Exam #1
Math 166 Section A2
27 September 1996

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 sine is equal to x.




B) Simplify the expression Cos(ArcTan(x)):
(You may find it helpful to sketch another triangle!)

Cos(ArcTan(x)) =   __________

2). Differentiate the following:

A)   d/dx   ArcSin( 2x)

B)    d/dx  Ln( x * Cos(x))

C)    d/dx   x^x

3). Integrate the following:

A)    Ü Ln(x)  /  x     dx

B)    Ü     1  /  (1 + x^2)    dx

C)    Ü     5 ^ (-2 x)    dx

4). Solving Simple Differential Equations

Find a solution y(x)=.....  for   y '  =   Cos(x)  /  ( y+1 ),   showing your work.

5). Exponential Functions

A function   f(x) = C E^(k x)   is graphed below.  Determine the value of C and  an expression for k.

C = ____________

k = ____________

7). Geometric Area Between Two Curves

A) Sketch the graphs of the functions y=2-x^2 and y=x.

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

C) Repeat part B, but use integration in the y direction.

7). The Volume of a Solid of Revolution

Consider the functions y = 4-x^2  and y=Sqrt(x) for x from 0  to 2 graphed below:

B)  Suppose the curve of this function is rotated about the x axis.  Write down an expression
for the volume of the resulting solid.  (You do not have to evaluate the integral!)

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