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Calculus I Exam 3-A and 3-B by L. Ballou - Prof. Lynda L. Ballou, Exams of Analytical Geometry and Calculus

New Mexico Institute of Mining & Technology (NMT)Analytical Geometry and Calculus
Prof. Lynda L. Ballou

Two calculus exams, exam 3-a and exam 3-b, from math 131 - calculus i by l. Ballou. The exams cover various topics including limits, derivatives, integrals, asymptotes, maximum and minimum values, and graphical analysis. Students are required to find linearizations, evaluate limits, find asymptotes, maximize and minimize functions, and graph functions satisfying certain conditions.

Typology: Exams

Pre 2010

Uploaded on 08/08/2009

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Exam 3-A
L. Ballou Name___________________________
Math 131 Calculus I November 12, 2008
Part 1: No Calculator! Show all work!
(5 points each)
1. Find the linearization of
( )
21 3.1
1
fx x
x
= + +โˆ’
โˆ’
at
0x=
2. For the function,
( )
24fx x=โˆ’+
on
[ ]
1, 0โˆ’
, show that there exists a
( )
1, 0cโˆˆโˆ’
such that
( )
'1fc=
.
3. Evaluate
2
0
1
lim
x
x
ex
x
โ†’
โˆ’โˆ’
4. Evaluate
(
)
2
ln
lim
x
x
x
โ†’โˆž
5. Evaluate
pf3
pf4
pf5
pf8

Partial preview of the text

Download Calculus I Exam 3-A and 3-B by L. Ballou - Prof. Lynda L. Ballou and more Exams Analytical Geometry and Calculus in PDF only on Docsity!

Exam 3-A

L. Ballou Name___________________________

Math 131 Calculus I November 12, 2008

Part 1: No Calculator! Show all work!

(5 points each)

1. Find the linearization of ( )

f x x x

at x = 0

2. For the function, ( )

2

f x = โˆ’ x + 4 on [ โˆ’1, 0 ], show that there exists a c โˆˆ โˆ’( 1, 0 ) such that

f ' ( ) c = 1.

  1. Evaluate 0 2

lim

x

x

e x

โ†’ x

  1. Evaluate

2 ln lim x

x

โ†’โˆž x

5. Evaluate ( )

1

0

lim ln

x

x

e x

โˆ’

โ†’

Exam 3-A

L. Ballou Name___________________________

Math 131 Calculus I November 12, 2008

Part 2: Show all work for full credit!

  1. (10 points) Evaluate

ln 2/ 1 ln( ) lim

x

x

x

โ†’โˆž

2. (10 points)If you graph ( )

5/3 2/ f x = x โˆ’ 5 x , describe (or show) what the graph will look like near

x = 0. Explain why!

3. (10 points)Find all asymptotes of ( )

2

2

x x f x x

, using calculus.

  1. (5 points) Find the absolute maximum and minimum of

3 2

y = x โˆ’ 2 x on [ โˆ’1,5]

  1. (10 points) Graph a function that satisfies the following:

a. lim ( ) 3

x

f x โ†’ยฑโˆž

= and f ( 0 ) = 0

b. ( )

2

lim x

f x โ†’+

2

lim x

f x โ†’โˆ’

2

lim x

f x โ†’โˆ’+

= โˆ’โˆž and ( )

2

lim x

f x โ†’โˆ’โˆ’

c. f ' ( x ) > 0 on ( โˆ’โˆž โˆ’, 2 ) ๏• ( โˆ’2, 0)and f ' ( x ) < 0 on ( 0, 2) ๏•( 2,โˆž)

d. f " ( x ) > 0 on ( โˆ’โˆž โˆ’, 2 ) ๏• ( 2,โˆž)and f " ( x ) < 0 on ( โˆ’2, 2 ).

  1. (10 points)Find the height and radius of the largest right circular cylinder that can be put in a

sphere of radius R.

Exam 3-B

L. Ballou Name___________________________

Math 131 Calculus I November 12, 2008

Part 1: No Calculator! Show all work!

(5 points each)

1. Find the linearization of f ( x ) = 1 + x + sin x โˆ’ 0.5at x = 0

2. For the function, ( )

2

f x = 2 x + 1 on [ 0, 2 ], show that there exists a c โˆˆ ( 0, 2)such that f ' ( ) c = 4.

  1. Evaluate 0 3

lim

x

x

e

โ†’ x

  1. Evaluate

lim ln 1 2 x x

x

โ†’โˆž (^) + e

  1. Evaluate 2

lim sec 7 cos 3 x

x x ฯ€ โˆ’ โ†’

5. (20 points)For ( ) ( )

2 3

x f x = x โˆ’ e

a. Find intervals where f is increasing and decreasing.

b. Find intervals where f is concave up and concave down.

c. Find the local maximum and minimum values of f.

d. Find the points of inflections of f.

  1. (10 points) Graph a function that satisfies the following:

a. lim ( ) 2

x

f x โ†’ยฑโˆž

= and f ( 0 )= 0

b. ( )

2

lim x

f x โ†’+

= โˆ’โˆž and ( )

2

lim x

f x โ†’โˆ’

c. f ' ( x ) > 0 on (1, 2 ) ๏• ( 2,โˆž)and f ' ( x ) < 0 on ( โˆ’โˆž,1)

d. f " ( x ) > 0 on ( 0, 2)and f "( x ) < 0 on ( โˆ’โˆž, 0 ) ๏• ( 2,โˆž).

  1. (10 points) Find the height and radius of the largest right circular cone that can be put in a sphere

of radius R.