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STUDENT NUMBER:
1 of 16
HAND IN
Answers recorded on exam paper.
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUEEN’S UNIVERSITY AT KINGSTON
MATH 121 - APR 2017
Section 700 - CDS Students ONLY
Instructor: A. Ableson
INSTRUCTIONS:
•
This examination is 3 HOURS in length.
•
Only
CASIO FX-991
, or
Blue Sticker
calculators are permitted.
•
Answer
all questions
, writing clearly in the space provided. If you need more room, continue your
answer on one of the blank pages at the back,
providing clear directions to the marker
.
•
For full marks, you must show all your work and explain how you arrived at your answers, unless
explicitly told to do otherwise.
•
Wherever appropriate,
include units in your answers
.
•
When drawing graphs,
add labels and scales on all axes
.
•
Put your
student number on all pages
, including this front page.
PLEASE NOTE:
Proctors are unable to respond to queries about the interpretation of exam questions.
Do your best to answer exam questions as written.
I
II
III
IV
V
VI
VII
VIII
IX
X
Total
20
8
10
10
8
10
8
8
8
10
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.
2 of 16
STUDENT NUMBER:
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) Which of the following differential equations is
not
separable?
(A)
dy
dx
=
x
(
y
+ 1)
(B)
dy
dx
= (
x
+ 1)(
y
+ 1)
(C)
dy
dx
=
x
+ 1
(D)
dy
dx
=
xy
+ 1
ANSWER:
(2) Consider the surface defined by
z
= (
x
2
+ 1) sin(
y
) +
xy
2
. Which of the following planes produces a
parabola where it intersects this surface?
(A)
x
= 0
(B)
y
= 0
(C)
z
= 0
(D) None of the above.
ANSWER:
(3) An airplane is flying at an airspeed of 400 km/h, while being pushed by a wind that is blowing at 50
km/h from the southwest. If the plane orients itself so as to be flying due east, what will the airplane’s
speed be, relative to the ground?
(A) Ground speed will be 450 km/h or more.
(B) Ground speed will be 425 km/h or more, and less that 450 km/h.
(C) Ground speed will be 400 km/h or more, and less that 425 km/h.
(D) Ground speed will be less than 400 km/h.
ANSWER:
STUDENT NUMBER:
3 of 16
(4) Which of the following statements about vectors is
not
necessarily true for all choices of vectors
~a
and
~
b
?
(A)
~a
+
~
b
=
~
b
+
~a
(B)
||
~
b
+
~a
||
=
||
~a
+
~
b
||
(C)
||
~a
+
~
b
||
=
||
~a
||
+
||
~
b
||
(D)
~a
·
~
b
=
||
~a
||||
~
b
||
cos(
θ
), where
θ
is the angle between
~a
and
~
b
.
ANSWER:
(5) Which of the following contour diagrams represents the contours of the surface
z
=
y
2
+
x
?
(A)
(B)
(C)
(D)
ANSWER:
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4 of 16
STUDENT NUMBER:
(6) For what value(s) of
t
are the vectors
<
6
,
4
,
-
3
>
and
<
2
,
(
t
2
+
2
3
t
+ 1)
, t
)
>
parallel?
(A)
t
=
-
1.
(B)
t
= 0.
(C)
t
= 1.
(D) All real values of
t
.
ANSWER:
(7) A thin 2 m long rod has lineal density
δ
(
x
) = 4
-
x
g/m, where
x
, in meters, is measured from 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:
(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 critical points
does
f
(
x, y
) have in region shown above?
(A) 6 or more
(B) 5
(C) 4
(D) 3 or fewer
ANSWER:
STUDENT NUMBER:
5 of 16
(9) Consider the differential equation
dy
dx
=
y
+ 2
x
. If we start approximating the solution to the equation
at
x
= 1 and
y
=
-
3, using Euler’s method with an interval size of Δ
x
= 0.5, our next predicted point
on the solution curve will be
(A) (
x, y
) = (1
.
5
,
-
4)
(B) (
x, y
) = (1
.
5
,
-
3
.
5)
(C) (
x, y
) = (1
.
5
,
-
1)
(D) (
x, y
) = (1
.
5
,
-
0
.
5)
ANSWER:
(10) You are at the point (2, -1, -1), standing upright and facing the
xz
plane.
The positive
z
axis is
upwards. You
•
take two steps forward,
•
turn left, and
•
take four steps forward.
What are the coordinates of your final position?
(A) (0, -5, -1)
(B) (0, 3, -1)
(C) (-2, 1, -1)
(D) (-2, 3, -1)
ANSWER:
6 of 16
STUDENT NUMBER:
Section II.
Areas and Volumes
1. Consider the finite region
R
enclosed within the
boundaries
y
= ln(
x
),
x
= 1, and
y
= 3.
(a) On the axes to the right, sketch those boundaries, then shade
in the region
R
. Clearly indicated the scales on your axes.
[/1]
(b) Consider the solid generated by rotating the region
R
around the
x
axis. Write down a definite integral
whose value is the volume of this solid.
[/3]
2.
Find the area enclosed between the graphs of
y
= sin(
πx
)
and
y
=
x
on the interval
x
= 0
. . .
3. A sketch may help.
[/4]
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STUDENT NUMBER:
7 of 16
Section III.
Improper Integrals
For each of the following integrals, use the definition of an improper integral to determine whether it
converges or diverges. If converges, find the value of the integral.
(a)
Z
∞
0
te
-
kt
dt
, where
k
is a positive constant.
[/5]
(b)
Z
π/
2
0
cos(
x
)
sin(
x
)
dx
.
[/5]
8 of 16
STUDENT NUMBER:
Section IV.
Learning
As you know, when a course ends, students start to forget the material they have learned.
One model
(called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional
to the difference between:
•
the material currently remembered, and
•
some positive constant,
a
.
(a) Let
y
(
t
) be the fraction of the original material remembered
t
weeks after the course has ended. Set
up a differential equation for
y
. Your equation will contain two constants; the constant
a
is less than
y
for all
t
.
[/3]
(b) Find the general solution to the differential equation.
[/5]
(c) Describe the practical meaning (in terms of the amount remembered) of the constant
a
in your solution
to part (b).
[/2]
STUDENT NUMBER:
9 of 16
Section V.
Slope Fields
Consider the slope field shown to the right.
(a) Circle the formula below for differential equation that
corresponds to the slope field. No explanation is required.
[/1]
dy
dx
= 2
x
+
y
dy
dx
=
x
+ 2
y
dy
dx
= 2
x
-
y
dy
dx
=
x
-
2
y
(b) For each of the following points, sketch the solution
that passes through it. Each graph sketch should extend
to boundary of the slope field shown.
[/3]
•
(0,-2)
•
(-1,0)
•
(0,1)
1
-
1
-
2
1
-
1
-
2
x
y
(c) Most of the solutions to the DE are fairly complicated functions. However, by studying the slope field
you can identify one simple linear function that is a solution. What is the formula for that linear solution?
[/2]
(d) Show that your solution from part (c) satisfies the differential equation you picked in part (a).
[/2]
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10 of 16
STUDENT NUMBER:
Section VI.
Functions of Two Variables
1. Find a formula for the linear function
z
=
g
(
x, y
) described by the table below.
[/4]
y
20
40
60
80
-30
3
6
9
12
x
-25
2
5
8
11
-20
1
4
7
10
-15
0
3
6
9
2.
The Dubois formula
s
(
w, h
) relates a person’s
skin surface area,
s
(in
m
2
) to weight
w
(in kg),
and height
h
(in cm).
A contour diagram of this
function is shown to the right.
(a) Find the value
s
(70
,
210) and then express the
meaning of the value in a sentence.
[/1]
(b) Estimate the values
s
w
(70
,
210) and
s
h
(70
,
210), being clear about how you are using the contour
diagram to arrive at your answer. (Note: there are several ways to estimate these derivatives; we will
accept any appropriate method.)
[/3]
s
w
(70
,
210) =
s
h
(70
,
210) =
(c) Write the meaning of each value in sentence form.
[/2]
s
w
(70
,
210) :
s
h
(70
,
210) :
STUDENT NUMBER:
11 of 16
Section VII.
Optimization
Consider the surface defined by
f
(
x, y
) =
x
3
+
y
2
-
12
x
-
6
y
-
24.
(a) Find the critical point(s) of
f
(
x, y
).
[/3]
(b) Classify each of the critical points as a local min, local max, or saddle point.
[/3]
(c) Does this surface have a
global
maximum? Support your answer with a brief explanation.
[/2]
12 of 16
STUDENT NUMBER:
Section VIII.
Surfaces
Consider the function
z
=
h
(
x, y
) =
x
-
√
y
, limited to the region 0
≤
x
≤
4, 0
≤
y
≤
4.
(a) On the axes below, draw and clearly label the contours at heights
z
= 0
,
2 and
-
1. Remember the
region is bounded to the domain shown (0
≤
x
≤
4, 0
≤
y
≤
4).
[/3]
0
1
2
3
4
0
1
2
3
4
x
y
(b) Evaluate the gradient at the point (3, 1), and draw the gradient vector on your contour diagram from
part (a).
[/2]
(c) Compute the directional derivative at (3, 1) in the direction given by the vector
h-
4
,
3
i
.
[/2]
(d) Without calculation, identify the (
x, y
) point where the global maximum of
h
occurs on the domain
shown, 0
≤
x
≤
4, 0
≤
y
≤
4. Explain your answer briefly.
[/1]
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STUDENT NUMBER:
13 of 16
Section IX.
A hiker is moving over a mountain whose height is given by
h
(
x, y
) = 1000
-
0
.
005
x
2
-
0
.
01
y
2
where
x, y
and
z
are measured in meters.
(a) If the hiker starts at (
x, y
) = (30
,
50), in which direction should they walk if they want to go in the
steepest uphill direction?
[/3]
(b) What is the slope in the direction found in (a)?
[/2]
(c) The hiker instead decides to follow a marked trail. Their path on the map is defined by the functions
x
(
t
) = 70
-
20
t
and
y
(
t
) = 40 + 5
t
, where
t
is measured in minutes.
At what rate is the hiker ascending or descending at
t
= 2? Include units in your answer. (Note that
at
t
= 2, the hiker is at coordinates (30, 50).)
[/3]
14 of 16
STUDENT NUMBER:
Section X.
Optimizing Production
A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on
the quantities,
q
1
and
q
2
, supplied by each factory, and is expressed by the joint cost function,
C
=
f
(
q
1
, q
2
) =
q
2
1
+
q
1
q
2
+ 2
q
2
2
+ 1000
The company’s objective is to produce 400 units, while minimizing production costs. Use the method of
Lagrange multipliers to determine how many units should be supplied by each factory.
[/10]
STUDENT NUMBER:
15 of 16
Space for additional work.
Indicate clearly which Section you are continuing if you use this
space.
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16 of 16
STUDENT NUMBER:
Space for additional work.
Indicate clearly which Section you are continuing if you use this
space.