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					                                                                  Practice Exam


                                                                      January 25, 2009

                                                                                                               π
1 Estimate the area under the graph f (x) = cos x from 0 to                                                    2   using four approximating rectangles and
midpoints. Sketch the graph and the rectangles.

Solution :
           π π     π 3π    π 5π    π 7π    π     π   π     3π  π     5π  π     7π
            f ( ) + f ( ) + f ( ) + f ( ) = cos ( ) + cos ( ) + cos ( ) + cos ( )
           8 16    8 16    8 16    8 16    8     16  8     16  8     16  8     16


2 Express the following limit as a definite integral in the interval [2, 6]:
                                                                            n
                                                                  lim            xi ln(1 + x2 ) ∆x
                                                                                            i
                                                                 n→∞
                                                                           i=1


Solution :
                                                         n                                                6
                                             lim              xi ln(1 + x2 ) ∆x =
                                                                         i                                    x ln (1 + x2 ) dx
                                             n→∞                                                      2
                                                     i=1



3 Evaluate the following integral:
                                                                                10
                                                                                      x − 5 dx
                                                                            0

Solution :
     10                    5                        10
                                                                                           x2     5            x2        10           25        25
          x−5 dx =             (5−x) dx+                 (x−5) dx = 5x−                               +           −5x         = 25−      +50−50− +25 = 25
 0                     0                        5                                          2      0            2         5            2          2
Note you can also solve it by drawing the function and interpreting the area as the sum of the areas of two
triangles each with base 5 and height 5.

4 Find the derivative of the following function:
                                                                                      x2
                                                                                                  1
                                                                  g(x) =                     √          dt
                                                                                     tan x       2 + t4

Solution :
                               x2                             tan x
                                         1                                 1                               2x        sec2 x
              g(x) =                √          dt −                   √          dt , therefore g (x) = √        −√
                           0            2 + t4            0               2 + t4                          2 + x8    2 + tan4 x




                                                                                        1
                                                        √
                  x                                t2       1+u4
5 If F (x) =      1
                      f (t) dt, where f (t) =      1         u   du,        find F (2).

Solution :
                                                                                     √                          √
                                                                       1 + (x2 )4      1 + x8                     1 + 28  √
 F (x) = f (x) , therefore F (x) = f (x) = 2x                              2
                                                                                  =2          , hence F (2) = 2          = 257
                                                                         x               x                          2

                                                  4
6 If f (1) = 12, f is continuous and              1
                                                      f (x) dx = 17, what is the value of f (4).

Solution :
                                 4
                      17 =           f (x) dx = f (4) − f (1) = f (4) − 12 , therefore f (4) = 17 + 12 = 29
                             1



7 Evaluate the following integral:
                                                                           ex
                                                                                dx
                                                                         ex + 1

Solution :
                                                                              ex              du
         Let u = ex + 1 then du = ex dx hence                                      dx =          = ln u + C = ln(ex + 1) + C
                                                                            ex + 1            u

8 Evaluate the following integral:
                                                                        π
                                                                            sin5 x dx
                                                                       −π

Solution :
                                                                                                       π
Since sin x is an odd function, then sin5 x is an odd function and therefore                           −π
                                                                                                            sin5 x dx = 0

9 Evaluate the following integral:
                                                                 10
                                                                      (3x2 + x − 10) dx
                                                             0

Solution :
             10
                                                x3   x2                      10             100
                  (3x2 + x − 10) dx = 3            +    − 10x                     = 103 +       − 100 = 1000 + 50 − 100 = 950
         0                                      3    2                       0               2

10 Who is the best superhero? (Circle the answer or write in another one)

                         Spider − M an , Superman , Batman , Captain America , Daredevil

                                 Green Lantern , W olverine , Buf f y , Angel , T im Gunn
                             Ozymandias , Rorschach , M r. F antastic , El Santo , Other


Solution :
Gauss




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