Question #1 Mu Bowl____________________________________Mu National Convention 2012
Let f ( x) 
1
.
x
Let B be the average value of the function on the interval (1, 9) .
Let A be the solid generated when the region bounded by y  f ( x) and the vertical lines x  1 , x  2 is rotated
about the x -axis.
Let C be the volume of y  f ( x) rotated about the y -axis , the line x 
Evaluate
1
, and y  1 .
4
A
 B  C .
ln 4
Question #1 Mu Bowl____________________________________Mu National Convention 2012
Let f ( x) 
1
.
x
Let B be the average value of the function on the interval (1, 9) .
Let A be the solid generated when the region bounded by y  f ( x) and the vertical lines x  1 , x  2 is rotated
about the x -axis.
Let C be the volume of y  f ( x) rotated about the y -axis , the line x 
Evaluate
A
 B  C .
ln 4
1
, and y  1 .
4
Question #2 Mu Bowl____________________________________Mu National Convention 2012
5
Evaluate:

5  4x  x 2 dx
2
Question #2 Mu Bowl____________________________________Mu National Convention 2012
5
Evaluate:

2
5  4x  x 2 dx
Question #3 Mu Bowl___________________________________Mu National Convention 2012
Evaluate each of the following limits. If the limit does not exist, assign it a value of -1. If the limit exists, assign
the value the limit converges to. Find the sum of all of the limit values.
1)
1
lim
x0
3  10
lim
2)
3)
4)
x
lim
x
1
( )
x
1
x
(100 xe  100 x)
( 16 x 2  x  4 x)
1
1
(  2
)
x0 x x  x
lim
Question #3 Mu Bowl____________________________________Mu National Convention 2012
Evaluate each of the following limits. If the limit does not exist, assign it a value of -1. If the limit exists, assign
the value the limit converges to. Find the sum of all of the limit values.
1)
1
lim
x0
3  10
lim
2)
3)
4)
x
lim
x
1
( )
x
1
x
(100 xe  100 x)
( 16 x 2  x  4 x)
1
1
(  2
)
x0 x x  x
lim
Question #4 Mu Bowl____________________________________Mu National Convention 2012
Evaluate each of the following integrals. If the integral converges, assign the integral the value of what the
integral converges to. If the integral diverges, assign the integral a value of -2. Find the sum of the values of all
of the integrals.

1)
20ln( x)
1 x dx
3
dx
2) 
x 1
0
1
3)
 ln xdx
0
Question #4 Mu Bowl____________________________________Mu National Convention 2012
Evaluate each of the following integrals. If the integral converges, assign the integral the value of what the
integral converges to. If the integral diverges, assign the integral a value of -2. Find the sum of the values of all
of the integrals.

1)
20ln( x)
dx
x
1

3
2)
dx
 x 1
0
1
3)
 ln xdx
0
Question #5 Mu Bowl____________________________________Mu National Convention 2012
Consider the parametric equations:
x  9  t2
y  t2  t3
.
Let S be the value of the derivative of the function when t  2 .
Let A be the value of the 2nd derivative of the function when t 
1
.
8
Let the interval ( X ,  ) be the values of t for which the function is concave upward.
Evaluate S  A  X .
Question #5 Mu Bowl____________________________________Mu National Convention 2012
Consider the parametric equations:
x  9  t2
y  t2  t3
.
Let S be the value of the derivative of the function when t  2 .
Let A be the value of the 2nd derivative of the function when t 
1
.
8
Let the interval ( X ,  ) be the values of t for which the function is concave upward.
Evaluate S  A  X .
Question #6 Mu Bowl____________________________________Mu National Convention 2012
Let A  the slope of the curve y3  xy 2  4 when y  2 .
Let B  the minimum value of the slope of the curve y  x5  x3  2 x .
The points on the curve x 2  y 2  4 closest to the point (6, 0) are expressed as (C ,  D ) .
Evaluate 4A  D  C  B
Question #6 Mu Bowl____________________________________Mu National Convention 2012
Let A  the slope of the curve y3  xy 2  4 when y  2 .
Let B  the minimum value of the slope of the curve y  x5  x3  2 x .
The points on the curve x 2  y 2  4 closest to the point (6,0) are expressed as (C ,  D ) .
Evaluate
4A  D  C  B
Question #7 Mu Bowl____________________________________Mu National Convention 2012
Diego is a master at sequences and series. When Pamela asks him to help him with the following question:

“ Consider the series
 (e
(3/ n )
n 1
 e(3/( n 1)) ) . If this series converges, find the sum. If it diverges, write DIVERGES”
Diego will put what as the correct answer?
Question #7 Mu Bowl____________________________________Mu National Convention 2012
Diego is a master at sequences and series. When Pamela asks him to help him with the following question:

“ Consider the series
 (e
n 1
(3/ n )
 e(3/( n 1)) ) . If this series converges, find the sum. If it diverges, write DIVERGES”
Diego will put what as the correct answer?
Question #8 Mu Bowl____________________________________Mu National Convention 2012
Daniel Kang is the integration master. When Daniel is asked to find
ln 2
1
 1 e
x
dx
0
his answer is F . What is the value of
eF ?
Question #8 Mu Bowl____________________________________Mu National Convention 2012
Daniel Kang is the integration master. When Daniel is asked to find
ln 2
1
 1 e
x
dx
0
his answer is F . What is the value of
eF ?
Question #9 Mu Bowl____________________________________Mu National Convention 2012
1
Evaluate:
8
 xx
0
x
dx
Question #9 Mu Bowl____________________________________Mu National Convention 2012
1
Evaluate:
8
0 x  x x dx
Question #10 Mu Bowl___________________________________Mu National Convention 2012
The region bounded by the curve y  x 2 and the line y  4 are rotated about the line y  4 . Its volume is expressed
as B .
The region bounded by the curve y  x 2 and the line y  4 are rotated about the x  axis . Its volume is expressed as
C .
The region bounded by the curve y  x 2 , x  2 , and y  0 are rotated about the line x  2 . This volume is expressed
as D .
Find B  C  D .
Question #10 Mu Bowl___________________________________Mu National Convention 2012
The region bounded by the curve y  x 2 and the line y  4 are rotated about the line y  4 . Its volume is expressed
as B .
The region bounded by the curve y  x 2 and the line y  4 are rotated about the x  axis . Its volume is expressed as
C .
The region bounded by the curve y  x 2 , x  2 , and y  0 are rotated about the line x  2 . This volume is expressed
as D .
Find B  C  D .
Question #11 Mu Bowl___________________________________Mu National Convention 2012
Evaluate
 tan 1 ( x)  x
(hint: use your knowledge of series)
x3
x0
lim
Question #11 Mu Bowl___________________________________Mu National Convention 2012
 tan 1 ( x)  x
Evaluate
(hint: use your knowledge of series)
x3
x0
lim
Question #12 Mu Bowl___________________________________Mu National Convention 2012
If the area enclosed by the inner loop of the graph r  1  2sin  can be expressed as R 
I C
, find the value
K
of RICK .
Question #12 Mu Bowl___________________________________Mu National Convention 2012
If the area enclosed by the inner loop of the graph r  1  2sin  can be expressed as R 
of RICK .
I C
, find the value
K
Question #13 Mu Bowl___________________________________Mu National Convention 2012
1
Using the Maclaurin series for

x
e x , approximate the value of e dx
using the first three terms.
0
Question #13 Mu Bowl___________________________________Mu National Convention 2012
1
Using the Maclaurin series for
e
x
, approximate the value of
x
e
 dx
0
using the first three terms.
Question #14 Mu Bowl___________________________________Mu National Convention 2012
Find the value of 2012th derivative of sin x at 
Question #14 Mu Bowl___________________________________Mu National Convention 2012
Find the value of 2012th derivative of sin x at 