Math 166
Final Exam 
Spring 1995

This Mathematica file contains the questions from last year's final exam.  You may use Mathematica, a calculator, or pencil and paper to answer any of the questions.

The symbol e is the base of the natural logarithm system.  
The symbol ^ is used to show exponentation, so that x^2 means the square of x.
The symbol Sqrt(x) means the square root of x.
The symbol arctan(x) means the inverse tangent of x.
The symbol ¯ is the best Mathematica could do for what we call "theta".

Problems 1 through 10 are worth 5 points each, and for them no partial credit will be assigned.  Please put your answers to these problems in the blanks provided.  Problems 11 through 17 are worth 10 points each, and you are to supply full solutions for them.

1) If f(x) = ln ( 1 + e^x), write f '(x).

2) Write the value of  cos( arctan (x/2)) as a function of x, without trigonometric functions.

3) Evaluate     Ü  arcsin(x)  dx.

4) Write the partial fractions decomposition of the rational function

   x+2  /  (  x^2   (x-3)  (x^2 + 9) ) .
   
 Do not try to calculate the constants of the decomposition.

5) If  f(x) = x e^(-x),  find a positive value of x for which  f ''(x) = 0.

º
 6) Evaluate    Ü         Sqrt( 1 + cos(¯))    d ¯
                          0

É
 7) If  S  =  Ö   2^(n+1) / 3^n,  what is the value of S?
                   n=0

8) Let C be the polar curve { r = e^¯ :  0  æ ¯ æ 2º }.  Calculate the length of C.

9) Evaluate the limit   lim  (1 - cos(2x)) / (x sin(x))
                                   x->0

10) If the region R bounded by the circle (x-1)^2 + y^2 = 1 is revolved around the y axis to generate a solid region, write a definite integral whose value is the volume of this solid region.  (Do not evaluate the definite integral).

11)  The curve {y = Sqrt(x) : 1 æ x æ 4} is revolved around the x axis to generate a surface S.  Calculate the area of S.

12) Calculate the area of the region in the first quadrant which lies inside both the polar curves r=sin(¯) and r=sin(2¯).

13) For each of these three infinite series, decide whether the series converges and give an explanation to support your answer:

É                               É                                       É
Ö  1 / (n^2 + n)       Ö  n^(0.9) / (n^2 + n)      Ö  n / (n^2 + 2^n)
n=1                           n=1                                    n=1

14) 
 a) Use a geometric series to write the sum of the first 5 nonzero terms of the Maclaurin series of f(x) = ln (1+x).
 
 b) Write the sum of the first 5 nonzero terms of  g(x) = Ü  ln(1+x^2)  /  x   dx.

15)
                                                                É
 a) Explain why the infinite series   T =  Ö  (-1)^(n+1)  n   /   (n! + n)  converges.
                                                               n=1
                                                               
                                                       N
 b) If we approximate T by TN =  Ö  (-1)^(n+1)  n   /   (n! + n)  so that | T - TN | < 10^(-4), 
                                                      n=1    
what is the smallest value of N we should use?

a
16) If  a  is a fixed positive constant,   evaluate  Ü     1 /  (  Sqrt(x)   Sqrt(a-x)  )   dx.
                                                                            0

17) The curve C = { x = sin(y) : 0  æ y æ 3 º /4} is spun around the y axis to create a glass-shaped vessel, pointed at the bottom, whose height is 3 º /4.  To what height B would one fill this glass in order that it be exactly half full?