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NAME (PRINT):Last/Surname First /Given Name

STUDENT #: SIGNATURE:

UNIVERSITY OF TORONTO MISSISSAUGA

APRIL 2010 FINAL EXAMINATION-Version A

MAT133Y5YCalculus and Linear Algebra for Commerce

Any Wilk, Shay Fuchs, Ken GiulianiDuration - 3 hours

Aids: Formula sheet provided (last pages of this paper)Calculator Model(s): Only TI-30X, TI-30XIIB or TI-30XIIS(with no lids or inserts)

The University of Toronto Mississauga and you, as a student, share a commitment to academicintegrity. You are reminded that you may be charged with an academic offence for possessingany unauthorized aids during the writing of an exam, including but not limited to any electronic

devices with storage, such as cell phones, pagers, personal digital assistants (PDAs), iPods,and MP3 players. Unauthorized calculators and notes are also not permitted. Do not have anyof these items in your possession in the area of your desk. Please turn the electronics off andput all unauthorized aids with your belongings at the front of the room before the examinationbegins. If any of these items are kept with you during the writing of your exam, you may becharged with an academic offence. A typical penalty may cause you to fail the course.

Please note, you CANNOTpetition to RE-WRITEan examination once you have begun writing.

INSTRUCTIONS:1. There are two parts to this examination:

PART I: MULTIPLE CHOICE 15 questions of equal value (3 marks each).PART II: Questions of varying value. In each question, you must show your workand write a concludingstatement(in the box provided). A correct answer obtained with false reasoning (or no reasoning) will notreceive any marks.

2. This examination has 24 different pages including this page. Make sure your copy of the examination has 24different pages and sign it at the top.

3. Budget your time. Good Luck!

Continued on page 2

PART I ******************PART II******************

Q1- Q15 Q1 Q2 Q3 Q4 Q5 Q6

Value 45 7 10 10 8 10 10

Score

TOTAL:

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MAT133Y5Y April 2010 Final examination-A Student number______________________________

*******************************************************************************************

PART I:MULTIPLE CHOICE [Total: 45 marks, 3 marks for each part]

For each question clearly circle the letter corresponding to your answer,

in the MULTIPLE CHOICE ANSWER TABLEprovided BELOW .

(ONLY ONE LETTER should be circled for each question).

NO marksfor incorrect answers or answers indicated outside the MULTIPLE CHOICE ANSWER TABLE]

PART I: MULTIPLE CHOICE ANSWER TABLE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A A A A A A A A A A A A A A A

B B B B B B B B B B B B B B B

C C C C C C C C C C C C C C CD D D D D D D D D D D D D D D

E E E E E E E E E E E E E E E

NO marksfor incorrect answers or answers indicated outside the MULTIPLE CHOICE ANSWER TABLE]

********************************************************************************************

Continued on page 3

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PART I:MULTIPLE CHOICE (continued) [Total: 45 marks, 3 marks for each part]For each question clearly circle the letter corresponding to your answer, in the MULTIPLE CHOICE ANSWERTABLE provided on page 2.

1. The solution to the logarithmic equation given by log(2 11) log3 log( 4)x x+ = + is

A) 1 B) -4 C) -1 D) 3/7 E) none of these

2. Which of the following options has the highest effective rate?

A) 10.25% compounded annually

B) 10.10% compounded semi-annually

C) 9.8% compounded continuously

D) 10% compounded quarterly

E) An effective rate of 10.3%

Continued on page 4

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MAT133Y5Y April 2010 Final examination-A

PART I: MULTIPLE CHOICE (continued) [Total: 45 marks, 3 marks for each part]For each question clearly circle the letter corresponding to your answer, in the MULTIPLE CHOICE ANSWERTABLE provided on page 2.

3. Which set of constraints will determine a boundedfeasible region?

A)

2 4

4 4

0

x y

y x

x

B)

2 4

4 4

0

x y

y x

x

C)

2 4

4 4

0

x y

y x

x

D)

2 4

4 4

0

x y

y x

x

E) none of these

Continued on page 5

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PART I: MULTIPLE CHOICE (continued) [Total: 45 marks, 3 marks for each part]For each question clearly circle the letter corresponding to your answer, in the MULTIPLE CHOICE ANSWERTABLE provided on page 2.

4.A linear system is represented by the following augmented coefficient matrix

1 2 1 1

1 5 4 7

0 4 4 8

.

This linear system has

A) no solutions B) exactly one solution C) exactly two solutionsD) infinitely many solutions E) none of these

5.According to the matrix equation1

1 32

z y xe x

y k

+ =

, the value ofzis:

A) ln3 B) ln4 C) 3 D) e2 E) e

4

Continued on page 6

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6.

1 if 0

3( )1

if 05

x

xf x

xx

+= < +

is continuous on the intervals

A) ( ) ( ),0 0, B) ( ) ( ) ( ), 5 5, 3 3, C) ( ) ( ) ( ), 5 5,0 0,

D) ( ) ( ) ( ), 3 3,0 0, E) none of these

7. If 3 1( ) xy x x += then ( )y x =

A) ( ) 3 12 ln xx x x ++ B) 3 11

3 3ln xx xx

+ + +

C)

3 13ln

xx

x

++ D) ( ) 3ln xx x E) ( ) 33 1 xx x+

Continued on page 7

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8. Given the demand equation 21 2p q= for10 90q . For which value of qis the demand elastic?

A) 49.0 B) 51.5 C) 42.3 D) 64.5 E) 87.3

9. The minimum average cost for [10, 20]x , if total cost is given by 3( ) 40 686C x x x= + + , is

A) 473.3 B) 208.6 C) 1870 D) 187 E) 5036

Continued on page 8

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10. Suppose that r(q) is the total daily revenue (in $) at a fast food restaurant when qhamburgers are sold per day

If 310000

dr qdq

= + , find the pricepthat should be charged if q= 200 hamburgers are to be sold per day.

A) p= $5.98 B)p= $2.99 C)p= $3.01 D)p= $4.50 E) none of these

11.The average value of2

2( )

1 3

xf x

x=

+

on the interval [0, 1] is

A) 1 B)2

3

C)ln 2

3

D)1

3

E) none of these

Continued on page 9

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12.The improper integral0

x

x

e

edx

e

A) equals 1 B) equals e C) equals1

e D) diverges to E) none of these

13.The area of the region enclosed by the curve lny x= , 0y = and 4x= is

A) 4 ln 4 3 B) ln 4 C) 4 ln 4 4 D)3

4 E) none of these

Continued on page 10

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PART II: You must show your workand write a concluding statement(in the box provided).

Question 1 [Total: 7 marks]

Anna makes 5 payments into an annuity. Each payment is $1000, placed into the annuity at the beginning of each

year. For the first 3 years, interest is earned at 5% compounded annually. In the fourth and fifth years, however, thinterest rate goes up to 6% compounded annually. Determine the value of the annuity after 5 years.Round your answer to two decimals.

Conclusion for Question 1: Round your answer to two decimals.


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From a 15 inch by 15 inch piece of cardboard, four congruent squares are to be cut out, one at each corner. The

remaining cross-like piece is then to be folded into an open-topped box. What size squares should be cut out tomaximize the volume of the resulting box?

15 in

15 in

x

x

x

x

x

x x

x


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Question 2 (continued) [Total: 10 marks]

Conclusion for Question 2:


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The demand function for a product is

210 800

30 200qp

q q+=

+ +.

Assume that the market equilibrium occurs when 10q= , and find the consumers surplus under market

equilibrium. Round your answer to one decimal.

Conclusion for Question 3: Round your answer to one decimal


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This page is intentionally left blank. Use it for rough work.


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Question 4 (continued)[Total: 8 marks]

For each part, clearly indicate your final short answer in the appropriate BOX provided.[8 marks, 1 mark for each part, with NO part marks for incorrect final answers.]

f) Does the given function have a horizontal asymptote, If so, where?

g) Does the given function have a vertical asymptote, If so, where?

h) Sketch the graph of the given function,on the co-ordinate system provided

5555 444 4 3 33 3 222 2 1111 1111 2222 3333 4444 5555 6666

2222

1111

1111

2222

3333

4444

5555

6666

x

y


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Question 5 (continued) [Total:10 marks]

(b)What will be the temperature of the pot of soup after 10 minutes? [2 marks

Round your answer to the nearest degree (integer).

Conclusion for Question 5 (b):

(c)What will be the temperature of the soup after a long time (i.e. what is lim ( )t

T t

)? [2 marks

Conclusion for Question 5(c):


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The revenue r(in dollars per square meter of ground) obtained from the sale of a crop of tomatoes grown in an

artificially heated greenhouse is given by

( )( , ) 5 1 xr x y y e= and the cost c(in dollars per square meter of ground) is given by

2( , ) 20 0.1c x y x y= +

wherexis the amount of fertilizer applied per square meter andyis the temperature (in C ) maintained in thegreenhouse.

(a)Write an expression, in terms ofxandy, for the profit Pper square meter obtained from the sale of the cro

of tomatoes. [2 marks

Conclusion for Question 6(a):

(b)Evaluate 5(ln ,5)4

xP and 5(ln ,5)

4y

P [2 marks

Conclusion for Question 6(b):


P(x,y) =

5(ln ,5)

4x

P =5

(ln ,5)4

yP =

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Question 6(continued) [Total: 10 marks]

(c)Given that the pairs

( , ) (ln 5,20)x y = and 5( , ) (ln ,5)4

x y =

are the only critical points of the profit function in part (a), use the second derivative test to determinewhether either of these points corresponds to a relative maximum profit per square meter. [6 marks

Conclusion for Question 6(c):


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MAT133Y5Y April 2010 Final examinationFORMULA SHEET

Your name and student # MUST be written in INK on this page if you wish to separate it from the rest!

STUDENT NAME: ________________________________ Student number______________________________

FORMULA SHEET (1)

Some facts that may be useful:For discrete periodic compoundingwith periodic rate r(where r is theAPRdivided by the number of interest periods in onyear):

(1 )nS P r= + is the compound amount when a principal amount Pis invested for nperiods at a periodic rate of r.

(1 ) nP S r = + is the present value (or principal amount) that needs to be invested at the periodic rate of rfor n

interest periods so that the compound amount is S.The effective rate rethat is equivalent to anAPR(a nominal rate) compounded n times a year is given by:

1 1

n

e

APRr

n

= +

For continuous compounding:

rtS Pe= is the compound amount Sof a principal amount Pafter tyears at an annual interest rate rcompoundedcontinuously.

rtP Se= is the present value Pof a compound amount Sdue at the end of tyears at an annual interest rate rcompoundedcontinuously.

The effective rate recorresponding to an annual rate of rcompounded continuously is 1r

er e=

Annuities:

1 (1 ) nrA R

r

+

= is the present value of an

ordinary annuity

(1 ) 1

n

rS Rr

+ = is the future value of an

ordinary annuity

11 (1 )1

nrA R

r

+ += +

is the present value

of an annuity due

1(1 ) 11

nrS R

r

+ + =

is the future value of

an annuity due)

A= Present value of an annuityS= amount (future value) of an annuityR= amount of each paymentn= number of paymentsr= periodic rate of interest

For a LOAN amortized over nperiods at a periodic rate r, we have:

Principal outstanding at beginning of period k:

Ak=( )

11 1

n kr

Rr

+

+

The interest portion of the kth payment is:

Akr=R ( )1

1 1 n kr

+

+

The principal contained in the payment of the kth period is:

R-R ( )1

1 1 n kr

+

+ = ( )1

1 n k

R r +

+

Total finance charges (interest paid): nRA

****************************************************

For continuous annuities:

0

( )T

rtA f t e dt= 0

( )( )

T

r T tS f t e dt

=

With the annual rate being r(compounded continuously), for Tyears,and where the payment at time tis at a rate off(t) dollars per year.

A= Present value of the continuous annuityS= accumulated (future value) of the continuous annuity


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MAT133Y5Y April 2010 Final examinationFORMULA SHEET (2)

Derivatives:

1[ln ]'x

x= [ ] 'x xe e= 1[ ]'r rx rx =

[ ]' lnx xa a a= (for a> 0)

Some general rules for differentiation:

The Product Rule:[ ( ) ( )]' '( ) ( ) ( ) '( )f x g x f x g x f x g x= +

The Quotient Rule:

( )2

'( ) '( ) ( ) ( ) '( )

( ) ( )

f x f x g x f x g x

g x g x

=

The Chain Rule:

[ ]'( ) '( ( )) '( )f g x f g x g x=

Elasticity of Demand:

The point elasticity of demand , which is anegative

number, is given by

p

p dqq

dp q dp

dq

= =

Economics stuff some useful relations:c

cq

= ,

r pq= , P r c=

Here,cstands for cost (total),

cstands for averagecost,pstands for the demand (price per unit), qis the

number of units of a product, ris revenue, Pis profit.

Integrals:

11 lndx x dx x C

x

= = +

x xe dx e C = + ,kx

kx ee dx C

k= +

(for 0 1)ln

xx a

a dx C aa

= + <

1

(for 1)1

rr x

x dx C rr

+

= + +

Average value of a function on[a,b]:

The average (or mean) value of a functiony=f(x) over

the interval [a, b] is denoted by y (or) fand is given

by

1( )

b

a

f f x dxb a

=

Some general rules for integration:

( ) ( )cf x dx c f x dx=

[ ( ) ( )] ( ) ( )f x g x dx f x dx g x dx+ = +

[ ( ) ( )] ( ) ( )f x g x dx f x dx g x dx =

( ( )) ( ) ( )f g x g x dx f u du = when we let ( )u g x=

( ) '( ) ( ) ( ) ( ) '( )f x g x dx f x g x g x f x dx= OR udv uv v du= (Integration by parts)

mat133y final 2010w

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