Math 166
Final Exam 
Spring 1996

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 arctsin(x) means the inverse sine 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.   Note that exact answers are expected for all problems, except those for which an approximate answer is requested.

1) Evaluate     Ü  (cos(x))^3  dx.

2) Evaluate     Ü  x cos(x)  dx.

3) Calculate the area of the region in the first quadrant, bounded by y=1-x^2 and y=1-x.

4) If C is the curve described by y = (2/3) x^(3/2), for 0 æ x æ 3, what is the length of C?

5) Find a value of x rounded to 3 decimals for which arcsin(x) - arcsin(-1/2) = º/4.

6) Evaluate    Ü  (4x+6)  /  (x^2 + 3x)   d x.

7) Evaluate    Ü e^(x/2)  /  Sqrt (1-e^(-x))  d x.

8) Write the Maclaurin series for 1 / (1+x^2).

9) If the region R = { (x,y) : 0 æ y æ sin(x), 0 æ x æ º/2} is revolved around the line y=-1 to create a solid region S, write a definite integral whose value is the volume of S.  (Do not evaluate your integral).

10) Evaluate   lim  (e^(2x) - 1)   /   ( Sqrt(x+4) - 2 )
                      x->0

11)  From the data in this sketch of a right triangle, express angle A as a function of x, and calculate dA/dx when x = 6.

p1=Line[{{0,0},{15,0},{15,15},{0,0},{15,9}}];
p2=Text["15",{7.5,-1}];
p3=Text["9",{16,4.5}];
p4=Text["A",{4,3}];
p5=Text["x",{16,12}];
Show[Graphics[p1],Graphics[p2],
  Graphics[p3],Graphics[p4],Graphics[p5]]

É
12)  Explain why the infinite series     Ö  (-1)^(n+1)  ln(n)  /   n  converges conditionally.
                                                            n=1

É
13) Evaluate    Ü     1  /  ( (x^2+1) (x+1) )      d x.
                          0

É
14)  a) Explain why the infinite series   S =  Ö n  (  1/Sqrt(2)  )^(n-1)  converges.
                                                                     n=1
       b) What is the exact value of S?

15) Calculate the area of the region lying inside both the polar curves r=1+cos(¯) and r=3*cos(¯).

16) a) Write the Maclaurin series for sin^2(x) = 1/2 - 1/2 cos(2x).

      b) If we were to use the first five terms of your answer to a to approximate sin^2(x) for 0æxæ1, find a bound for the error in this approximation.

17) Let R be the region bounded by the graphs of y=ln(x) and y=(x-1)^2.  If R is revolved about the y axis to create a solid region T, approximate the volume of T.