University of Pennsylvania Math 104 Final Exam Fall 2017

First and Last Name ____________________________________(PRINT) Penn ID_________________ Professor (circle one): Ghini-Bettiol Sergel Block Gressman Rimmer

Recitation number _____________________________

There are fifteen questions on this examination. No calculators are allowed, but you may use one standard sized 8.5”X11” sheet with notes handwritten on both sides. Show your work in the space provided, and then transfer your answers carefully to this sheet. It is important to show your work because we will be going back over it – you might gain additional partial credit for substantial progress toward the solution of a problem, or you might lose credit for an unsubstantiated correct answer. Please put away and silence (don’t set to vibrate) all electronic devices (computers, tablets, cell phones, mp3 players), use of these are forbidden during the examination period. Good luck!

My signature below certifies that I have complied with the University of Pennsylvania's Code of Academic Integrity in completing this examination. In particular, all the work on this test is my own. _________________________________________ Signature

1. (A) (B) (C) (D) (E) (F) 9. (A) (B) (C) (D) (E) (F)

2. (A) (B) (C) (D) (E) (F) 10. (A) (B) (C) (D) (E) (F)

3. (A) (B) (C) (D) (E) (F) 11. (A) (B) (C) (D) (E) (F)

4. (A) (B) (C) (D) (E) (F) 12. (A) (B) (C) (D) (E) (F)

5. (A) (B) (C) (D) (E) (F) 13. (A) (B) (C) (D) (E) (F)

6. (A) (B) (C) (D) (E) (F) 14. (A) (B) (C) (D) (E) (F)

7. (A) (B) (C) (D) (E) (F) 15. (A) (B) (C) (D) (E) (F)

8. (A) (B) (C) (D) (E) (F)

1. Find the volume of the solid generated by revolving the region bounded

above by sin and bounded below 0 for 0 about the line .y x y x x

(a) 2 (b) 22 (c) 24 (d) 2

2

(e)

2

4

(f) None of these

2. Find the volume of the solid generated by revolving the region bounded

above by sec and bounded below 0 for 0 about the axis.3

y x y x x

(a) (b) 2 (c) 3 (d) 3 (e) 4 (f) None of these

4

2

13. Let + . Find the arclength for 1 2.

16 2

xy x

x

(a) 5

7 (b)

6

7 (c)

5

6 (d)

7

4 (e)

7

16 (f) None of these

4. Evaluate

2 2

21

1.

x xdx

x x

(a) 0 (b) 1 (c) 4

1 ln3

(d) 2 (e) 8

2 ln3

(f) None of these

5. Evaluate

2

2

1

ln 1 .x x dx

(a) 0 (b) 1 (c) ln 2 (d) 1

2 (e) 1

ln 22

(f) None of these

6. Evaluate

3

3/220

.25

dx

x

(a) 0 (b) 1

100 (c)

3

100 (d)

5

100 (e)

7

100 (f) None of these

7. Let be the solution to the initial value problemy x

2 sin with 0dy

x y x x ydx

What is 2 ?y

(a) (b) 2 (c) 4 (d) 0 (e) 2 (f) 4

8. Consider the initial value problem

21 2 with 0 2.dy

x y ydx

What is the lim ?x

y x

(a) 2e (b) /22e (c) /42e (d) 1 (e) 0 (f) e

9. Let

2 2 / 0

0 0

r bCr e rf r

r

Find so that is a probability density function pdf

for the random variable , is a constant .

This is used to model the distance between the nucleus and the electron

in a hydrogen atom. With 0,

C

r b

b it is called the Bohr length.

Find the mean of this pdf.

(a) 3

, mean4

bC b (b) 2

4, meanC b

b (c) 24

, meanC bb

(d) 3

4 3, mean

2C b

b (e) 2

2

4 3, mean

2C b

b (f) 34 3

, mean2

C bb

10. Find the limit of the sequence

ln 3 lnna n n n

(a) 0 (b) 1 (c) ln 3 (d) 3 (e) (f) the limit does not exist

11. Determine which of the following series are convergent.

For full credit be sure to explain your reasoning and tell what test was used.

2

42 2 2

2ln2

2 !

nn

n n n

n n nI ne II III

n n

(a) only I (b) only and I II (c) only and I III

(d) only II (e) only and II III (f) only III

12. Determine whether the following series convergent absolutely ,

converge conditionally , or diverge . For full credit be sure

to explain your reasoning and tell what test was used.

A

C D

2

2 2

1 2 1

3

n nn

nn n n

(a) both A (b) one , the other A C (c) one , the other A D

(d) both C (e) one , the other C D (f) both D

13. Find the interval of convergence of the power series

3

2

2 5.

nn

n

x

n

(a) 9112 2, (b) 911

2 2, (c) 9112 2, (d) 9 11

2 2, (e) 9 112 2, (f) 9 11

2 2,

0

14. Use the Taylor polynomial of degree 3 for ln 1

3centered at 0 to approximate the value of ln .

2

f x x

x

3ln

2

(a) 2

3 (b)

3

2 (c)

15

4 (d)

5

12 (e)

9

24 (f)

11

24

30

15. Let be the unique function that satisfies 0 0 and

1sin for all . Find the Taylor Series of centered at 0.

F x F

F x x x F x xx

(a)

6 3

0

1

2 1 !

n n

n

x

n

(d)

6 2

0

1

2 1 !

n n

n

x

n

(b)

6 2

0

1 6 3

2 1 !

n n

n

n x

n

(e)

6 2

0

1 6 2

2 1 !

n n

n

n x

n

(c)

6 3

0

1

6 3 2 1 !

n n

n

x

n n

(f)

2 3

0

1

6 3 2 1 !

n n

n

x

n n

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Scrap Paper If you use this page and intend for it to be graded, then you must indicate so on the page with the original problem on it. On this page, make sure you label your work with the corresponding problem number.

Do NOT rip this page off.