STUDENT NUMBER:
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DEPARTMENT OF MATHEMATICS AND STATISTICS
QUEEN’S UNIVERSITY AT KINGSTON
MATH 121 - On Campus - DEC 2013
VERSION 1
INSTRUCTIONS:
• Answer all questions, writing clearly in the space provided. If you need more room, continue
your 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 do
otherwise.
• 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 exam
questions. Do your best to answer exam questions as written.
FOR MARKER’S USE ONLY
Section Possible Received
I
20
II
10
III
8
IV
10
V
10
VI
15
VII
10
VIII
10
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
Dec 2013
STUDENT NUMBER:
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Section I. Multiple Choice (10 questions, 2 marks each)
Each question has four possible answers, labeled (A), (B), (C), and (D). Choose the most appropriate answer. Write your answer in the space provided, using UPPERCASE letters. Illegible
answers will be marked incorrect. You DO NOT need to justify your answer.
(1) Find one solution to the equation 4e3x+1 = 8.
2
(A) ln
−1
3
ln(2)
3
ln(2)
(C)
−1
3
ln(2) − 1
(D)
3
(B) 1 −
ANSWER:
(2) Suppose C(r) is the total cost of paying off a car loan borrowed at an annual interest rate
of r%. What are the units of C ′ (r)?
(A) dollars
(B) dollars/year
(C) dollars/(percent interest)
(D) (percent interest)/year
ANSWER:
(3) Find the value of the integral
Z
0
1
x
dx.
x2 + 1
(A) 2 ln 2
1
(B) ln 2
2
π
(C)
4
π π
(D) −
4
6
ANSWER:
MATH 121
Dec 2013
STUDENT NUMBER:
(4) Consider the integral I =
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Z
2
√
1 + x3 dx. Of the following constraints, select the one that is
0
both true and the most restrictive. Note: you do not need to evaluate the integral to answer
this question.
(A) 2 ≤ I ≤ 6
(B) 1 ≤ I ≤ 6
(C) 1 ≤ I ≤ 8
(D) 0 ≤ I ≤ 8
ANSWER:
(5) The function f (x) is defined as follows:


1 + 2x x < 3
f (x) = 3
x=3

 2
x −2 x>3
Which of the following statements best describes the function around x = 3?
(A) The function is continuous at x = 3; the limit lim f (x) exists.
x→3
(B) The function is NOT continuous at x = 3; the limit lim f (x) exists.
x→3
(C) The function is continuous at x = 3; the limit lim f (x) does NOT exist.
x→3
(D) The function is NOT continuous at x = 3; the limit lim f (x) does NOT exist.
x→3
ANSWER:
(6) The cost of producing q items in a factory is given by the function
C = f (q) = 125 + 3q
C is in thousands of dollars, and q is the number of units. Find the inverse of this cost
function.
1
125 + 3q
1
=
125 + 3C
C − 125
=
3
q − 125
=
3
(A) f −1 =
(B) f −1
(C) f −1
(D) f −1
ANSWER:
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STUDENT NUMBER:
2 /b
(7) Let f (x) = e−(x−a)
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. Where does f (x) have its global maximum?
(A) x = a
(B) x = b
(C) x = b − a
a
(D) x =
2
ANSWER:
(8) The air in a factory is being filtered so that the quantity of a pollutant, P (in mg/liter), is
decreasing according to the function P = P0 e−kt , where t is time in hours. If 20% of the
pollution is removed in the first 10 hours, which value below is closest to the percentage of
the pollution left after 30 hours?
(A) 1% is left.
(B) 40% is left.
(C) 50% is left.
(D) 60% is left.
ANSWER:
(9) The anti-derivative of f (x) = 3x + 1 + x2 is given by which function below?
(A) ln(3) · 3x + x + 2x3 + C
3x
+ x + 2x3 + C
(B)
ln(3)
x3
(C) ln(3) · 3x + x +
+C
3
x3
3x
+x+
+C
(D)
ln(3)
3
ANSWER:
(10) If the function g(x) has an average value of 20 over the interval x = 1 to x = 5, which of the
following statements is true?
Z 5
(A)
g(x) dx = 80
1
(B)
(C)
(D)
Z
Z
Z
5
g(x) dx = 20
1
5
g(x) dx = 5
1
5
g(x) dx = 4
1
ANSWER:
MATH 121
Dec 2013
STUDENT NUMBER:
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Section II. Dosage Response
For some positive constant C, a patient’s temperature change, T , in degrees Celsius, due to a
dose of medication of D mg, is given by
T =
C D
−
2
3
D2.
(a) What dosage maximizes the temperature change? Your answer may depend on C.
dT
(b) The sensitivity of the body to the drug is defined as
. What dosage level is associated
dD
with the maximum sensitivity? Again, your answer may depend on C.
MATH 121
Dec 2013
STUDENT NUMBER:
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Section III. Differentiability
Consider the function
(
1 + 2 cos(πx)
g(x) =
−1
for 0 ≤ x ≤ 1
for x > 1
(a) Sketch the graph of g(x) on the axes below. Indicate the vertical scale on the graph.
0
1
2
(b) Use the definition of continuity to decide whether g(x) is continuous at x = 1.
(c) Is g(x) differentiable at x = 1? Support your answer.
MATH 121
Dec 2013
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Section IV. Investments
(a) Saving 1000 dollars per year for 20 years, at 5% interest, will give a final savings amount of
Z 20
(1000) e0.05(20−t) dt dollars. Evaluate this integral to compute the final savings. Give
0
your answer rounded to the nearest dollar.
(b) Savings can be increased by saving more and more each year. If an investor saves S(t) =
1000 + 100t dollars/year for 20 years, their final savings amount is given by
Z
20
(1000 + 100t) e0.05(20−t) dt. Evaluate this integral to compute the final savings, rounding
0
your answer to the nearest dollar.
MATH 121
Dec 2013
STUDENT NUMBER:
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Section V. For time, t, in hours, 0 ≤ t ≤ 1, a bug is crawling at a velocity, v, in meters/hour
given by
1
v(t) =
.
1 + 2t
(a) Write an integral that represents the total distance traveled by the bug.
(b) Use the trapezoidal rule with 4 intervals (n = 4) to estimate the distance that the bug crawls
during this hour.
(c) Sketch the graph of v(t) on the axes below, and then add a representation of the trapezoidal
rule’s estimate of the distance traveled.
0
0.5
1
(d) Find the exact distance traveled by the bug between t = 0 and t = 1.
MATH 121
Dec 2013
STUDENT NUMBER:
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Section VI. Volume
Water is flowing into and out of a tank at a rate given by r(t) in liters/minute (t is in minutes).
When r(t) is positive, water is flowing into the tank; when r(t) is negative, water is flowing out
of the tank.
A graph of r(t) is shown below, along with some of the areas between the graph and the value
r = 0.
3
3
2
2
1
1
Area = 5.6
r(t)
r(t)
Area = 0.7
0
0
−1
−1
−2
−2
−3
0
−3
0
Area = 0.4
Area = 4.8
2
4
6
8
10
t
2
4
6
8
10
t
Let V (t) be the volume of water in the tank. The tank initially contains 2.5 liters of water, so
V (0) = 2.5.
(a) Find the all the critical points of V (t).
(b) Compute the volume of water in the tank at each critical point.
(c) Classify each of the critical points as a local minimum, local maximum, or neither.
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(continued)
3
3
2
2
1
1
Area = 5.6
r(t)
r(t)
Area = 0.7
0
0
−1
−1
−2
−2
−3
0
−3
0
Area = 0.4
Area = 4.8
2
4
6
8
10
2
4
t
6
8
10
t
(d) At what time in the interval 0 ≤ t ≤ 10 is the tank volume at its global minimum, and
what is that volume? Remember to support your answers.
(e) At what time in the interval 0 ≤ t ≤ 10 is the tank volume at its global maximum, and
what is that volume?
(f) Draw a detailed graph of the volume over time, V (t), indicating the results from your answer
from parts (a)-(e),
0
MATH 121
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4
6
8
10
Dec 2013
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Section VII. Optimization
A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other
is bent into an equilateral triangle.
If you want to have the total area of both figures as a minimum, how much wire should be used
for the square and triangle, respectively? Indicate how you know your answer is minimum for
the area.
MATH 121
Dec 2013
STUDENT NUMBER:
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Section VIII. A population reproduces according to the logistic map, defined by
At+1 = 0.8At (1 − At )
where A is the proportion of the largest possible population, and t is in generations. Because A
is a fraction, 0 ≤ A ≤ 1.
(a) Assuming that the initial population level is 90% of its maximum (A0 = 0.9), predict the
population level for the next four generations (up to and including A4 ).
(b) Using your answer to part (a), sketch a graph of At vs. time. Be sure to label your axes.
(c) Sketch the graph of At+1 vs. At , and use cobwebbing to graphically estimate the values of
A1 , A2 , A3 and A4 .
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At+1 = 0.8At (1 − At ),
0 ≤ At ≤ 1
(d) Use the updating formula to find all equilibrium value(s) of At .
(e) Classify each of the equilibrium level(s) you found in part (d) as either stable or unstable,
using the slope test for equilibria.
MATH 121
Dec 2013
STUDENT NUMBER:
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Section IX. Tangent Lines
The relation
x2 − 4x + 4y 2 + 8y = 17
produces an ellipse on the xy plane.
(a) The point (x, y) = (5, 1) is a point on the ellipse. Find the equation of the tangent line
to the ellipse at that point.
(b) Find the x coordinate(s) of all the points on the ellipse where the tangent lines are
horizontal.
(c) Find the y coordinate(s) of all points on the ellipse where the tangent lines are vertical.
MATH 121
Dec 2013