Exam 2
Math 16C, Summer Session I 2020
Due August 1, 2020 at 6:00pm PST (GMT -7)
”Bears are hype in some instances, but not so much in others.” - Raymond
Chou
1 Those Lagrange Multipliers Sure Can Multi-
ply All Right
Use the method of Lagrange Multipliers for the following optimization prob-
lems. (20 points total)
1.1 (9 points): People like it when things are optimal. Optimize the func-
tion f(x, y, z) = x2y2z2 according to the constraint 2x + 3y + 5z = 9.
1.2 (9 points): Boxes are pretty cool. Consider a lidless box with dimensions x,
y, and z. If this box must have surface area of 60 sq. units, find the dimensions
that give the maximum volume.
1.3 (2 points): A rectangular prism in four dimensions with side lengths x,
y, z, w has volume V = wxyz. If this rectangular prism is subject to the con-
straints xyz+wxy+wxz+wyz = 10 and xy−yz+zw−wx = 20, write, but do
not solve a Lagrange multiplier function for this scenario. (Find L(x, y, z, w);
I think I once used the notation F(x, y, z, w) for this as well)
2 Dubious Double Integrals
So I heard you like double integrals, so I put some double integrals on the exam.
(10 points total)
2.1 (3 points):
∫ 1
0
∫ x
x2
xy dy dx
2.2 (3 points):
∫ 2π
π
∫ π
0
(sin(x) + cos(y)) dy dx
2.3 (4 points):
∫ 4
1
∫√x
0
3
2e
y
√
x dy dx
1
3 Impossible Integrals and Rambunctious Re-
versals
XXXXXXXXXXpoints): The following integral, as is, is impossible to compute:∫ 1
0
∫ 1
x
ey
2
dy dx
I don’t like giving impossible problems, and as it turns out, there is a way to
solve this problem. Reverse the order of integration and solve this integral.
(You may use a graphing utility to visualize the region if necessary)
3.2 (3 points Extra Credit): Solve the following integral by any means.∫ 8
0
∫ 2
3
√
x
1
y4 + 1
dy dx
4 Abho
ent Areas and Vile Volumes?
So now we’re going to compute some volumes over some areas! (10 points)
4.1 (5 points): Compute the volume bounded between the xy-plane and the
surface z = 1(x+1)2(y+2)2 in the region x ≥ 0, y ≥ 0.
4.2 (5 points): Compute the volume of the region bounded below by the xy-
plane and above by z = 2y1+x2 , whose shadow on the xy-plane is bounded by the
curves y =
√
x, y = 0, x = 25.
5 Convergence Conundrums
Determine whether the following series converge or diverge using the tests we
learned. Please explicitly state which test you use (20 points total)
4.1 (4 points):
∑∞
n=0
1
7n
4.2 (4 points):
∑∞
n=1
n
n+1
4.3 (4 points):
∑∞
n=0
1
n!
4.4 (4 points):
∑∞
n=0 n(
4
5 )
n
4.5 (4 points):
∑∞
n=1
1
n2−1
4.6 (2 points Extra Credit):
∑∞
n=2
ln(n)
n1.5
2
6 Turmoil and Taylor Series
So now comes the part where we get to be tortured by Taylor series. Please
show your work. (Make a chart!) (20 points)
XXXXXXXXXXpoints): Compute the Taylor polynomial for f(x) = sin(x) with a = 0.
Write the first five terms, then express concisely in sigma notation.
XXXXXXXXXXpoints): Compute the Taylor polynomial for ln(x) with a = 1. Write
the first five terms, then express concisely in sigma notation.
7 Miscellaneous Mischief
I needed to fill 10 more points worth of problems. (10 points)
7.1 (3 points): Does the sum
∑∞
n=0 3(
99
100 )
n converge? If so, can you com-
pute its sum? If you can, compute it.
7.2 (2 points): Does the sum
∑∞
n=0
n0.3
n+1 converge? If so, can you compute
its sum? If you can, compute it.
7.3 (5 points): Use the ratio test to determine whether the sum
∞∑
n=0
(−1)nn!3n
(2n)!
converges or not.
3