Docstoc

mu advanced calculus test

Document Sample
mu advanced calculus test Powered By Docstoc
					                          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
                          Advanced Calculus Test
                  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
                          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 )  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
                          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

                  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
                          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

  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

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:7
posted:9/23/2011
language:English
pages:5