Multivariable Calculus Second Exam
Name: Recitation:
Multivariable Calculus Math 215 Winter 2003
Second Exam March 20, 2003
Please do all your work in this booklet and show all the steps. Write your final answer in the corresponding box. Calculators and note-cards are not allowed.
Problem Possible points Score 1 10 2 15 3 15 4 20 5 10 6 20 7 10 Total 100
2
Problem 1. (10 pts.) (a – 3 pts.) Compute the gradient of the function f (x, y) = x3 y − xy 2 .
(a) Answer: (b – 3 pts.) Find the maximum rate of change of this function at the point P (3, 2).
(b) Answer: (c – 4 pts.) Find the rate of change of this function f (x, y) at the point Q(1, −2) in the direction of the vector v = h2, 4i.
(c) Answer: ˛ ˛ ˛ ˛10 ˛
x2 Problem 2. (15 pts.) Find and classify all critical points of the function f (x, y) = xy + y − − 2x + 3. 2 2
3
2
Answer: ˛ ˛ ˛ ˛15 ˛
4
Problem 3. (15 pts.) Using Lagrange multipliers, find the points on the unit circle centered at C(3, 4) that are closest and farthest from the origin.
(closest) Answer: (farthest) Answer: ˛ ˛ ˛ ˛15 ˛
5
Problem 4. (20 pts.) Z
1
Z
1
y y dx dy representing the double integral of the function f (x, y) = over x x 0 y the region D. Make a careful sketch of the region of integration and then evaluate this integral by switching the order of integration.
(a – 10 pts.) Consider the iterated integral
√
(ba) Answer: (b – 10 pts.)
Use polar coordinates to find
ZZ p
x2 + y 2 dA, where D is the unit disk centered
1
D
at the point (1, 0).
0.5
0
0.5
1
1.5
2
–0.5
–1
(b) Answer: ˛ ˛ ˛ ˛20 ˛
6
Problem 5. (10 pts.) Find the center of mass of the homogeneous (i.e., the density function σ(x, y) = σ is constant) lamina bounded by the standard parabola y = x2 and two straight lines as shown on the picture to the right.
4
3
2
1
–2
–1
0
1
2
x
Answer: ˛ ˛ ˛ ˛10 ˛
7
Problem 6. (20 pts.) Z
1
Z
2−2z
Z
4−y 2
f (x, y, z) dx dy dz and then (a – 10 pts.) Sketch the region of integration E for the iterated triple integral 0 0 0 ZZZ rewrite this integral as f (x, y, z) dz dy dx (i.e., find the appropriate limits of integration). E
(a) Answer: ZZZ (b – 10 pts.) Consider the triple integral f (P ) dV that can be written in spherical coordinates as the following iterated E π Z 2π Z 2 Z 3 integral: ρ2 sin φ dρ dφ dθ. What is the function f (P )? Describe (or sketch) the region of integration 0
π 6
0
E. Evaluate this integral and explain what it represents.
(b) Answer: ˛ ˛ ˛ ˛20 ˛
8
Problem 7. (10 pts.) The graph below is a plot of some level curves for a function f (x, y), along with arrows representing the gradient ∇(f ) (adjacent level curves represent the same change in f ). D is the region bounded by (and including) the oval constraint curve shown, which is given by g(x, y) =a constant. On the graph, please carefully mark and label the following: (a – 2 pts.) An example of a critical point that is a local minimum. (b – 2 pts.) An example of a critical point that is a local maximim. (c – 2 pts.) An example of a critical point that is a saddle. (d – 2 pts.) Location of the absolute maximum of f (x, y) on the region D. (e – 2 pts.) Location of the absolute minimum of f (x, y) on the region D.
6 4 2 0 y
–2 –4 –6 –8 –10 –10
–8
–6
–4
–2
0
2
4
6
x
˛ ˛ ˛ ˛10 ˛
Multivariable Calculus Math 215 Winter 2003
Second Exam March 20, 2003
Please do all your work in this booklet and show all the steps. Write your final answer in the corresponding box. Calculators and note-cards are not allowed.
Problem Possible points Score 1 10 2 15 3 15 4 20 5 10 6 20 7 10 Total 100
2
Problem 1. (10 pts.) (a – 3 pts.) Compute the gradient of the function f (x, y) = x3 y − xy 2 .
(a) Answer: (b – 3 pts.) Find the maximum rate of change of this function at the point P (3, 2).
(b) Answer: (c – 4 pts.) Find the rate of change of this function f (x, y) at the point Q(1, −2) in the direction of the vector v = h2, 4i.
(c) Answer: ˛ ˛ ˛ ˛10 ˛
x2 Problem 2. (15 pts.) Find and classify all critical points of the function f (x, y) = xy + y − − 2x + 3. 2 2
3
2
Answer: ˛ ˛ ˛ ˛15 ˛
4
Problem 3. (15 pts.) Using Lagrange multipliers, find the points on the unit circle centered at C(3, 4) that are closest and farthest from the origin.
(closest) Answer: (farthest) Answer: ˛ ˛ ˛ ˛15 ˛
5
Problem 4. (20 pts.) Z
1
Z
1
y y dx dy representing the double integral of the function f (x, y) = over x x 0 y the region D. Make a careful sketch of the region of integration and then evaluate this integral by switching the order of integration.
(a – 10 pts.) Consider the iterated integral
√
(ba) Answer: (b – 10 pts.)
Use polar coordinates to find
ZZ p
x2 + y 2 dA, where D is the unit disk centered
1
D
at the point (1, 0).
0.5
0
0.5
1
1.5
2
–0.5
–1
(b) Answer: ˛ ˛ ˛ ˛20 ˛
6
Problem 5. (10 pts.) Find the center of mass of the homogeneous (i.e., the density function σ(x, y) = σ is constant) lamina bounded by the standard parabola y = x2 and two straight lines as shown on the picture to the right.
4
3
2
1
–2
–1
0
1
2
x
Answer: ˛ ˛ ˛ ˛10 ˛
7
Problem 6. (20 pts.) Z
1
Z
2−2z
Z
4−y 2
f (x, y, z) dx dy dz and then (a – 10 pts.) Sketch the region of integration E for the iterated triple integral 0 0 0 ZZZ rewrite this integral as f (x, y, z) dz dy dx (i.e., find the appropriate limits of integration). E
(a) Answer: ZZZ (b – 10 pts.) Consider the triple integral f (P ) dV that can be written in spherical coordinates as the following iterated E π Z 2π Z 2 Z 3 integral: ρ2 sin φ dρ dφ dθ. What is the function f (P )? Describe (or sketch) the region of integration 0
π 6
0
E. Evaluate this integral and explain what it represents.
(b) Answer: ˛ ˛ ˛ ˛20 ˛
8
Problem 7. (10 pts.) The graph below is a plot of some level curves for a function f (x, y), along with arrows representing the gradient ∇(f ) (adjacent level curves represent the same change in f ). D is the region bounded by (and including) the oval constraint curve shown, which is given by g(x, y) =a constant. On the graph, please carefully mark and label the following: (a – 2 pts.) An example of a critical point that is a local minimum. (b – 2 pts.) An example of a critical point that is a local maximim. (c – 2 pts.) An example of a critical point that is a saddle. (d – 2 pts.) Location of the absolute maximum of f (x, y) on the region D. (e – 2 pts.) Location of the absolute minimum of f (x, y) on the region D.
6 4 2 0 y
–2 –4 –6 –8 –10 –10
–8
–6
–4
–2
0
2
4
6
x
˛ ˛ ˛ ˛10 ˛
