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

                                                                                
  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
                          Advanced Calculus Test
                  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
                          Advanced Calculus Test
                  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
                          Advanced Calculus Test
                  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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