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Math 132 Final Exam: Calculus and Series, Exams of Calculus

Dr. Bhim Rao Ambedkar UniversityCalculus

This is the Exam of Calculus which includes Taylor Series, Proctor, Multiple, Borrow, Bubble Corresponding, Partial Credit, Evaluate, Converge, Oscillates etc. Key important points are: Derivative, Calculate, Respect, Sin, Arcsin, Cos, Tan, Series Converges, Determine, Reasoning

Typology: Exams

2012/2013

Uploaded on 02/26/2013

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Math 132
Fall 2007 Final Exam
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Download Math 132 Final Exam: Calculus and Series and more Exams Calculus in PDF only on Docsity!

Math 132 Fall 2007 Final Exam

  1. Calculate d

ฯ€ฯ€ฯ€ฯ€ 2 cos( x )sin( x )^3 x.

a) 1 b)

c)

d)

e)

f)

g)

h)

i)

j)

  1. Let F( x )==== d

x

2 5 ++++ t^4 1 ++ ++ t^3

t. Calculate the derivative D( F )( 2 ) of F at 2.

a) 4 b) 5 c) 6 d) 7 e) 8 f) โˆ’โˆ’โˆ’โˆ’ 4 g) โˆ’โˆ’โˆ’โˆ’ 5 h) โˆ’โˆ’โˆ’โˆ’ 6 i) โˆ’โˆ’โˆ’โˆ’ 7 j) โˆ’โˆ’โˆ’โˆ’ 8

  1. Calculate d

8 x^2 ++++ 2 x ++++ 6 ( 1 ++++ x )( 1 ++++ x^2 )

x.

a)

ln 2( ) b)

ln 2( ) c) ln 2( ) d) 2 ln 2( ) e) 3 ln 2( )

f) 4 ln 2( ) g) 5 ln 2( ) h) 6 ln 2( ) i) 7 ln 2( ) j) 8 ln 2( )

  1. Calculate d

e x^2 ln ( x ) x.

a)

e^3 b)

( 2 e^3 โˆ’ 1 ) c)

( e^3 โˆ’ 2 ) d)

( e^3 โˆ’ 1 ) e)

( 2 e^3 + 1 )

f)

( e^3 + 2 ) g)

( e^3 + 1 ) h)

( 2 e^3 + 1 ) i)

( e^3 + 2 ) j)

( e^3 + 1 )

  1. If y 0( )==== 0 and dy dx = cos( x ) 1 โˆ’โˆ’โˆ’โˆ’ y^2 , then what is y( x )?

a) sin( ฯ€ฯ€ฯ€ฯ€ cos( x )) b) sin( sin( x )) c)

cos ฯ€ฯ€ฯ€ฯ€^ cos(^ x )๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท 2 d)^ cos^ (^ sin(^ x^ ))-1^ e)^ arcsin(^ x )

f) arcsin( arcsin( x )) g) sin( tan( x )) h) tan( sin( x )) i) arcsin( tan( x )) j) arcsin( arctan( x ))

  1. Consider the following three statements about a series โˆ‘โˆ‘โˆ‘โˆ‘ n == == 1

an with positive terms:

I: The series converges because lim ==== n โ†’โ†’ โ†’โ†’โˆžโˆžโˆžโˆž

an 0.

II: The series converges because lim ==== n (^) โ†’โ†’ โ†’โ†’โˆžโˆžโˆžโˆž

a (^) n ++++ 1 bn^ 1.1^ and^

n ==== 1

bn converges.

III: The series converges because lim ==== n โ†’โ†’โ†’โ†’โˆžโˆžโˆžโˆž

a (^) n ++++ 1 an^1.

For each statement, determine whether the reasoning is correct or incorrect.

a) I: correct, II: correct, III: correct b) I: correct, II: correct, III: incorrect c) I: correct, II: incorrect, III: correct d) I: correct, II: incorrect, III: incorrect e) I: incorrect, II: correct, III: correct f) I: incorrect, II: correct, III: incorrect g) I: incorrect, II: incorrect, III: correct h) I: incorrect, II: incorrect, III: incorrect i) Wrong answer j) Bonus wrong answer

  1. Consider the three series

I: โˆ‘โˆ‘โˆ‘โˆ‘ n ==== 0

โˆžโˆžโˆžโˆž (^) n^5

3 n^

, II: โˆ‘โˆ‘โˆ‘โˆ‘

n ==== 0

โˆžโˆžโˆžโˆž 10 n n!

, and III: โˆ‘โˆ‘โˆ‘โˆ‘ n ==== 2

n ln( n )

and the statements

( C ) The series converges ( D ) The series diverges

For each series, decide which of statements (C), (D) is correct.

a) I: C, II: C, III: C b) I: C, II: C, III: D c) I: C, II: D, III: C d) I: C, II: D, III: D e) I: D, II: C, III: C f) I: D, II: C, III: D g) I: D, II: D, III: C h) I: D, II: D, III: D i) Wrong answer j) Bonus wrong answer

  1. Consider the two series

I: โˆ‘โˆ‘โˆ‘โˆ‘

n ==== 1

( โˆ’โˆ’โˆ’โˆ’ 1 ) n^

n ๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท 1 ++++ n

n and II: โˆ‘โˆ‘โˆ‘โˆ‘ n ==== 0

โˆžโˆžโˆžโˆž (^) ( โˆ’โˆ’โˆ’โˆ’ 1 ) n n

1 ++++ n

and the statements

( AC ) The series converges absolutely ( CC ) The series converges conditionally ( D ) The series diverges

For each series, decide which of statements (AC), (CC), (D) is correct. a) I: AC, II: AC b) I: AC, II: CC c) I: AC, II: D d) I: CC, II: AC e) I: CC, II: CC f) I: CC, II: D g) I: D, II: AC h) I: D, II: CC i) I: D, II: D j) Wrong answer

  1. Consider the two series

I: โˆ‘โˆ‘โˆ‘โˆ‘

n ==== 0

1 ++ ++ n^3 10 ++++ 100 n^2 ++++ n^3

n and II: โˆ‘โˆ‘โˆ‘โˆ‘ n (^) == == 1

3 ++++ n ๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท๏ฃท 3 n

n

and the statements

( C ) The Root Test establishes convergence ( D ) The Root Test establishes divergence ( F ) The Root Test is not conclusive.

Apply the Root Test to series I and II and for each, decide which of statements (C), (D), (F) is correct.

a) I: C, II: C b) I: C, II: D c) I: C, II: F d) I: D, II: C e) I: D, II: D f) I: D, II: F g) I: F, II: C h) I: F, II: D i) I: F, II: F j) Wrong answer

  1. Let f( x )====

x^3 e ( 2 x^2 )

. What is f

a) 20 b) 40 c) 60 d) 80 e) 100 f) 120 g) 140 h) 160 i) 180 j) 200

  1. Calculate the interval of convergence of โˆ‘โˆ‘โˆ‘โˆ‘ n ==== 0

โˆžโˆž โˆžโˆž (^) ( โˆ’โˆ’โˆ’โˆ’ 1 ) n^ ( x ++++ 3 ) n

n ++++ 1 4 n^

. Let R be the radius of

convergence. You will need to calculate the sum of four integers and it might help to record them as you go.

Let c be the base point of the power series. ( c = ________ )

Set ฯฯฯฯ = R if R is an integer and -1 otherwise. ( ฯฯฯฯ = ________ )

Set ฯƒฯƒฯƒฯƒ = 1 if the left endpoint belongs to the interval of convergence and 0 otherwise. ( ฯƒฯƒฯƒฯƒ =

________ )

Set ฯ„ฯ„ฯ„ฯ„ = 3 if the right endpoint belongs to the interval of convergence and 0 otherwise. ( ฯ„ฯ„ฯ„ฯ„ =

________ )

What is the value of c ++++ ฯฯฯฯ ++++ ฯƒฯƒฯƒฯƒ ++++ฯ„ฯ„ฯ„ฯ„?

a) -4 b) -3 c) -2 d) 2 e) 3 f) 4 g) 7 h) 8 i) 10 j) 11

  1. Let T( x ) be the degree 2 Taylor polynomial of ln( x ) with base point 2. What is T 3( ) โˆ’โˆ’โˆ’โˆ’ln 2( )?

a)

8 b)^

4 c)^

8 d)^

2 e)^

f)

g)

h) 1 i)

j)

  1. What is the coefficient of x^5 in the Maclaurin series of

8 x 4 โˆ’โˆ’โˆ’โˆ’ x^2

a)

16 b)^

16 c)^

8 d)^ โˆ’โˆ’โˆ’โˆ’^

8 e)^

f) โˆ’โˆ’โˆ’โˆ’

g)

h) โˆ’โˆ’โˆ’โˆ’

i) 2 j) โˆ’โˆ’โˆ’โˆ’ 2

  1. What is the coefficient of x^4 in the Maclaurin series of

( 1 ++++ x^2 )

a) โˆ’โˆ’โˆ’โˆ’

9 b)^

9 c)^ โˆ’โˆ’โˆ’โˆ’^

6 d)^

6 e)^ โˆ’โˆ’โˆ’โˆ’^

f)

g) โˆ’โˆ’โˆ’โˆ’

h)

i) โˆ’โˆ’โˆ’โˆ’

j)