A ) and det (A2).

(

2 x 1 -2 4 -1

5. Let B = 4 0 x

603

~3 )\ 1 .

9

rH \ ~ )

[11]

Find, by inspection, two vaiues of x for which B- 1 does not exist. Give reasons for your claims.

6. Find the solution to the differential equation x dv

(x 2 + eY + - )~ = 1 - 2x y- In y y dx

which satisfies y(O) = l.

2

[5J

[6]

e'J J.) ~ I j f ,I 6 '/ D () 3 0 J / I .r >0 ~ I I I o~ ::=-

t?!d4

(Ii) 01.«:- (j! A.) ~ ~ r. (3 V

( -1 >j" :; J I 'f! J o'03,.l I I S ';l. . I I I t,.. ; _ 0 ~ '/I... C<..

36

I oI.u-( 4 ~ ) = (y §i)',

sO'. ~ :t;> -it I ft- ~ .,. - Pt- SO c-t..f g = 0

.,f' 1:::. ;) ( d~ C f ;: , ~. C J >-0 o&r- tf.:1 c) <

S~' ... -<- ,8 -( ..t-t:,.--..,If . c:::~~ oed /,~ 0) g -t-lOea .4fc!?-~ If ~ .1 ~ l';. ~ .

b. ~ (.£'-:;f-~JJ-I):::- j + ~L ) it (r.+ef/+~)-d-:t+-d, ,,!) [ ..;;,.., M oc:51 + ~"< ~ ~ /J,l- ,

f(r','!}:: J~"J +,,)1] -I};U":- X k J + 20 -X- -,l 1ft} ~',..<-<- -( :;. AJ ~ .~ ~ t;)::: e '1) $.1:>

1 ) r

5' ~'c-... ,.;, )(...t:...-.J +-). 2-!/ - ~ f- e 1;=- c / C ~ K....-.

PcdI'J- -.£ ':::-0) J'::: / AAK i ~ A ~"'-J(J..,/ /tJ~ ,...."

)( -IZ"'-J +-): (; - y- + -e 11;:: e ·

DEPARTIvIENT OF MATHEMATICS AND APPLIED MATHEMATICS MATHEMATICS 2080"\V

CLASS TEST 4 - MONDAY, 15 OCTOBER 2007

Time: 1 ~ hours Fun Marks: 50

Notes: (i) Only approved ca, jcuia t oTs may be used. (i i) Full answers are expected.

1. Consider the differential equation

I'() 1 I'· :3 () 4 lj' X - - 11 lx) - - 'Ij x = - x > O . • \ 2' '" x 2 " x' (* )

(i) ShoVl that Yl(X) = x 3 and Y2(X) = ~ arc linearly independent solutions of t he homoge­neous differential equation associated with (*) .

(ii) Use the method of variation of paramet.ers to constr\!~t a pair of equations which can be solved to find, a particular solution to (*).

(iii) Solve the system you found in (ii) to obtain a particular solution, and hence write down all the solutions to (*).

[101

2. Use the met hod of undetermined coefficients to find two particular solutions to t.he differential equation

3.

(D2 - 3D)y(x) = 12xe3x .

('. -'h h (1) , ( 2) \. f h . (3 2 ) I) ::" ow t at \ 1 allCI \ _ 7 aIf~ eIgenvectors 0 t e matnx 7 _ 2 .

Oi) Use diagonalization to solve the system of linear first-order differential equations

xi (t) = 3Xl (t) + 2X2(t) + 2

x; (t) = 7Xl(t)- 2X2(t) - 7

[7]

[12]

4. (i) Use the diagonaiization process to identify the conic

7x2 + 2.1/ + 12xy = 22,

by reducing it to a standard form.

(ii) Sketch the conic in (i) with respect to the x- and y-axes. Shov,r also the directions of the new axes that you introduced in (i).

(iii) Decide whether your transformation represents a rotation or a reflection combined with it rotation. Give the equation of the line in which the reflection ta.kes place and also the angle through which the axes are rotated .

(iv) Use your results from (i) and (ii) to sketch the conic

2 ,'2 , . 8 12 7x+ 2y I 12::r;y - Vf3 .r + Vi3 11 = 24.

'1004) ( 2 1 1 3 5. Let A = I 1 2 2 _~ .

\ :3 0 0 -3

(i) Without doing any calculations, "vrite down the value of detA. Give a reason for your answer .

(i i) Civf'n that. ,\ = :-3 and A = -5 are eigenvalues of A, find all the eigenvalues of A.

(iii) Is A a diagonalizable matrix'? Explain.

(iv) Give a basis for the eigenspace E:; .

2

[7]

( ;i~)

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS MAM2080W' - CLASS 'rES,!' 4 - l\rl0NDAY, 06 October, 2008 MAM2084S - CLASS TEST 2 - MOND_AY, 06 October, 2008

Time: 1. 1. hours Full Marks: 51 4

Notes: (i) Only approved calcu.lators 111ay be used. (ii) Full a.nswers are expected.

1. Given that Yl(:r) = xandY2(x) = ~~ a.re solutions of the homogeneous differential equation x

associated \vith x2 y" ( x) + xy' ( x) - y (x ) = x In x ..... ..... (*)

use Variation of ParaIl1eters to obtain a system of equations that can be used to find a particular solution to (*). Hence find the general solution to (*). [9)

2. Use the definition to shov·/ t hat the set of functions {:r, ~} is linearly independent on the x

interval (0, (0). (3]

3. Use the method of undetennined coefficients to find a particular solution to the ditIerential equation

(D2 - 9)y(x) = (24x + 20)e-3x .

[6J

4. Let P be the 3 x 3 matrix which represents a reflection of )R3 in the plane x - 2y + 3z = O. \rVrite do\vn all the eigenvalues and the corresponding eigenvectors of P. [4]

5. The ~3 x 3 ITlatrix A is such that det A :::: O. If A has a repeated eigenvalue and the sum of the diagonal elernents of A are 6; \vhat are t he possible eigenvalues of A? Give full reasons.

6. lTse the diagonalization process to reduce the conic \vith equation

2X2 + 2y2 - 4xy -t- J2x + V2y = 2

to a standard forrn yvith respect to a. new set of axes. Sketch the conic \vith respect to the x - and y-- axes, sho'iving the directions of the nC'w axes .

[3]

[11]

7. Use t he diagonalization process to solve the systen1 of linear first·oorder differential equations

x~(t) = 21;1(t) - X2(t) + 2X3(t) + et .

x~( t) = Xl(t) + 2X3(t) + et

x~(t) = -2Xl (t ) + 2X2(t) - 3x;~(t)

1

[15]

I.

3_

) /Y/l-I'( cJ-O?~ -S CL~ /e'TT~

6 ()c~ /CJ<f &7t., ~ eo ? -_.'-----_. --------------------

-- J../J\..

l (/ ::

5"0 ~(J[}:>- / (!!. .. Y.).2.. a,-p( 1/ it) ~ - x1- A 'L + x2. 7 T .S' ~

.- .J.,t),

/Z.! -1f> -" : Lbv ~ + : _ -: ..R."-A ) c,,·./ .-d.. J..R~ --,o1..J. '"",-

..£;? '1} ex I ; A J( I- _if -t 2- I.e- J/;I-- - 3:c .JZ.cX_, A If G-- R , !J )L Ci-t / ~

k ~· l-4. f:J-_ 0 .L.r ouw 1)L - /

:t & ( -0 oOJ. .)

PJ~~( cL-/-;1=o /?:.Jf .t . .=-:;" : J- t( f- % -= <>

I -=)c/ ~,& -!-o ~~, OAli-Y' 4~~) 5 /

.-d" l' l' ;L,.i. j j? ..L..~? ~~-o-...u...:f-- 0.-..

11 6 F (4 -r- 8.-» x e _ 1 A. , . '

(?>-.J)tfl+ J) [(A.~f ~j.~) e _1j '" e- 3 A.(JJ - (.)2> [' A:>Lf t.l 'j

.: e-J"(J-()[A--j-~/Jf..-]::- e-J~/_6A-I.J-Il.A f-.J-Il)

.4tu~ AA.)'Q ~ - i,2 8 _- ~ lr- ~ ~ -- t; r4 ·t4-.;l .. 5 =...2-0

- J. -s·

6, .

f . !-/~/; ~/ \-.z .2

-~-+---::> .x...

-~-'0/ ~l:: ;~ -/-~ I :(i - }')I:~ _ sO_r/;=--(t-(l(j+-~)

'"' . .j. . ?

..<"d- ) I -.2 - -I 1~· .1 I eJ -/ (0 I d-

UNIVERSITY OF CAPE TOWN

University Examinations-November 2007

DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS

MATHEMATICS 2080W - MAIV12080W - Paper 2

Time: 2~ hours Full Marks: 100 Marks Available: 103

If you would like to access your examination script(s) please see the noticeboard in the Mathematics Building for application details or alternatively go to http://www.rnth.uct.ac.za

This paper con8i8ts of 2 pages. A table of Laplace transforms is attached. Only approved calculator8 rnay be used. Full answers are expected. Marks will be deducted for incomplete solutions.

----------------

(i) Is ~X a linear subspace of R4 in the case where c = I ? Explain.

(ii) In the case where c = 0, find a basis for, and give the dimension of the linear subspace X . Give reasons for your claims.

(iii) Is { ( 1 ) , ( J 1 ) , ( i 1 ) , ( I ) } a basis for ~4 ? Explain.

2. Let A = (~ =~ ~~ ~). ° ° 0 4

(i) Evaluate the cofactor of the element in the third row and second column of A.

(ii) Calculate det A.

(iii ) Is A an invertible matrix? Explain.

[2 , 5, 3]

(iv) Vlithout further calculations, write down the sum of, and the product of the eigenvalues of A. Explain how you arrived at these results.

(v) Find all the eigenvalues of A in the case where x = -5.

(vi) Is A a diagonalizable matrix in the case where x = -5? Why/why not?

(vii) Obtain a basis for each eigenspace of A in the case where x = -5.

[2, 4, 1, 3, 5, 3: 7J

1

3. The surface S has cartesian equation 2X2 + y2 + Z2 + 2yz + V"iy - V2z = 2.

(i) Use the diagonalization process to reduce S to a standard fonn with respect to a new set of axes.

(ii) Identify the surface S, and sketch it with respect to the X -, y- and z-axes. Show also the directions of the new axes.

(iii) Does the orthogonal nlatrix that you introduced in (i) represent a pure rotation or a rotation cOlubined with a reflection? Give reasons to support your clairns .

[9, 4, 2]

4. Which of the following statements are true and which are false? Prove those that are true, otherwise provide a counterexample.

(i) T he sum of two n x n orthogonal matrices is an ort hogonal lllatrix.

(ii) Eigenvectors corresponding to distinct eigenvalues of a square rnatrix are linearly inde­pendent.

[6]

5. Consider the differential equation

y"(t) + 2y' (t) + y(t) = (2 -- t) e- t (t)

(i) \iVrite down the general solution to the horIlogeneous differential equation associated with

(t) . (ii) Use the method of undetermined coefficients to find a particular solution to (t)·

(iii ) Use variation of paralueters to find a particular solution to (t) . Show your working.

(iv) Use the Laplace transform to find the solution to (t) which satisfies the initial conditions y(O) = 1, y'(O) = O.

[2, 6, 8, 5]

6. Find the general real solution to each of the following:

'7 , .

(i) (D 3 + 8)y(t) = 0,

(ii) (D4 - D2)y(t) = 0,

Y dy (iii) v2 + 2 + (1 + 2xy + arctan x) ·d~ = O.

v l+x . x

{

82 - 8 -" 8 } (i) Obtain £-1 (82 + 23 + 5)(3 _ 1) .

(ii) Find £ {l'(t-u)3COS2udu}.

{I if 0 < t < 2

(iii) Express f(t) = t if t ?::: 2

£ { f(t)}.

[3, 3, 5]

in tenns of the unit-step function, and hence find

[8, 3, 4]

2

UNIVERSITY OF CAPE TOWN

University Examinations- October 2008

DEPARTMENT OF MATHEMArrICS AND APPLIED MATHEMATICS

MATHEMATICS 2080W - lVIJ\M2080W - Paper 2

MATHEMATICS 2084S - MAM2084S - Paper 1

Time: 2~ hours Full Marks: 100 Marks Available: 103

If you would like to access your examination script(s) please see the noticeboard in the Mathematics Building for application details or alternatively go to http://www.mth.uct.ac.za

This paper consists of 3 pages.A table of Laplace tTansjorms is attached. Only approved calculators may be used. Full answers are expected. !vI arks will be deducted fOT incornplete soZ,utions.

------------------------

(

a + 1 b

1. Let A = ~ b : 1

a b

(i) Evaluate det A.

c c

c+l c

~ ) , a, b, c, d E JR.

d+4

(ii) Use your result from (i) to decide if A is invertible when a = b = c = 1 and d = -16. Explain.

(iii) Use Cranler's Rule to find the value of X2 if A (

~~'4231 ) ( 3;2 )

and a = b = c = 1 and d = O.

(i) Find a cartesian equation(s) for the linear subspace A of JR4 generated by X.

(ii) Write down a basis for, and give the dimension of A. Justify your claims.

(iii) Extend the basis you find in (ii) to a basis for ]R4 .

1

[5,2,3]

[5,3,2]

3. Find bases for the eigenspaces of the matrix

A=(l 2 -1

1) 2 0 3 2 0 0

and decide whether A is diagonalizable. Explain . [8]

4. Let A = 2 2 0 . (322) 204

(i) Show that ( ! ) is an eigenvector of A. What is the corresponding eigenvalue?

(ii) Use the fact that det A = 0 to find all the eigenvalues of A.

(iii) Use the above results to reduce the surface with equation

3x2 + 2y2 + 4Z2 + 4x y + 4x Z + 2y - Z = 6

to a standard form with respect to a new set of axes. Sketch this surface with respect to

the X- , y- and z- axes , and show the directions of the new axes. [2 ,3,9]

5. Show that the transforrnation represented by the rnatrix ~ (i -;2 =~) 3 2 1 2

represents a rotation. Find also the axis of rotation, and the angle through which this rotation takes place. [7]

6. Which of the following statements are true and which are false? Prove those that are true, otherwise give a counter-example.

(i) The set X = {~ E ]R3 : .1:1 + X3 = :c~} is a linear subspace of ]R3.

(ii) If ;r and yare eigenvectors corresponding to the distinct eigenvalues ,,\ and J.L of a matrix A, then {;r,1{} is a linearly independent set .

(iii) The eigenvalues of a real syrnmetric rnatrix are real.

(i v) Let ~v be the Wronskian of the functions Y1 (x) and Y2 (x) . If VV f- 0 for some Xo E I, then {Yl(X) , Y2(X)} is linearly independent on the interval 1.

[12]

2

7. Find the general solution to each of the following differential equations:

(i) 6xy + sin y + (1 + 3x2 + X cos y)y'{x) = 0,

(ii) (D2 - 4D)y(x) = 2 + 48x2,

(iii) (D - 3)2y(x ) = 10e3x ,

(iv) x 3y"(x) - 2x y(x) = 1, given that Yl(X ) = x2 and Y2(X) = -~ are both solutions of the x

associated homogeneous differential equation.

8. (i) Use the Laplace transform to solve the initial-value probleIl1

y"(t) +- 2y'(t) + y(t) = e-t , y(O) = -3, y'(O) = 2.

.. {O if 0 < t < 2 (11) Express j(t) = t if t ~ 2

in terrrlS of the unit-step function, and hence find .c{f (t)}.

(iii) Find £{f; u~~ cos(t - u)du}.

(iv) Find £-1 {' ___ s _}. 3 2 + 48 + 20

(v) If y(t) = £-1 {e:;}, sketch the graph of y(t) and find y(5). )

3

[5,7,5,7]

l'v1Al'v12080W - MATHEMATICS 2080W

Table of Laplace Transforms

~_f_(t_) __ --+-

t n

cos at

sin at

t cos at

t sin at

I I I u(t - a)f(t - a)

Lf(n)~

£. {f(t)} = F(s)

1

s-a

s

8 2 - a2

(82 +- a2)2

2as

F(s - a)

e-as F(s) I

sn F(s) - 8 n - 1 f(O) - 8,,-2 f'(O) - . .. - 8f(n-2)(0) - f(n-I)(oJ

L

Department of Mathematics and Applied M athematics

MAMI03W and ENDI07W

Exercise Sheet on First Order Linear Differential Equations

Solve

(i) dy -x - +- 2y = e dx

(ii) 2 dy x/2 ---- -y= e dx

(iii) dy sin x

y(l) = 0 x---- +- 3y = --dx x 2 '

(iv) dy .

x-- + Y = Slnx dx

(v) dy

y(5) = 2 (y-x)d-=y, :£

( vi) ? dy (r~ -(:.c --- l)L d +- 4 :£ - 1 y = x+- 1 ,x

(Answer: y = e- X +- Ce-2x ).

(Answer: y = ~eX/2 +- CeX/2).

(A cos x cos 1.

nswer: y = --3- +- -3-) ' x x

c cosx (Answer: y = - - --).

x x

y 8 (Answer: x = - +- -).

2 y

lx3 - X C (Answer: y - 3 + ) . - (x - . 1) 4 (x - 1) 4 .

(vii) x ~.~ - if = x2 sec x t.an x, J; > 0, y( IT /3) = 2 (Answer: y = x2 sec x + (~~ - 2):c2) ,

2. (a) A large tank is filled with 500 litres of pure water. Brine containing 2 kg of salt per litre is pumped into the tank at the rate of 5 l/min. The well-stirred solution is pUlllped out at the sanle rate. Find the nun1ber of kilograms of salt y(t) in the tank any time t. Hence deduce the amount of salt (in kg) in the tank after 25 minutes (Answer: y(t) = 1000 -- lOOOe- t / 100 kg and y(25) ~ 221.20 kg)

(b) A large tank is partially filled with 100 litres of fluid in which 10 kg of salt is dissolved. Brine containing 1/2 kg of salt per litre is pumped into the tank at the rate of 6 l/min. The well-stirred solution is pumped out at the same rate. Find the nU111ber of kilograllls of salt y(t) in the tank after 30 minutes. (Answer: About 64.38 kg)

1

Department of Mathematics and Applied Mathematics

MAMI03W and ENDI07W

Exercise Sheet on Second Order Linear Differential Equations

Find the general solution of each of the following differential equations. If initial conditions are given, find the solution that satisfies the stated conditions.

I.

2.

3.

4.

5.

y" -- y' - 2y = 0

y" + 6y' + 25y = 0

y" - 2y = 0

y" - 2y' + y = 0

y" + 2y' + 2y = 0

(Answer: y(x) = CI e-x + c2e2x ).

(Ans\ver: y(x) = e-3x (cI cos( 4x) + C2 sin( 4x)]).

(Answer: y(x) = CI e-xvi2 + c2exv'2).

(Answer: y(x) = (CI + xC2)e X).

(Answer: y(x) = e-X(cI cosx + c2sinx]).

6. d2y dy

4 dx2 + 4 dx + y = 0, y(O) = 1, y'(O) = 1 ~1

(Answer: y(x) = h + -x1J e- x

/2

). L 2

7. 4y" + y = 0

8. y" - y' - 12y = 0, y(O) = ;3, y'(O) = 5

d2y dy 9. -d 2 + 2 -d + 5y = 0) y (0) = 2, y' (0) = 6

x ·x

10. y" + 4y' + 5y = 0, y(O) :::: 1, y'(O) = 0

II. y" + 2y = 0

12. (D2 + 5.D + 6)y(x) = 0

13. d2y dy - - 4- + 4y(t) = 0 dt2 dt .

14. (D2 + 2D + 4)y(x) = 0

15. d2 y dy

15- + 13-- - 44y(t) = 0 dt2 dt <

16. (D2 + 16)y(t) = 0

OX) . (X) (Answer: y(x) = Cl cos C2 + C2 SIn '2 ,).

(Ans\ver: y(x) = e-3x + 2e4X).

(Answer: y(x) = e-x [2 cos(2x) + 4 sin( 4x) J).

(Answer: y(:;;) = e-2x [cos x + 2 sin x] ). (Answer: y(x ) = Cl cos(.~\/2) + C2 sin (:r}2) ) 0

(Answer: y(x) = Cl e-·2x + C2e-3x).

(Answer: y(x) = e- .3x [CI cos( v'3 x) + C2 sin(}3 x)]).

(Answer: y(t) = Cl cos(4t) + C2 sin(4t)).

17. y"(x) -- 6y'(x) + 10y(x) = 0, y(O) = 0, y'(O) =--:: 4 (Answer: y(x) = 4e3;r; sinx).

18. (D - 1)2y(t) = 0, y(O) = 2, y'(O) = 5 (Answer: y(t) = et (2 + 3t)).

d2 y dy r.-- _, r;:; ;;:; .;;:; \ 19. dx

2 +6 dx + 12y = 0, y(O) = 2y3, y'(O):::: 0 (Ans.: y(x) = e 3X[2 \/3 cos( y3x) +6s1n(y3x)]).

1