Friday, February 25, Practice test.  

The real test is Monday, February 28, 2005 




1.  Find the multiplicative constants A and B so that

                    -1   2            x  3
      F(x) =  A  Tan  ( x ) + B (x + e  )


                  x             x  2     x
      F'(x) =  -------  + (x + e  )  (1+e  )
                   4
                1+x





2.  Find where the curves  f(x) = 10/x and g(x) = 7-x intersect.

Then find the area of the bounded region determined.




                   4         3
3.  Find   d/dx   INT    ln(t ) dt
                 Sin[x]





                               (2/x)
4.  Find    Limit  (1 + Sin[x])
            x->0





                                           2
                                          x
5.  Use substitution to evaluate  INT  --------- dx
                                             6
                                        1 + x






      x=1
6.    INT   ArcTan[x] dx   
      x=0





                      3       4
7.  Find      INT  Sin [t] Cos [t] dt



                                                4
8.  The region bounded by  y = Sqrt[x] and y = x  is rotated

about the y-axis.  Find the volume of the solid generated. 




9.  The signum function sgn(x) is defined by sgn(x) = 1 for x>0,

sgn(0) = 0, Sgn(x) = -1 for x < 0.  Let a < 0 and b > 0.

       b
Find  INT  sgn(x) dx in terms of a and b.  Justify your answer.
       a



                                   x
10.  The region determined by y = e, y=x, and x=0, x=2 is

rotated about the x-axis.  Set up, DO NOT EVALUATE the definite

integral for this volume.




                           dx
11.  Evaluate  INT  ----------------  using triangle substitution.
                     2       2
                    x  Sqrt[x - 9 ]

Draw and label the triangle.




          x=+Infinity        dx
12.  Find      INT     ------------ 
          x=-Infinity          2
                            1+x




                x=2Pi   Cos[x]    
13.  Find       INT   --------  dx
                x=0   1+Sin[x]




     x=4       dx
14.  INT  --------------
     x=2   Sqrt[4x-x^2]




15.  Let R be the region in the first quadrant below the curve

      -2/3 
y = x      and to the left of x=1.

(a)  Find the area of R.


(b)  Find the volume of the solid generated by revolving R

about the x-axis. 



16.  Show that if p(x) is any polynomial, then

                      p(x)
        Lim        ------------  = 0
    x->Infinity         x
                       e




17.  Find     Limit   2 x Ctn[x] 
               x->0


                        Tan[3x]
18.  Find   Limit      --------
           x-> Pi/2     Tan[x]