MATH 1B MIDTERM 1 (PRACTICE 3)

PROFESSOR PAULIN

DO NOT TURN OVER UNTILINSTRUCTED TO DO SO.

CALCULATORS ARE NOT PERMITTED

THIS EXAM WILL BE ELECTRONICALLYSCANNED. MAKE SURE YOU WRITE ALLSOLUTIONS IN THE SPACES PROVIDED.YOU MAY WRITE SOLUTIONS ON THEBLANK PAGE AT THE BACK BUT BESURE TO CLEARLY LABEL THEM

FormulaeZ

tan(x) dx = ln | sec(x)|+ C

Zsec(x) dx = ln | sec(x) + tan(x)|+ C

Z1

1 + x2dx = arctan(x) + C

Z1p

1� x2dx = arcsin(x) + C

d tan(x)

dx= sec

2(x)

d sec(x)

dx= tan(x) sec(x)

1 = sin2(x) + cos

2(x) 1 + tan

2(x) = sec

2(x)

cos2(x) =

1 + cos(2x)

2sin

2(x) =

1� cos(2x)

2

|EMidn | K(b� a)3

24n2|ESn |

K(b� a)5

180n4

Name:

Student ID:

GSI’s name:

Math 1B Midterm 1 (Practice 3)

This exam consists of 5 questions. Answer the questions in thespaces provided.

1. Compute the following integrals:

(a) (10 points) Zln(x)2 dx

Solution:

(b) (15 points) Ztan

5(x) sec�3

(x) dx

Solution:

PLEASE TURN OVER

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7Cx Luca g CK flu Laida alucard Jj Kdoc7 x 1x g ca x afula

J ucsc 2dx K Lucic 2Zulu Cnc 1 2x C

oddJ fans x see x die JsinsCd cos 2cal dxet u Casca DI sincx doc dda

sin Csc

J sins cos 2cal doc f l u'T u Z du

Ju Zz t uz du U t

Zu f us c Ccos se Zens x j s3Gc7 C

Math 1B Midterm 1 (Practice 3), Page 2 of 5

2. (20 points) Find the arc length of the the curve y = ln(cos(x)) between 0 and⇡3 .

Solution:

PLEASE TURN OVER

7cal tu casual 7 coal siYf tancas

Are length Vite da V c doc

secco dse it a ui co 3

Ln I see Cx fan xo

Ln I see Tg tan II Tu secco tantotu z Fs I

tI

Math 1B Midterm 1 (Practice 3), Page 3 of 5

3. (25 points) Compute the following integral:

Zx2

+ 3x+ 3

(x+ 1)3dx

Solution:

PLEASE TURN OVER

X 13 3 A B c A x 1112 Bex 111 Ct t

1 3 x 11 Cx1112 x 111 1 3

Axl c EA c D a A c Btc

x 1113

A x 13 3ZA 113 3 13 1 3 11

t1,3

At Btc 3 C L

I I

f x43 da lula 111 12 EX1,13

Math 1B Midterm 1 (Practice 3), Page 4 of 5

4. (a) (10 points) Use the Trapizoidal Rule with n = 4 to approximate the definite integral

Z 8

0

f(x) dx,

where f(x) takes the following values:

x 0 1 2 3 4 5 6 7 8

f(x) 0 2 4 3 1 4 5 5 3

Solution:

(b) (15 points) Assuming that |f 00(x)| 2, for all 0 < x < 8, how large an n would we

need to choose to guarantee that

|ETn | 0.01

Solution:

PLEASE TURN OVER

Xo x xz x acec s ODx

4 2

Ty 1 1Caco 276C 27 seal 27Gcs f Cxc

E o 2 4 t 2 I 2 S t 3 23

I f x E 2 ou co s IETUI EZic

12 n 2

Need Go choose a Saar that 2 83E O D i

IZ u

u Z z2 S soo

12

Need a to guarantee 1Etal E o o12

Math 1B Midterm 1 (Practice 3), Page 5 of 5

5. (25 points) Evaluate following improper integral:

Z 0

�1

(x+ 1)5

p(�x2 � 2x)

dx

If it is divergent, write divergent and explain your reasoning.

Solution:

END OF EXAM

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