Calculus II Final

7/24/2019 Calculus II Final

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Spring 2011 Math 262 ELAC FINAL Name: ______________________

Email/Phone: ________________Show all necessary work neatly, clearly, systematically, and understandably. For all answers, simplify as much as possible.

Total points available: 131 points (11 extra points included).

1. (15:5,5,5)For each of the following series, find its radius of convergence and interval of convergence:

a.

( )!"

=

#0

4

n

n

n

x

b.( )

!"

= +

#

0 12

3

n

n

n

n

x

c.( )

!"

=

+

2 ln

3

n

n

n

x


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2. (15:5,4,6)Find the limit:

a. !"#$

%& '

()1lim

1x

x

ex

b.11

sinlim

30!+

!

"

x

xx

x

c. ( )xx

xx

1

2

0

531lim ++!

3. (13:5,5,3)Find the derivative:

a. )xy 2coslog3=

b. xxysin

=

c.

21tan xy

!

=

4. (6:3,3)Consider ( ) ( )

2

41ln

x

xxf

+

= .

a. Write the Maclaurin series of ( )xf ingeneral term. Start with

( ) ( )!"

=

+

+

#=+

0

1

1

11ln

n

nn

n

x

x

b.

Use this series to find( )

( )014

f .


3/6

5. (30:6,4,4,5,6,5)Evaluate:

a. ! dxxex

3sin

b. ! dxxx ln

c. ! dxxx sectan33

d. ! dxxarcsin

e. !" 22 xx

dx

f.

! "++

dx

xx

x

2

54

2


4/6

6. (9:4,5)Find the length of each of the following curves:

a. ( )23

22

3

1+= yx , where 10 !! y

b. !2"= er , for 0!"

7. (8:4,4)Sketch the curve and find the area enclosed by !2cos2 +=r .


5/6

8. (8)Find the surface area generated by revolving the curve 3 3xy = , 20 !! y about the y-axis.

9.

(4)Find the sum of the series ( ) ( )!

"

#

+

#1

12

!2161n

n

n

n

n

$

10.

(22:4,3,4,4,7)Let!"#

=

=

tytx

3

3

sincos .

a. Find the x-ycoordinates of the points where the parametric equations will have horizontal orvertical tangents.


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Continue from #10 Let!"#

=

=

ty

tx3

3

sin

cos.

b. Find the second derivative for the parametric equations.

c. Sketch the graph

d. Find the perimeter of the region bounded by the curve.

e. Find the area of the region bounded by the curve.

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Calculus II Final