8/20/2019 Test1 Year 2010

1/12

Code: L725

Department of Mathematics

LJniversity

of Toronto

MAT 135Y

Term-Test

ffL

Thursday, November 11, 2010

Time allowed: 90 minutes

PleasePRINT

in

INK or

BALL-POINT PEN:

(Please

PRINT full name

and UNDERLINE

surname):

NAME OF STUDENT:

STUDENT NO.:

SIGNATURE OF STUDENT

(in

INK or BALL-POINT

PEN):

TUTORIAL

CODE

(..e.,

M4A, R5D, etc.):

TUTORIAL TIME

(".S.,

T4,

R5,

F3, etc.):

NAME

OF

YOUR TA:

NOTE:

1. Before

you

start, check

hat this test

has 12

pages.

There are NO blank

pages.

2.

This test has two

parts:

PART

A

[50

marks]:

10

multiple

choice

questions

PART B

[50

marks]:

7 written

questions

Answers to both PART A and PART B are

to be

given

in this booklet.

No

computer

cards

will

be used.

3.

No

aids allowed.

No calculators

DO NOT TEAR

OUT

ANY

PAGES.

FOR MARKERS

ONLY

QUESTION

MARK

PAFUT

A

150

B1

l7

B2

lB

B3

l8

B4

/ c

B5

l. ?

/ t

B6

l7

B7

l8

TOTAL

ln0

Page1

of

12

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Code: L725

PART A

[50

marks]

Please

ead carefullv:

PART A consists

of

10

multiple-choice

questions,

each

of

which

has exactlv

one correct arswer.

Indicate

your

arrswer o

each

question

by completely

filling in

the appropriate

circle with

a

dark

pencil.

MARKING

SCHEME:

5

ma,rks for

a correct answer,

0

for

no answer or a

wrong answer. You

ale not

required to justify your arurr/ers n PARjT A. Note that for PART A, only your final answers (as

indicated by the circles

you

darken)

countl

your

computations

and

answers indicated

elsewhere will

NOT count.

DO NOT TEAR

OUT

ANY PAGES.

1. Find

the value

of l mrz

-

3r

-

4

 

n -+4 f r2 -2 r -8

@

O

@

o

@

2. Findthevalueof

m

ffi

1

t

o

6

3

4

_t

3

undefined

@

@

e

o

@

 t

I

5

a

D

_1

2

_3

2

undefined

Page2

of72

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3.

Find

the value

@o

@

undefined

@*

o*

oi

Code: L725

of lim

(sin

3r)

(cos

2r)

(csc

6r)

;;o

(sin

4r)(cos

5r)(csc *)

curve

A

-

2r3

-

4r

*

5

at

the

point

(1,

3) will

intersect

.

The

line

tangent

to the

the

y-axis

at

the

point

@

@

@

o

@

(0,0)

(0,

3)

(0,

3)

(0,

1)

(0,

2)

Page

3 of12

8/20/2019 Test1 Year 2010

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5.

Letf(*)- [crz-2r

1f

8/20/2019 Test1 Year 2010

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Code: L725

7

If

f

(r)

-

(ln

)r"

",

then

'

(r)

-

-e

tre

,

e'

1

e

?

e

B. It

f

(r)

-

cos(2rr),

find

f(at)(#),

i.e., ind

the

value

of the

61"t

derivative

of

f

atr-h.

O

-26ontr

@

-26on6o

@

_267n6r

O

261n6t

@

26onGL

@

@

@

e

@

Page

5

of. 2

8/20/2019 Test1 Year 2010

6/12

Code:

L725

9.

Let

(*)

:

tre*2

nd

9

:

f

(f(")).

Then

g'(l

-

(2" '+

1)"" '

2(2e2

1)""'

3(2e2 I)e"2

2(2e2

1)

"L+e2

3(2e2 1)sL+"2

@

@

@

@

e

Page

6 of 12

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Code:

L725

10.

Find

the

value

of

lrg

cos2

-

cosz(2r)

.

,/,

1

r

t/Z

3

1

,

undefined

e

o

@

Page

7

of 12

8/20/2019 Test1 Year 2010

8/12

[7]

Code:

L725

PARI B

[50

marks]

Please ead

carefullv:

Present

your

complete solutions to

the

following

questions

in

the spaces

provided,

in

a neat and

logical fashion,

showing all

your

computations

a.nd

ustifications.

Any answer in PART

B without

proper

ustification

may receive ittle

or

no

credit.

Use

he back

of each

page

or

rough work only. If

you

must continue

your

formal

solution on the back

of a

page,

you

should indicate

clearly, in LARGE

Ietters, SOLUTIONCONTINUEDON THE BACK OF PAGE

In this case,you may get

credit for

what

you

write on the

back of that

page,

but

you

may also

be

penalized

for mistakes

on

the back of that

page.

MARKS

FOR EACH

QUESTION

AR^E NDICATED

BY

II.

DO NOT TEAR OUT ANY

PAGES.

1. Let

f

(a)

:

x2+3. Find

/'(c)

from

first

principles

i.e.

by using

onlv the definition

of the

derivative)

Page

8

of 12

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Codez 7725

2.

Use

arry

suitable method to frnd

ftfor

eachof the following.

There is no

need o simplify

your

fi.nalanswers

or this

question.

arctan

\ a )

Y :

2+es " -

t/,1

( b ) y

- ( r '

+8 ) t o ( " -5 ) t .

t/rl

3.

For

this

question,

simplify

your

final

answers

as much

as

possible.

(a)

Find

f '( i l

i t

f

(r)

-

ln(sin'

).

Rr-,

2J3

-

r t r

- ' -

t 4 l

L

' J

(b)

Find

'(4)

it

f

(r)

:

2fr.

t+rn-

,

LL

:

t/,1

Page

of12

8/20/2019 Test1 Year 2010

10/12

[5]

Code: L725

4. Find

\il:n

J

r'z

-

2r

+

t

-

r) .

5.

Find

the horizontal

asymptotes

(if

any) of each

of the following

functions.

Remember

to

justify

yor[

aflswers.

3lu

{al t lt l

:

-----r----:-.

2x -8

t4l

t3l

(u)

(")

-

2e-"

I.

,\,

l.IVt-)

_

|

d

Page

10

of 12

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Code:

L725

6.

Find the

tangent

ine

to the curve 2A*sin(2g)

-

r-

2

at the

point

(2,0).

[7]

(\,*.

Z=

(x

-L)

Page11

of 12

8/20/2019 Test1 Year 2010

12/12

7

' l { -

I

-

tsl

Code:

1225

,

7.

(NS g:

This is not zrn

easy

question.

Very

little or

no credit

will be

given

unless

your

solution is completelv

correct.)

A circle s

centered t the

poi:rt

(6,0)

andhas adius

5. Find

the slope

fthe

line

passing

hrough

the

point

1-f,

O1 na a.ngento the upper

half

of the

circle.

o

Page

12 of72