Exam #1
Math 166 Section A2
27 September 1996


Name:______________________

No calculators, computers, or outside references are allowed.

( Sqrt(x) means the square root of x. 
  Ln(x) is the natural logarithm of x,
  E means the base of the natural logarithm system "e".)

1). Inverse Trigonometric Functions

A)  Sketch a right triangle which contains an angle A whose tangent is equal to x.






B) Simplify the expression Tan(ArcSin(x)).  You may find it helpful to sketch a triangle.

     Tan( ArcSin(x) )  =   _______________

2). Differentiate the following:

A)   d/dx   ArcTan( Sin(x) )



B)    d/dx  Cos( Ln( x+1) )



C)    d/dx   x^x

3). Integrate the following:

A)    Ü  (E^x - E^(-x))  /  (E^x + E^(-x))^2    dx

B)    Ü  1  /  Sqrt( 1 - x^2)   dx

C)    Ü   3^(-2 x)  dx

4). Solving Simple Differential Equations

Show the steps necessary to write a solution for the following:


y ' /  (1 + x^2)  = y - 2






y(x)  =  ____________________________

5). Exponential Functions

The population of Dhaka was 4.22 million in 1990 and is expected to be 6.49 million
in the year 2000.  Supposing that this population follows a simple exponential growth
pattern,  pop(t) = C * E^(k*T).

What is the value of C?         C = ____________(in millions)


What is an expression for k?   (Show the steps in your reasoning)  




k = ____________

6). Geometric Area Between Two Curves

A) Sketch the graphs of the functions y = -x^2 +2x + 5  and  y = 6/x.

B) Write an expression for the unsigned or "geometric" area caught between these two curves.
You do not need to carry out the integration.

7). The Method of Disks

Use the method of disks to answer the following questions:
Consider the functions  y = 2*Sqrt(x)  and   y=x^2 / 4    for x from 0  to 4

A)  Suppose the curves of these functions are rotated about the x axis.  Write down an expression
for the volume of the resulting solid.  (You do not have to evaluate the integral!)








B)  Now suppose the curves are rotated about the y axis.  Write down an expression
for the volume of the resulting solid.  (You do not have to evaluate the integral!)