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2. Find the slope of the tangent line to the polar curve r

cos 2

when

3

1

3. Use the integral test to determine if

n 2

and justify your answer.

n ln n

2

is convergent or divergent. Explain

x 1

4. Find the radius of convergence of

n 3

5n n

n

. Justify your answer.

5. Use one of your comparison tests to decide if

n

answer.

n 1

converges. Justify your

2n 2 7

3 n

6

6. Find the area of the shaded region of the graph of r 1 sin

an , sn is the n th partial sum of s , and the terms of the series

7.(24 pts) Suppose S

n 1

n 1

are defined by an

2 3

5n

. (a. and b. are 5 points each, c. and d. are 7 points each)

a. Write out the first 3 terms of the sequence an (you don’t need to simplify your work).

a1

a2

a3

b. Write out the first 4 terms of the sequence sn (you don’t need to simplify your work).

s1

s3

s2

s4

c. Determine if the sequence an converges or diverges. If it converges, find the limit.

Justify your answer.

d. Determine if the series S converges or diverges. If it converges, find the limit.

Justify your answer.

x2

8. (5 points each) The figure below shows the graphs of f x 2 x , and g x

.

4

Also shown are regions R1 , R2, and R3. Use the figure to answer the following questions.

In each case set-up your integral answer but do not evaluate it.

a. Set up the integral or integrals to find

the area of the region R2.

b.

Set up the integral or integrals needed to find the volume formed by rotating R1

.about

the y-axis.

c. Set up the integral or integrals needed to find the volume formed by rotating R2 .about

the x-axis.

8) For work with sequences and series

i) For showing the convergence or divergence of a sequence show the

steps you used to find the limit. Then state your conclusion.

ii) If you are finding the sum of a geometric series,

ar n 1 or

n 1

ar n , make

n 0

sure you show how you found the value of a and r .

iii) For testing the convergence or divergence of a series

i) Name the test you are using and explain how your results from using

that test justifies your answer.

ii) For a geometric series make sure you show the value of r and explain

what that means for the convergence of the series.

9) You may print a copy of the test and do your work on it or copy and do the

problems, in order, on notebook paper. I want at most two problems on

one letter size sheet of paper. Space your work out so I can comment if

needed. You should have plenty of room to show your work. Do not cramp

your work into a tiny spot. If I cannot follow your work, it is wrong. Expand

it so I can follow your thinking on the problem. Some problems will be very

short. Others may take a lot of space. That is fine. Two problems at most

per side of paper!

10)Make sure what you write is readable on the pdf you upload! Write neatly

using a pencil or pen that is dark enough to show up well on the scanned

pdf. When you scan your exam, lighting is important. Check your scanned

test to verify that it is complete and legible before you upload it to

Canvas.

11)When you scan your work make sure your scanner is in “batch mode”. That

way I get one pdf file not a separate file for each page. The pages should be

the same order as the pdf test document.

12)Late work is penalized at 10% ( a letter grade ) per day.

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