12/08/2021

MATH 12002, CALCULUS I, TEST 4 NAME:

To the Student:

� Make sure that your camera and microphone work. Absence of a working camera could invalidate your

test.

� As soon as we begin, download the test 4 �le from Blackboard. Once in place, you should

1. Turn your camera on and point it to show the area on your desk that you are going to be writing

on.

2. Turn your microphone o¤. Activate it only when you need to communicate with the instructor.

3. Keep item (1) on for the duration of the test.

� If you have access to a printer and scanner, print a copy of the test. Present the problems neatly in

the space provided. You can use your own scrap paper. Once you are done, you have half an hour to

make a clean scan of your completed exam into a pdf �le and email it to me. Do not send me any

scrap paper you might have used.

� If you do not have access to a printer or scanner, use your own paper to neatly write your solutions

to the problems in order. Once you are done, you have half an hour to take well-lit and clear pictures

of your work, paste them to a Microsoft Word �le, in order, and email the �le to me. Do not send me

any scrap paper you might have used.

� In either case, name the �le by last name �rst name course number assignment number. For example,

Smith George MATH 12002 Test 4.pdf or Smith George MATH 12002 Test 4.docx

� All answers should simpli�ed and exact. Do not use decimal numbers. Answers with no work will

receive no credit. Each part of each question is worth 10 points adding to a total of 180 points for the

test.

� Good Luck!

1

1. Let f(x) = x2 + 1, P = f�2;1;2;5g. Find the left and right Riemann Sums for f over the partition

P. What is kPk ?

2. Find

Z 1

0

(x2 +x)dx by taking the limit of Right Riemann Sums, over a uniform partition of [0;1].

3. Find lim

n!1

�

2n

nX

k=1

cos

k�

2n

2

4. By interpeting the integral in terms of area �nd

Z p2

�

p

2

p

2�x2dx

5. For the following, compute dy

dx

:

(a) y =

Z x

0

1

t3 +1

dt

(b) y =

Z x2+1

0

p

costdt

(c) y =

Z xex

1

sin�

�

d�

(d) y =

Z y sinx

0

et

2

dt

3

6. Compute the given integral.

(a)

Z 2

1

(3t2 �4t+7)dt

(b)

Z

tan4 xsec2 xdx

(c)

Z e

1

(x�

1

x

)dx

4

(d)

Z

x

p

x2 +1dx

(e)

Z

x

p

x�1dx

(f)

Z e2

e

lnx

x

dx

(g)

Z

x2

3

p

x3 +8

dx

5

7. Find the area of the region bounded by the graphs of f(x) = 8�x2 and g(x) = x2.

8. Compute the value of limx!0+

R p

x

0

t4 sintdt

x3

9. Suppose that f is continuous and

R 2

0

f (x)dx = 3. Compute

R 2

0

�

1+

R x

0

f (t)dt

�2

f (x)dx.

6

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