STUDENT NUMBER: Page 1 of 14

DEPARTMENT OF MATHEMATICS AND STATISTICS

QUEEN’S UNIVERSITY AT KINGSTON

MATH 121/124 - APRIL 2013

INSTRUCTIONS:

• Answer all questions, writing clearly in the space provided. If you need more room, continueyour answer on the back of the previous page, providing clear directions to the marker.

• Show all your work and explain how you arrived at your answers, unless explicitly told to dootherwise.

• Only CASIO FX-991, Gold Sticker or Blue Sticker calculators are permitted.

• Write your student number clearly at the top of each page.

• You have three hours to complete the examination.

• Wherever appropriate, include units in your answers.

• When drawing graphs, add labels and scales on all axes.

PLEASE NOTE: Proctors are unable to respond to queries about the interpretation of examquestions. Do your best to answer exam questions as written.

FOR MARKER’S USE ONLY

Section Possible Received

I 20

II 10

III 10

IV 10

V 15

VI 10

VII 10

VIII 8

IX 7

TOTAL 100

This material is copyrighted and is for the sole use of students registered in MATH 121/124 and writing

this examination. This material shall not be distributed or disseminated. Failure to abide by these

conditions is a breach of copyright and may also constitute a breach of academic integrity under the

University Senate’s Academic Integrity Policy Statement.

MATH 121/124 April 2013

STUDENT NUMBER: Page 2 of 14

Section I: Multiple Choice (10 questions, 2 marks each)

Each question has four possible answers, labeled (A), (B), (C), and (D). Choose the most appro-priate answer. Write your answer in the space provided, using UPPERCASE letters. Illegibleanswers will be marked incorrect. You DO NOT need to justify your answer.

(1) If f(x, y, z) = 3xz + sin(x) + ln(xz), then∂f

∂z=

(A) 3x+1

xz

(B) 3x+1

z

(C) 3z + cos(x) +1

z

(D) 3z +1

xz

ANSWER:

(2) How are the vectors ~u = 2~i+ 6~j − 3~k and ~v = 4~i− 2~j − ~k related?

(A) They have an angle >π

2radians between them.

(B) They have an angle <π

2radians between them.

(C) They are perpendicular.

(D) They are parallel. ANSWER:

(3) Consider the slope field shown below.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Which of the following differential equations matches the slope field?

(A)dy

dx= y − x2 (C)

dy

dx= y2 − x

(B)dy

dx= x2

− y (D)dy

dx= x− y2

ANSWER:

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STUDENT NUMBER: Page 3 of 14

(4) For the differential equationdQ

dt= 5(Q− 2), which of the following statements is true?

(A) Q(t) will be increasing for all t > 0 if Q(0) = 0.

(B) Q(t) will be decreasing for all t > 0 if Q(0) = 3.

(C) Q(t) will be neither increasing nor decreasing over time if Q(0) = 2.

(D) Q(t) will behave according to all of the above.

ANSWER:

(5) In which direction does the function f(x, y) =3

xy+ 1 increase most quickly at the point

(1, 1)?

(A) ~i+~j

(B) ~i−~j

(C) −~i−~j

(D) ~i

ANSWER:

(6) A thin 2 m long rod has lineal density δ(x) = x + 2 g/m, where x, in meters, is measuredfrom the left end of the rod. Where is the center of mass of the rod?

(A) Undefined/not enough information.

(B) In the range 0 ≤ x < 1.

(C) At x = 1.

(D) In the range 1 < x ≤ 2.

ANSWER:

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STUDENT NUMBER: Page 4 of 14

(7) Let f(x, y) = 2x2+4y3+2xy+1. Determine which of the following shapes would be obtainedby intersecting the surface defined by f(x, y) with the plane y = 3.

(A) A parabola facing up.

(B) A parabola facing down.

(C) A straight line with slope 3.

(D) A cubic with 3 distinct roots.

ANSWER:

(8) The contour diagram for the function f(x, y) is shown below.

Assuming the shape of function is well described by the contour diagram, how many criticalpoints does f(x, y) have in region shown above?

(A) 3

(B) 4

(C) 5

(D) 6

ANSWER:

(9) For what values of t are the vectors

< 6, 4,−3 > and < 2, (t2 +2

3t+ 1), t) >

parallel?

(A) t = −1.

(B) t = 0.

(C) t = 1.

(D) All real values of t.

ANSWER:

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STUDENT NUMBER: Page 5 of 14

(10) Match the function z =sin(x2 + y2)

x2 + y2to its graph below.

A B

C D

ANSWER:

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STUDENT NUMBER: Page 6 of 14

Section II: Volume of RevolutionConsider a cone with base radius r and height h.

h

r

(a) Write an integral that represents the volume of the cone, based on the volume slicing method(i.e. Riemann sums method) used in the course.

(b) Evaluate the integral you found in part (a).

(c) Now consider the volume remaining if the top part of the cone is cut off at h/4 from the top.Write, but do not evaluate a single integral that represents the volume of this truncatedcone.

h/4

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STUDENT NUMBER: Page 7 of 14

Section III: Application to Economics

(a) A parent saving for their child’s university expenses opens a bank account earning 4.5%annual interest, compounded continuously. At what (constant, continuous) rate must aparent deposit money into such an account in order to have $150, 000 in the account exactly10 years from today?

(b) If the parent decides instead to deposit a lump sum now in order to attain the goal of$150, 000 in 10 years, how much must be deposited now? Assume the same account (4.5%interest continuously compounded) is used.

(c) State whether the integral

∫∞

1

e−x

xdx converges or diverges. Support your answer either

with a calculation or by applying the Comparison Test.

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STUDENT NUMBER: Page 8 of 14

Section IV: ContoursIn this question you are dealing with the surface defined by z = f(x, y) = y − 2x + 2.

(a) On the axes below, draw and clearly label the contours for the heights z = 0, z = 1, z = 2,and z = 3.

−2 −1 0 1 2

−2

−1

1

2

(b) Evaluate the gradient of f(x, y) at the point (x, y) = (1, 0), and draw the gradient vector onthe contour diagram.

(c) Compute the directional derivative at (x, y) = (1, 0) in the direction ~v = 2~i+ 3~j.

(d) Without calculation, identify the (x, y) point where the global minimum of the functionoccurs on the domain shown, −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2. Explain your answer briefly.

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STUDENT NUMBER: Page 9 of 14

Section V: Differential Equation ModelingConsider a can of cola taken from the fridge and placed outside. The can starts at 4 degreesCelsius, and the outside temperature remains at a constant 28 degrees Celsius.Newton’s Law of Heating and Cooling states that the rate of temperature change of an object isproportional to the difference between the temperature of the object and the temperature of itsenvironment.

(a) Assuming the Law above governs the rate of change of the temperature of the can, writeout the differential equation that dictates the rate of change of temperature of the can overtime. Use the symbols H for the can temperature, and k as the constant of proportionality.

(b) Use separation of variables to find the solution to this differential equation that satisfies thegiven initial temperature of the can (at t = 0, H = 4).

More space is provided on the next page if required.

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(more space for Section IV (b))

(c) Using either the differential equation, or the solution to the DE, find the equilibrium tem-perature of the can. Remember to show all your work.

(d) If it takes 15 minutes for the can to reach 10 degrees Celsius, how long will it take to reacha temperature of 20 degrees Celsius?

(e) Consider a modified scenario where

• the can at 4 degrees is taken from the fridge and put outside;

• after 15 minutes the can reaches a temperature of 10 degrees, at which point

• the can is put back in the fridge (4 degree environment).

Sketch the graph of the resulting temperature of the can over time on the axes below.

Time (min)

Tem

pera

ture

(de

g C

)

0 15 30 45

2

4

6

8

10

12

14

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Section VI: Linearization

The contour diagram to the rightshows the monthly payments, $P ,required to repay a 5-year car loanas a function of the interest rate, r,and the amount you borrow, A.

4040

50

5050

60

60

60

70

70

70

80

80

80

90

90

90

100

100

100

110

110

110

120

120

120

130

130

130

140

140

140

150

150

160

160

170

180

r (%)

A (

$)0 5 10 15

2000

3000

4000

5000

6000

7000

8000

(a) Estimate the derivatives∂P

∂rand

∂P

∂Aat r = 4% and A = $6000, based on the contour

diagram. Be clear about how you are arriving at your estimates. (Note: there are severalways to estimate these derivatives; we will accept any appropriate method.)

(b) Construct a linear approximation for the function P (r, A) around r = 4%, A = $6000.

(c) Use your answer to Part (b) to estimate the monthly payments required if the interest ratewere 3.7% and you borrowed $6,200. Give your answer to the nearest dollar.

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Section VII: Constrained Optimization

Find the maximum and minimum values of z = x3 + y on the circle x2 + y2 = 4.

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Section VIII: Critical Point Analysis

Consider the function f(x, y) = 4y2 + 2kxy + kx2, where k is a constant.

(a) Show that (0, 0) is a critical point of f(x, y).

(b) Determine the values of k (if any) for which the critical point at (0, 0) would be each of thefollowing:

(a) a local minimum,

(b) a local maximum,

(c) a saddle point, or

(d) unclassifiable using the Second Derivative Test,

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STUDENT NUMBER: Page 14 of 14

Section IX - Production

A factory will make q units of a product, given W units of labour and K units of capital,according to the formula

q = 500W 1/3K2/3

Currently, the budget for the factory specifies W = 64 units and K = 27 units.The budget for the factory specifies that

• the number of labour units is growing at 3 units per year, and

• the number of capital units is shrinking by 2 units per year.

Using the chain rule, determine the rate at which the current production level is changing withtime, and indicate clearly whether the production is increasing or decreasing.

MATH 121/124 April 2013