Math 224 Final ExamFall Quarter 2016

Dec 7, 2016

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Instructions

• This examination consists of 15 pages, not including this cover page. Verify that your copy of thisexamination contains all 15 pages. If your examination is missing any pages, then obtain a new copyimmediately.

• This examination consists of 10 questions for a total of 200 points.

• You have two hours to complete this examination.

• Do not use books, notes, calculators, computers, tablets or phones.

• Write legibly and only inside of the boxed region on each page.

• Cross out any work that you do not wish to have scored.

• Show all of your work. Unsupported answers may not earn credit.

Dec 7, 2016 Math 224 Final Exam Page 1 of 15

1. Determine whether each of the statements below is TRUE or FALSE. Indicate your answer by circlingthe appropriate choice. No explanation or justification is necessary.

(a) (3 points) If f(x) =R

x

0 e

t

4dt, then f is an increasing function.

True False

(b) (3 points) IfR 4

1 f(x) dx = �2, thenR 2

1 xf(x2) dx = �1.

True False

(c) (3 points) The improper integralR11

1x

⇡ dx converges.

True False

(d) (3 points) The series1X

n=1

1

2nhas value 2.

True False

(e) (3 points) If f(x) =P1

n=03n

n! x2n, then f

(7)(0) = 37.

True False

Dec 7, 2016 Math 224 Final Exam Page 2 of 15

2. (20 points) Let f be a di↵erentiable function. The following table gives some values of f(x).

x 0 1 2 3 4 5 6f(x) 2 3 5 7 11 13 17

Define the function g by g(x) =

Zx

2

x+1

tf(t) dt. Find g

0(2). Your answer should be a numerical expression,

but can be left unsimplified.

Page 2

Dec 7, 2016 Math 224 Final Exam Page 3 of 15

3. (15 points) Suppose f is a function satisfying

f

0(x) =1

x ln xand f(e) = 1.

Find f(x).

Page 3

Dec 7, 2016 Math 224 Final Exam Page 4 of 15

4. This problem has two parts; the second is on the next page. Evaluate the following integrals.

(a) (18 points)

Z 1

0

x

2e

2xdx

Page 4

Dec 7, 2016 Math 224 Final Exam Page 5 of 15

(b) (18 points)

Z1

x

2px

2 � 4dx

Page 5

Dec 7, 2016 Math 224 Final Exam Page 6 of 15

5. (20 points) Consider the region in the fourth quadrant (i.e. the portion of the xy-plane where x � 0and y 0) enclosed by the curves y = �x

2, y = �4, and x = 0. Find the volume of the solid obtained

by revolving this region around the line y = 4. Your answer should be a numerical expression, but doesnot have to be simplified.

Page 6

Dec 7, 2016 Math 224 Final Exam Page 7 of 15

6. Determine whether each of the following series converges or diverges.

(a) (9 points)1X

n=3

2 + cosn

n

3 + n

2 + 3

(b) (9 points)1X

n=1

sin

✓n

2 � 1

n

2 + 1

◆

Page 7

Dec 7, 2016 Math 224 Final Exam Page 8 of 15

7. (20 points) Find the radius and interval convergence of the following power series.

1X

n=1

(x� 1)n

3npn

Page 8

Dec 7, 2016 Math 224 Final Exam Page 9 of 15

8. (a) (8 points) Find the Taylor polynomial of degree 3 centered at 2 of the function f(x) = e

�x.

(b) (8 points) Find an interval centered at 2 on which the error in approximating e

�x by the Taylorpolynomial found above is at most 1

104 .

Page 9

Dec 7, 2016 Math 224 Final Exam Page 10 of 15

9. This question has two parts; the second is on the next page.

(a) (10 points) Find the Maclaurin series of the function f(x) = 1(1+x)2 . Hint: You can find a way to

do this by somehow manipulating the series1

1� y

=1X

n=0

y

n.

Page 10

Dec 7, 2016 Math 224 Final Exam Page 11 of 15

(b) (10 points) Evaluate the sum1X

n=1

(�1)n�1n

2n�1

by plugging in an appropriate value of x into the series found in part (a). Don’t forget to verifythat the value you plug in does lie within the interval of convergence.

Page 11

Dec 7, 2016 Math 224 Final Exam Page 12 of 15

10. Represent each of the following as a power series centered at 0.

(a) (10 points) x

2e

x

5

(b) (10 points)

Zsin x

x

dx

Page 12

Dec 7, 2016 Math 224 Final Exam Page 13 of 15

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Page 13

Dec 7, 2016 Math 224 Final Exam Page 14 of 15

YOU MUST SUBMIT THIS PAGE.

If you would like work on this page scored, then clearly indicate to which question the work belongs andindicate on the page containing the original question that there is work on this page to score.

Page 14

Dec 7, 2016 Math 224 Final Exam Page 15 of 15

DO NOT WRITE ON THIS PAGE.

Page 15