NAME:

104 FALL 2011: CalculusMIDTERM III

INSTRUCTIONS:

1. WRITE YOUR NAME at the top, where it says “NAME.”

2. If you want a ‘preliminary’ exam grade, FILL IN the [SINGLE] letter of your solutionON THIS PAGE. Please mark your answer clearly.

3. To obtain credit, you MUST SHOW YOUR WORK. Prof. Ghrist may give partialcredit based on your work, even if your filled-in answer is wrong. Likewise, a rightanswer with poor/no work will not receive full credit.

4. There is to be no cheating. No kinda-sorta-cheating. No anything that even comesclose. If in doubt, don’t do it, for your honor (and likely more) is at stake. Play fair.

5. There are to be NO calculators, cell phones, or books, or notes during this exam.Use a pencil, an eraser, your sheet, and your head: but not in that order.

6. You have 50 minutes to complete the exam. Do the exam quickly, but read carefully!You gain nothing by doing a different problem than what is asked.

7. If you have a question, re-read carefully. These questions should not require clarifi-cation, and you will not be given hints or explanations. If you suspect an error inthe question or a true ambiguity, please raise your hand and someone will help youdirectly.

8. Best wishes.

ANSWERS:

PROBLEM 1 (A) (B) (C) (D) (E) (F) (G) (H)PROBLEM 2 (A) (B) (C) (D) (E) (F) (G) (H)PROBLEM 3 (A) (B) (C) (D) (E) (F) (G) (H)PROBLEM 4 (A) (B) (C) (D) (E) (F) (G) (H)PROBLEM 5 (A) (B) (C) (D) (E) (F) (G) (H)PROBLEM 6 (A) (B) (C) (D) (E) (F) (G) (H)PROBLEM 7 (A) (B) (C) (D) (E) (F) (G) (H)PROBLEM 8 (A) (B) (C) (D) (E) (F) (G) (H)

1

PROBLEM 1: Compute the mean µ of the following PDF (probability density function)

ρ(x) =3

2

√x

over the domain 0 ≤ x ≤ 1.

(A) µ =1

2

(B) µ =4

15

(C) µ = 1

(D) µ =5

3

(E) µ =2

5

(F) µ =2

3

(G) µ =3

5

(H) none of the above

2

PROBLEM 2: With which sequence (bn)∞n=1 could you best use the comparison test

to prove whether∞∑n=1

√n− 1

3n2 + 2n+ 4

converges or diverges? And...which is it?

(A) bn =1

3n2; converges

(B) bn =1

3n2; diverges

(C) bn =1

3n; converges

(D) bn =1

3n; diverges

(E) bn =n− 1

3n2 + 2n+ 4; converges

(F) bn =n− 1

3n2 + 2n+ 4; diverges

(G) bn =1

3n3/2; converges

(H) bn =1

3n3/2; diverges

3

PROBLEM 3: What is the radius of convergence of the following power series? Showwork!

∞∑n=0

n!

1 · 3 · 5 · · · (2n− 3) · (2n− 1)xn

(A) 0

(B)1

3

(C)1

2

(D) 1

(E) 2

(F) 3

(G) ∞(H) none of the above

4

PROBLEM 4: An aerosol spray releases spherical droplets. Each droplet has a radiuschosen uniformly at random between 1 and 3 micrometers. What is the average volumeof an aerosol droplet (units being cubic micrometers, naturally)?

(A)26π

3

(B)40π

3

(C)13π

3

(D)80π

9

(E) 9π

(F)80π

3

(G)20π

3

(H) none of the above

5

PROBLEM 5: Exactly one of the following series converges conditionally: which is it?CIRCLE ONLY ONE ANSWER! Explain your answer, but, please, use your headso as not to waste time on the others...

(A) 1 +1

3+

1

5+

1

7+

1

9+ . . .

(B)∞∑n=1

(−1)n

(C)∞∑n=1

n!

nn

(D)∞∑n=1

cos(nπ)

n

(E)∞∑n=1

(−1)n2n

n!

(F)∞∑n=1

(−1)n21/n

(G)∞∑n=1

(lnn)n

(n+ 1)n

(H)∞∑n=1

√n2 + 1

(n+ 2)2

6

PROBLEM 6: Which of these integrals computes the y-coordinate, y, of the centroidof the region x2 + y2 ≤ 1; x, y ≥ 0? CIRCLE ONLY ONE ANSWER, and give yourreasons.

(A)

∫ 1

x=0

1

2x2√1− x2dx

(B)4

π

∫ 1

x=0

2√1− x2dx

(C)1

π

∫ 1

x=0

x√1− x2dx

(D)

∫ 1

x=0

1

2

√1− x2dx

(E)4

π

∫ 1

x=0

1

2(1− x2)dx

(F)4

π

∫ 1

x=0

1

2x2√1− x2dx

(G)

∫ 1

x=0

1

2(1− x2)dx

(H) none of the above

7

PROBLEM 7: One of the following series diverges. Which is it? CIRCLE ONLY ONEANSWER, and give your reasons.

(A)∞∑n=1

cosn

n!

(B)∞∑n=1

n+ 1

n2 + 2

(C)∞∑n=1

(−1)nn22n

n!

(D)∞∑n=2

2n

(lnn)n

(E)∞∑n=1

(−1)nsin 1

n

n

(F)∞∑n=1

n3

3n

(G)∞∑n=1

(−1)n3n+ 1

n2 + 2

(H)∞∑n=1

cos(nπ)

n

8

PROBLEM 8: Compute

limn→∞

ln(3n2 + 1)− ln(3 + 2n2)

(A) ∞(B) − ln 3

(C)1

3

(D) ln 3− ln 2

(E)2

3

(F) 0

(G)3

2

(H) the limit does not exist

9