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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  2e  4y
ˆ           ˆ                 B) 2e  1x  e  4y
ˆ          ˆ          C) e  4x  2e  1y
ˆ           ˆ
D) 2e  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
=====================================================================

                                       
7. Let f (t )  sin tx  cos ty  tz . 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

 2  8  8                                   2  4  4                   2  8  8
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
12. Evaluate:    
0
1  x 2 dx

                                              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 )  tx  t 2 y  cos tz . What is the speed of the particle at time t 
ˆ       ˆ        ˆ                                                                             ?
2

A)        2 1                             B)                             C)      2 1
D)       2 2                              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
( x , y ) ( 2 ,1)   xy  xy
3

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

                             L z 2
17. Let L  3                                       xy cos z
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

1 z2 2y          2
z 3e y
20. Evaluate:           dxdydz
0 0 1
x

(e  2) ln 2                       (e  2) ln 2                     (4e  5) ln 2
A)                               B)                               C)
4                                  2                                8
(4e  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
0
1  x dx

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 2  y 2 z . What is the mass of the cylinder?

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

28. A particle’s coordinates are described as a function of time: x(t )  e t , y (t )  2e 3t . Let 
2

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  ?

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

( x 2  4)( x 2  x  12)
29. Let f ( x)                                  . How many asymptotes does f (x) have?
( x 2  5 x  6)( x  4)

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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