Name:
Midterm 2 Make up
• You will have 110 minutes to complete this exam.
• A blank page has been attached to the test; it may be detached and used as scratch
paper.
• During this exam, you may only use the scratch paper provided and writing utensils.
No calculators, cell phones, books, notes or other resources will be permitted.
• Unless otherwise indicated, you must justify your solution to receive full credit on
a problem. Any of your classmates should be able to understand how you arrived at
your solution.
• Partial credit will be granted for work that demonstrates understanding of key concepts.
• Circle your final solutions.
• You can do it!
Point distribution
Multiple Choice: 3 points × 10 problems = 30 points
Free Response: 10 points × 5 problems = 50 points
Total: 80 points
1
Multiple Choice
You do NOT need to justify your solutions in this section. Simply fill in the bubble for
the correct answer to each question.
Problem 1 (3 points). Let g(x, y) be a differentiable function with continuous second derivatives. Suppose the second derivatives of g at the point (0, 2) are the following:
gxx (0, 2) = −2 gyy (0, 2) = 1 gxy (0, 2) = 3.
What can we conclude about the function g?
A g has a local maximum at (0,2).
B g has a local minimum at (0,2).
C g has a saddle point at (0,2).
D There is not enough information to draw a conclusion.
Problem 2 (3 points). Let R be the region in the first quadrant between the circles centered
at the origin with radii 1 and 3. Find the value of the following integral:
Z Z
cos(x2 + y 2 ) dA
R
π
(sin 3 − sin 1)
8
B π2 (sin 3 − sin 1)
C π4 (sin 9 − sin 1)
D π(sin 9 − sin 1)
A
E None of the above
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
Page 2 of 12
Problem 3 (3 points). Find the centroid (center of mass) of the lamina pictured below.
You may assume that the lamina has a uniform density. The upper boundary of the lamina
is given by the curve y = 4 − x2 .
4
)
A (0, 2 25
4
, 0)
B (2 25
C (0, 2)
D (2, 0)
E (0, 54 )
F) ( 45 , 0)
Problem 4 (3 points). Let f (x, y) = y 3 − 2x2 . What is the derivative of f in the direction
of h1, 1i at the point (−1, 3)?
A 9
B √92
C − √92
D -9
E None of the above
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
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Problem 5 (3 points). We are interested in finding the maximum and minimum of the
function f (x, y) subject to the constraint g(x, y) = c.
In the figure below, the dashed curves are level curves of f and the solid ellipse is the curve
g(x, y) = c.
Which of the points A, B, C and D are solutions to the system of equations that arise when
we use the method of Lagrange multipliers to solve this constrained optimization problem?
A Only A
B Only C
C B and C
D A, B and D
E None of the above
Problem 6 (3 points). Which of the following represents the volume under the surface
z = xy and above the triangle with vertices (1, 1), (4, 1) and (1, 7)
Z 4 Z −2x+9
Z 4 Z −2x+9
xy dy dx
xy dy dx
A
B
1
0
1
Z 7 Z (9−y)/2
C
Z 7 Z (9−y)/2
xy dy dx
1
1
0
D
xy dy dx
1
1
E None of the above
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
Page 4 of 12
Problem 7 (3 points). Consider the function f (x, y) = exy and let R be a rectangle in the
first quadrant. Where should the sample point in a given subrectangle be taken to make the
Riemann sum
X
f (x∗i , yj∗ )∆x∆y
i,j
as small as possible?
y j+1
y j+1
A
D
yj
yj
xi
xi+1
y j+1
xi
xi+1
xi
xi+1
y j+1
B
E
yj
yj
xi
xi+1
xi
xi+1
y j+1
C
yj
Problem 8 (3 points). Which of the following integrals corresponds to the volume of a
hemisphere of radius 2?
Z 2 Z √4−x2 Z √4−x2 −y2
I.
1 dz dy dx
√
2
0
− 4−x√
0
Z 2π Z 2 Z 4−r2
II.
1 dz dr dθ
Z0 2π Z0 π Z0 2
III.
ρ2 sin ϕ dρ dϕ dθ
0
0
0
A Only I
B Only II
C Only III
D I and II
E I and III
F) II and III
G I, II, III
H None
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
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Problem 9 (3 points). What is the projection of the point with spherical coordinates
, 3π
into the xz-plane?
1, 3π
2
4
The points below are given in rectilinear coordinates.
1
1
√
√
A 0, 0, − 2
B 0, 0, 2
C
√1 , 0, √1
2
2
D
1, 0, √12
E None of the above
Problem 10 (3 points). Suppose u is a function of x, y, and z, while x, y and z are each
functions of s and t. Here are some values of u, x, y, and z, and their derivatives, depending
on their inputs:
∂u ∂u ∂u
∂x ∂x ∂y ∂y ∂z ∂z
s t x y
u
z
x y z
∂x ∂y ∂z
∂s ∂t ∂s ∂t ∂s ∂t
1 0 2 −1
2
3
1
0 1 0 1
1
3 −1
2
0
4
2
0 1 2 −1 −1 −3
0
1 0 1 0 −1
2
2
1 −1 −2
5
2 1 3
0
1 −2 −1
1 1 2 1
3
1
0
1
1
2
1
2 2 2 −3
0
3
3
2 2 1 0
2
2 −1
1 −1
3
4
∂u
at the point (s, t) = x(1, 1).
∂s
A 1
B 0
C -2
D -3
Use this information to find
E None of the above
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
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Free Response
For each of the questions in this section, must provide justification for your answers.
Problem 11 (10 points). Use the second derivative test and the method of Lagrange multipliers to find the maxima and minima of the function f (x, y) = x2 + y 2 over the region
bounded by x4 + y 4 = 2.
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
Page 7 of 12
∂z
Problem 12 (10 points). Find ∂y
where z is defined implicitly by the equation
z = exyz .
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
Page 8 of 12
Problem 13 (10 points). Show that every normal line to the sphere x2 + y 2 + z 2 = r2 passes
through the origin.
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
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Problem 14 (10 points). Consider the following double integral.
Z 0 Z arcsin y
2
cos(x)e− cos x dx dy
−1
−π/2
a) Sketch the region of integration.
b) Evaluate the integral.
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
Page 10 of 12
Problem 15 (10 points). Let E denote the region inside the ellipsoid 3x2 + 3y 2 + z 2 = 1
and inside of the cone 3z 2 = x2 + y 2 .
Set up a triple integral in either spherical or cylindrical coordinates for the volume of
E. Do not evaluate this integral.
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
Page 11 of 12
Scratch Page
Fall 2017 Calculus III (Math 123 Sec. 004) Corrin Clarkson
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