# What is the area of the solid formed by taking a sphere of radius by vmarcelo

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```									                          Advanced Calculus Test
2007 Mu Alpha Theta National Convention
=====================================================================

For all questions, answer E. "NOTA" means none of the above answers is correct.

 /3

 cos
2
1. Evaluate:                    xdx
0

4  3 3                               2  3 3                            4  3 3
A)                                     B)                                  C)
24                                     12                                 12
2  3 3
D)                                     E) NOTA
6

2. What is the area enclosed by the polar curve r  cos 3 ?

                                                       
A)                    B)               C)                 D)               E) NOTA
12                      8              4                  2

3. Let f ( x , y )  xe y  xy 2 . What is  f ( 2 ,1) (that is, the gradient of f at (2,1))?

A) e  1 x   2 e  4  y
ˆ              ˆ           B)  2 e  1 x  e  4  y
ˆ            ˆ        C) e  4  x   2 e  1 y
ˆ              ˆ
D)  2 e  4  x  e  1 y
ˆ           ˆ           E) NOTA

4. What is the surface area formed by rotating the curve y  x 2 on the domain x  [ 0 , 2 ]
around the y-axis?

17 17  1                              17 17                               34 17
A)                                     B)                                  C)
12                                    12                                 3
34 17  2
D)                                     E) NOTA
3

5. What is the volume of the solid formed by taking a sphere of radius 3 and excluding any
volume within 60 degrees of a plane passing through the sphere’s center?

A) 18  2  3                   B) 9 3                 C) 18 
D) 18  3                         E) NOTA
                                                                         
6. Given that F ( x , y , z )  ( x  y ) x  z 2 y  xy z z , what is   F (a.k.a. div F ) evaluated at
ˆ       ˆ        ˆ
the point ( 2 ,  1,1) ?

A) 0                              B) 3                    C) x  y  2 z
ˆ ˆ       ˆ
D) 2 x  z
ˆ ˆ                          E) NOTA
2007 Mu Alpha Theta National Convention
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                                                                    
7. Let f ( t )  sin t x  cos t y  t z . What is f ' ' (   / 4 ) ?
ˆ         ˆ     ˆ

2               2                                           2            2                           2           2
A)            x
ˆ                   ˆ
y                          B)            x
ˆ           ˆ
y                 C)        x
ˆ           y z
ˆ ˆ
2                2                                          2               2                          2            2
2               2
D)            x
ˆ                   y z
ˆ ˆ                        E) NOTA
2                2

8. What is the volume of the solid formed by taking an ellipse with semimajor axes of
length 4 and semiminor axes of length 2 centered at ( 4 , 6 ) with its major axis parallel to
the line y  5 x  17 and rotating it around the line y  x ?

A) 20  2 2                                     B) 40  2 2                            C) 60    2
2
D) 80  2 2                                     E) NOTA

9. What is the approximate value of ln 2 found using the first 4 nonzero terms of the Taylor
polynomial for ln x centered at x  1 (round to the nearest thousandth)?

A) .583                           B) .693                    C) .783                   D) .833           E) NOTA



x
2
10. Evaluate:                         cos xdx
 /2

         8  8                                                     4  4                                   8  8
2                                                            2                                           2

A)                                                           B)                                          C)
4                                                               4                                       4
         8  8
2

D)                                                           E) NOTA
4

11. Let f ( x , y )  x 2 y and consider the triangular region A with vertices ( 0 , 0 ) , (1,0 ) , and
( 0 ,1) . Evaluate:                      f ( x , y ) dA
A

1                                    1                      1                         1
A)                                B)                         C)                        D)                E) NOTA
60                                    30                     10                        5

1

      1  x dx
2
12. Evaluate:
0

                                                             2                      
A)                                B) 1                       C)                        D)                E) NOTA
4                                                                4                    2
2007 Mu Alpha Theta National Convention
=====================================================================

13. Let the position function of a particle, f (t ) , be defined as follows:

f ( t )  t x  t y  cos t z . What is the speed of the particle at time t 
ˆ      ˆ
2
ˆ                                                                                                        ?
2

A)       
2
1                                  B)                                    C)        
2
1
        2
2
D)                                                    E) NOTA

x log   a
x
14. For positive integers a and b evaluate: lim
x    x log       x
b

1
A)                                B) log a b                      C) log b a                 D) a log b a               E) NOTA
log      a
b

15. The centroid of the region bound by the curves y                                                              x    and y  x is the point ( a , b ) .
a
What is ?
b

1                                       2                         5
A)                                B)                              C)                         D) 5                       E) NOTA
5                                       5                         2

xy
16. Evaluate:                  lim
 xy
( x , y )  ( 2 ,1 )        3
xy

1                                       2                         1
A)                                B)                              C)                         D) 1                       E) NOTA
5                                       5                         2


L z 2

17. Let L                                                       xy cos z
3
3         . Evaluate:                                            dxdydz
3                                   0 0 0

3                                       3                         3                           3
A)                                B)                              C)                         D)                         E) NOTA
6                                      4                         3                               2

2

18. Evaluate:  ln xdx
1

A) ln 2  2                                                       B) 2 ln 2  2                                         C) ln 2  1
D) 2 ln 2  1                                                     E) NOTA
2007 Mu Alpha Theta National Convention
=====================================================================

19. The coordinates of a particle are given as a function of time: x ( t )  t 2 , y ( t )  t 3 . What
5
distance does the particle traverse from time t  0 to t                                    ?
3

37                           19                        19                    38
A)                           B)                        C)                    D)             E) NOTA
54                               27                    18                    3

2   y                2
1 z 2        3       y
z e
20. Evaluate:                      x
dxdydz
0 0 1

( e  2 ) ln 2                                         ( e  2 ) ln 2                       ( 4 e  5 ) ln 2
A)                                                     B)                                   C)
4                                                        2                                  8
( 4 e  5 ) ln 2
D)                                                     E) NOTA
4

21. Consider the function f ( x , y )  ( x  1) 2  y 2  1 with a domain consisting of a disk of
radius two centered at the origin. What is the absolute maximum value of f minus the
absolute minimum value of f?

A) 0                         B) 1                      C) 8                  D) 9           E) NOTA

22. A square with sides of length 3 and centered at ( 5 ,9 ) is rotated about the line y  b to
form a solid with a volume of 72  . What is the sum of all possible values of b?

A) 5                                                   B) 10                                C) 18
D) Cannot be determined                                E) NOTA

1

23. Evaluate:        x      1  x dx
0

1                                4                     14                    16
A)                           B)                        C)                    D)             E) NOTA
5                            15                        15                    15

24. A cylinder’s base is a circle of radius 2 centered at the origin in the x-y plane extending a
height of 4 in the positive z direction. The density of the cone is given:
 ( x , y , z )   x  y z . What is the mass of the cylinder?
2   2

128                                                   256 
A)                           B) 64                    C)                    D) 256        E) NOTA
3                                                       3
2007 Mu Alpha Theta National Convention
=====================================================================

y
25. Consider the curve y  x 2 from ( 0 , 0 ) to ( 3,9 ) denoted C and the function f ( x , y )                                 .
x
What is  f ( x , y ) ds ?
C

37        37  1                              54       37
A) 3                                             B)                                            C)
12                                       3
54           37  1
D)                                               E) NOTA
3

26. What is the area of the cardioid described by the polar equation r  1  cos  ?

A)                              B) 2           C) 3                       D) 4             E) NOTA

z                                     
27. Let z  xy 2 sin xy . What is                        evaluated when x                     and y  2 ?
y                                     3

A)
2       3  2                            B)
2        3                          C)

2 3 3  2       
3                                              3                                               3

D)

2 3 3                                   E) NOTA
3

2

28. A particle’s coordinates are described as a function of time: x ( t )  e t , y ( t )  2 e 3 t . Let 
be defined as the acute angle between the x-axis and the tangent line to the particle’s
trajectory at time t  2 . What is tan  ?

2                             2               2
4e                               3e              3e
A)                               B)              C)                          D) 3e 2           E) NOTA
3                              4               2

( x  4 )( x  x  12 )
2               2

29. Let f ( x )                                         . How many asymptotes does f ( x ) have?
( x  5 x  6 )( x  4 )
2

A) 1                             B) 2            C) 3                        D) 4              E) NOTA

V
30. Given the relation V   r 2 h , what is                              when r  2 and h  3 ?
r

A) 4                            B) 8           C) 12                      D) 16            E) NOTA

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