MSTCALCULUS

Workbook

Paul Yiu

Department of Mathematics

Florida Atlantic University

Last Update: November 17, 2009.

Student:

Fall 2009

1

Name:

Problem A1. Explain how you would choose an initial value andfind a rational approximationr of

√5 satisfying|r2 − 5| < 1

1030 in a fewiterations. How many iterations do you need? There is no needto findthe rational approximately explicitly.1

1I modify the problem slightly so that you do not have to compute the rational approximation explicitly.Note that there was a typo in the original version: Find a rational approximationr of

√5 satisfying|r2−5| <

1

1030. Justify your answer.

2

Name:

Problem A2. Start with two positive numbersa0 anda1, and iterateaccording to the rule

an =√

an−1 +√

an−2.

Does the sequence(an) converge? If so, what is the limit? (No proof isrequired).

3

Name:

Problem A3. Does the sequence(

n2

2n

)

converge? If so, what is the

limit? 2

2Study the proof of Example 2.5(b) in Supplement 2.

4

Name:

Problem A4. (a) Prove that12· 3

4· · · 2n−1

2n< 1√

2n+1.

(b) Find limn→∞12· 3

4· · · 2n−1

2n.

5

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Problem A4. (a) Prove that12· 3

4· · · 2n−1

2n< 1√

2n+1.

(b) Find limn→∞12· 3

4· · · 2n−1

2n.

Solution. (a)(

1

2· 3

4· · · 2n − 1

2n

)2

=1

2· 1

2· 3

4· 3

4· · · 2n − 1

2n· 2n − 1

2n

<1

2· 2

3· 3

4· 4

5· · · 2n − 1

2n· 2n

2n + 1

=1

2n + 1.

The result follows by taking square roots.(b) Since

0 <1

2· 3

4· · · 2n − 1

2n<

1√2n + 1

andlimn→∞1√

2n+1= 0, it follows that

limn→∞12· 3

4· · · 2n−1

2n= 0.

Here is an alternative (inductive) proof of (a).Clearly, 1

2< 1√

3.

Assume12· 3

4· · · 2n−1

2n< 1√

2n+1. Then

1

2· 3

4· · · 2n − 1

2n· 2n + 1

2n + 2

<1√

2n + 1· 2n + 1

2n + 2=

√2n + 1

2n + 2=

√2n + 1

√

(2n + 2)2

<

√2n + 1

√

(2n + 1)(2n + 3)

=1√

2n + 3.

Therefore, by induction, (a) is true for every positive integern.

6

Name:

Problem A5. (a) Leta andb be given positive numbers. Why doesthe expression

√

a +

√

b +

√

a +√

b + · · ·

(in whicha andb alternate indefinitely) define a real number?

(b) Identify the numbers√

1 +

√

7 +

√

1 +√

7 + · · ·

and√

7 +

√

1 +

√

7 +√

1 + · · ·.

7

Name:

Problem A6. Show that regardless of initial value, the sequence(xn) defined recursively by

xn+1 = 1 − 1

xn

does not converge.

8

Name:

Problem A7. Let λ ∈ (0, 1) be a fixed number. Givena, b, define asequence(xn) recursively by

x1 = a, x2 = b, xn = λxn−1 + (1 − λ)xn−2.

Show that the sequence(xn) converges and find its limit.

9

Name:

Problem A7. Calculate without any devices or computer programsthe square root of3 correct up to6 decimal places.

10

Name:

Problem B1. A circle is approximated by a regular octagon ob-tained by cutting out corners from its circumscribed square. What is theapproximate value ofπ?

11

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Problem B2. Let an andbn respectively denote the perimeters of acircumscribing regular6 · 2n-gon and an inscribed regular polygons of acircle of unit diameter. Prove that

an+1 =2anbn

an + bn

, bn+1 =√

an+1bn,

witha0 = 2

√3, b0 = 3.

12

Name:

Problem B3. Evaluate the integral3

∫ 1

0

x4(1 − x)4

1 + x2dx.

3You will love the answer, which is quite easy to find if you makeuse of

x4(1 − x)4

1 + x2= x

6 − 4x5 + 5x

4 − 4x2 + 4 − 4

1 + x2.

13

Name:

Problem C1. (a) Given the precise values ofsin 30◦ = 12, cos 30◦ =√

32

, sin 45◦ = cos 45◦ =√

22

, find the sines and cosines of15◦ and75◦.

(b) Prove thattan(A + B) = tan A+tan B

1−tan A tan B.

14

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Problem C2. Find an angleα such thata sin x+b cos x = c sin(x+α) for an appropriatec, and make use of it to find the integral

∫

dx

a sin x + b cos x.

15

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Problem C3. Make use of

sin x =2t

1 + t2, cos x =

1 − t2

1 + t2, dx =

2dt

1 + t2

4 wheret = tan x

2to evaluate the integral

∫

dx

a sin x + b cos x.

4My apology for having omitted earlier the factor2 in the numerator ofdx.

16

Name:

Problem C4. Maxima and minima without calculus:Find the area of the largest rectangle that can be inscribed in a semicircleof radiusa.

17

Name:

Problem C5. Maxima and minima without calculus:The sideXY of the right triangleOXY is tangent to the circle of radiusr. What is the minimum possible of the triangle?

Y

XO

r

r

18

Name:

Problem C6. Find the maximum volume of a right circular coneinscribed in a sphere of radiusr.

19

Name:

Problem C7. Show that(a−x)(x−b)x2 has maximum(a−b)2

4abwhenx =

2ab

a+b.

20

Name:

Problem D1. (a) Leta1, a2, . . . ,an be given positive numbers. Findthe value ofx which makes

f(x) :=1

n+1(a1 + a2 + · · · + an + x)

n+1√

a1a2 · · ·anx

minimum.

(b) Make use of (a) to give a direct, inductive proof of AGI.

21

Name:

Problem D2.What is the common limit of the sequences(xn) and(yn) defined by

xn+1 =2xnyn

xn + yn

, yn+1 =xn + yn

2,

which initial valuesx1 = a andy1 = b. Justify your answer.

22

Name:

Problem D3.Let an = 1

n+1+ 1

n+2+ · · · + 1

2n. Is the sequence(an) convergent or

divergent? Justify your answer.

23

Name:

Problem D4. Evaluate the following limits:

1. limn→∞n

√2n =

2. limn→∞n

√(2n)!

n=

24

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Problem D5. Evaluate the following limits.

1. limn→∞(√

n2 + 1 − n) =

2. limn→∞(√

n2 + n − n) =

25

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Problem D6. Show thatlimn→∞n

√n + 1 = 1.

26

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Problem E1. The circle through the three pointst1, t2, t3 on theparabolay2 = 4ax intersects it at a fourth point. What is this point?

27

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Problem E2. Use integral calculus to find the area bounded by theparabola and the line joining the pointst1 andt2 on the parabola.

O

t2

t1

y2

= 4ax

28

Name:

Problem E3. (a) Find the equation of the normal to the parabolay2 = 4ax at the point(at2, 2at).

(b) A circle with center(p, q) passes through the origin and meets theparabola again at three different points with parameterst1, t2, t3. Showthat

at3i = 2pti − 4ati + 4q, i = 1, 2, 3.

(c) Hence show that the normals at these three points are concurrent.

29

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Problem E4. Find the minimum normal chords of the parabolay2 = 4ax.

30

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Problem E5. Calculate the length of the semicubical parabolaQ

between the pointsQ(0) andQ(s).

31

Name:

Problem F1. (a) Show that the trisectrix has polar equationr =a cos t − a

4 cos t.

(b) Calculate the area enclosed by the loop of the trisectrix.

32

Name:

Problem F2. Consider the unit circlex2 + y2 = 1. Show that Theenvelope of the line joining(cos t, sin t) to (cos 2t, sin 2t) is a cardioid.

33

Name:

Problem F3. If b > a, the conchoid

(x, y) = (a tan θ + b sin θ, b cos θ)

with length−b has a loop. What is the area enclosed by the loop?

A

34

Name:

Problem F4. (a) Locate the points of inflection of the witch ofAgnesi.

a

(b) Calculate the area under the witch of Agnesi (and above the x-axis).

35

Name:

Problem F5. Consider the circle through the origin and with center(

at, a

t

)

on the rectangular hyperbolaxy = a2.(i) Find the equation of the circle.

(ii) Find the envelope of the circle as a parametrized curve.

(iii) Find the Cartesian equation of the envelope.

36

Name:

Problem F6. Calculate the area enclosed by the polar curveρ2 =b2 − a2 sin2 θ, a < b.

37

Name:

Problem G1. Calculate the length of the arc from(0, 0) to P (t) =(at2, 2at) of the parabolay2 = 4ax.

38

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Problem G2. Calculate the perimeter of the astroid

x2

3 + y2

3 = a2

3 .

39

Name:

Problem G3.

1. Find the area enclosed by the cardioid.

A

2. The cardioid is revolved about the horizontal axis. Find thevolumegenerated.

40

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Problem G4. If a > b, calculate the area of the smaller loop of thelimaconρ = a cos θ + b.

A

41

Name:

Problem H1. (a) Use Newton’s binomial theorem to find the ex-pansion of 1√

1−x2to the term ofx10.

(b) Make use of (a) to find the series expansion ofarcsin x.

42

Name:

Problem H2. Suppose we construct a sequence of rectangles asfollows. We begin with a square of area one. We then alternateadjoininga rectangle of area one alongside or on top of the previous rectangle.Find the limiting ratio of length to height.

43

Name:

Problem H3. (a) Define a function on(1,∞) by

f(x) =x

log x.

Show thatf(x) has a local minimum atx = e. Explain whyf(x) ≥ e

for all x ∈ (1,∞).(b) Let b > 1. Show that the function

g(x) =xb

bx

defined forx > 1 is increasing on(1, b

log b) and decreasing on( b

log b,∞).

(c) Using (a) and (b), deduce that if1 < a < b < e, thenab < ba.

44

Name:

Problem H4. Show that

g(z) :=z

ez − 1+

z

2

is an even5 function. Make use of this to show that the Bernoulli numberBn = 0 for an odd integern > 1.

5Erroneously written “odd” before; my apology.