7/28/2019 Test 2012-02

1/16

Code:41

DEPARTMENT OF MATHEMATICSUniversity of Toronto

MAT136H1STerm Test

Winter 2012Time allowed: 1 hour, 45 minutes

NAME OF STUDENT:(Please PRINT full nameand UNDERLINE surname):

STUDENT NUMBER:

SIGNATURE OF STUDENT:

TUTORIAL CODE(e.g.,M4A, R5D, etc.):

TUTORIAL TIME(e.g.,T4,R5,F3, etc.):

NAME OF YOUR TA:

NOTE:

1. Before you start, check that

this test has 14 pages.There are NO blank pages.

2. This test has two parts:PART A [48 marks]: 12 multiple choice questions.PART B [52 marks]: 7 written questions.Answers to both PART A and PART B are tobe given in this booklet.No computer cards will be used.

3. No aids allowed.NO CALCULATORS!

4. DO NOT TEAR OUT ANY PAGES

FOR MARKERS ONLYPART A /48

B1 /7

B2 /7

B3 /7

B4 /7

B5 /8

B6 /8

B7 /8

TOTAL /100

Page 1 of 14

7/28/2019 Test 2012-02

2/16

Code:41

PART A [48 marks]

Please read carefully:PART A consists of 12 multiple-choice questions, each of which has exactly one correct answer. Indica

your answer to each question by completely filling in the appropriate circle with a dark pencil.MARKING SCHEME: 4 marks for a correct answer, 0 for no answer or a wrong answer. You are not requirto justify your answers in PART A. Note that for PART A, only your final answers (as indicated bthe circles you darken) count; your computations and answers indicated elsewhere will NOcount.DO NOT TEAR OUT ANY PAGES

1. If f(x) = 8x3 4x + 5 and f(1) = 6, then f(1) =.A

O

6

BO 0C

O1

DO

4E

O 5

2. If 32 {5f(x) + 4g(x)}dx = 5 and

32 g(x)dx = 5, then

32 f(x)dx =.

AO

6B

O 0C

O5

DO 3E

O8

Page 2 of 14

7/28/2019 Test 2012-02

3/16

01234565

789!"#!$%&'!"#7

$()*))'+'+&)&$)

&$()8

,-.

/.

7.

.

.

-8

:

;?!

@

)-

,

:

/

:

A

)

A)-

:

A

:

7/28/2019 Test 2012-02

4/16

Code:41

5. Find the area of the region enclosed between the graphs ofy = 3x and y = 3x26A

O

272

BO

24

5C

O

252

DO

263

EO

253

6. Let R be the region bounded by the curves y = 4x and y = 2x2. Find the voluof the solid generated by revolving R about the y-axis.

AO

143

BO

133

CO

163

DO

173

EO

132

Page 4 of 14

7/28/2019 Test 2012-02

5/16

Code:41

7.20|x |dx =

AO

22 2B

O

22

CO

2 + 1D

O 2

EO 22

8. Water flows from the bottom of a storage tank at a rate r(t) = (200 4t) literminute, where 0 t 50. Find the amount of water that flows from the tankduring the first 10 minutes.

AO

1,650 litersB

O1,700 liters

CO

1,750 litersD

O1,600 liters

EO 1,800 liters

Page 5 of 14

7/28/2019 Test 2012-02

6/16

Code:41

9. Find the value of

limn

(1

n + 2n2+

2

2n + 2n2+

3

3n + 2n2+ . . . +

n

n2 + 2n2).

AO

1 2 ln 3 + 2 ln 2B

O 1 + 3 ln 3 3 l n 2C

O 1 + 2 ln 3 2 l n 2D

O 1 + 2 ln 3 ln 2E

O1 2 ln 3 + ln 2

Page 6 of 14

7/28/2019 Test 2012-02

7/16

Code:41

10. Let f be a continuous function on [3, 6] such thatthe average value off(x) on [3,1] is 5,the average value off(x) on [

1, 2] is 4,

the average value off(x) on [2, 6] is 2.Let g(x) = 2 + f(x). Find the average value of g(x) on [3, 6].

AO

174

BO

173

CO

133

DO

132

EO

163

Page 7 of 14

7/28/2019 Test 2012-02

8/16

Code:41

11. Evaluate87

x

x6dx.

Hint: the formula

dxx2

a2

= ln

|x +

x2

a2

|+ C may be useful.

AO

4 7 + 6 ln 3 3 ln(4 + 7)B

O8 7 + 6 ln 3 3 ln(4 + 7)

CO 8 7 6 l n 3 3 ln(4 + 7)D

O 4 +

7 6 l n 3 3 ln(4 + 7)E

O 8 +

7 6 l n 3 3 ln(4 + 7)

Page 8 of 14

7/28/2019 Test 2012-02

9/16

Code:41

12.21 ln(x +

x2 1)dx =

AO

2 ln(1 +

2) 1B

O 2 ln(1 + 2) 1C

O

2 ln(1 +

2) 2

DO ln(1 +

2) 1

EO

2 ln(1 +

2)

Page 9 of 14

7/28/2019 Test 2012-02

10/16

Code:41

PART B [52 marks]

Please read carefully:Present your complete solutions to the following questions in the spaces provided, in a neat and logical fas

ion, showing all your computations and justifications. Any answer in PART B without proper justificatimay receive very little or no credit. Use the back of each page for rough work only. If you must continyour formal solution on the back of a page, you should indicate clearly, in LARGE letters, SOLUTIOCONTINUED ON THE BACK OF PAGE . In this case, you may get credit for what you write the back of that page, but you may also be penalized for mistakes on the back of that page.MARKS FOR EACH QUESTION ARE INDICATED BY [ ].DO NOT TEAR OUT ANY PAGES.

1. Findxe5xdx.

[7]

2. Find

(2x3 + cosx)(x4 + 2 sin x)45dx

Note: This is a very easy problem!

[7]

Page 10 of 14

7/28/2019 Test 2012-02

11/16

Code:41

3. Find

tan74 x sec4 xdx.

[7]

4. Find

dx

(25x2)3/2 .

[7]

Page 11 of 14

7/28/2019 Test 2012-02

12/16

Code:41

5. Find

5x2+3x+64x(x2+16)

dx.

[8]

Page 12 of 14

7/28/2019 Test 2012-02

13/16

Code:41

6. Find

dxx+x

x

.

[8]

Page 13 of 14

7/28/2019 Test 2012-02

14/16

Code:41

7. Note: This is not an easy question and will be marked very strictly. Very littleno credit will be given unless your solution is completely correct.

Given that10 (arcsinx)6dx = k, find the value of

10 (arcsinx)8dx in terms ofk

Simplify your final answer as much as possible.

[8]

Page 14 of 14

7/28/2019 Test 2012-02

15/16

HAT 136FART A8. lSoo +.JF s.2 6.9 ?.?TLlL. JT I LttJZ) - I (A.rr cLef'o;lL , - X5zq e5^

-t- 25. tcgz.. f"t ,r = rcf t2-,a.i,*'1 . T-gs-^.a*{t:c3+.2-.-",).lr={2131 c.',x)dr

l't/I^ * 5( 2 *3 + "fr:O( *++2a;^-)."d r. = !t ,^t"d,t^ =_(r,f+2--a..i^>c)'- tC. .g2-t t6 , Lez(+ 6) \ \'--

B t. J *',{e**?^ et, --) (*o,J*r. ao.a>.) -o-o.trc dzc = Sco g c(o. - z'olao,Czs- ^")tu: (.5 - zs,aqr%)t/^ ={" s( t-'*rt"rr-'=frs *t"19r- -s3r,'t 2e .u.'-i-- 5 ft;ry -- J f*#fo- : *'J aa-t et'e.(gr.L+3x*6F) ?( (*'+t6)

86. /.Xq=,JT. T4-*'.,\'l.{c-r^.-.- f {x :-f) fid'hrr^JT l

: 2( a-rdr 2 r.^ o{ ,.,,* = oLZ

7/28/2019 Test 2012-02

16/16

7.1*- (o'*i'.,,-x 8lzc7 w-r,-\tI rl-Jxo? \tr: 8(o,-r-a;+rt ,:e\xJt-)co --, d-tr87. l,L Arr

)*:q,L=(3.-)*-slg#u*=({)t-8t"+ti T (o^..j^ ^)6 dr,) =({)t < 56.&

6

- y*) t* = 2o o * *2_-t'1f'=fgo o-+-=9tlt#' - .{^ in* Ln- n-?co t1 ?, it2"rn n+"o ^,h, h*,r_) i= |Alt ^E(#a,.=Jiffi:l:#65 (*t3)dw, A ntt-- - ^\e tr !a^(x lr;o:,,)g ="r*t:F r,e- ).;* t *fJG fu 94lr \.r tJ-

At