Math 104 Final Examination - Spring 2001

Show all work. No credit will be given for answers without proper mathematical justification. All questions are worth 10 points.

1. a) Algebraically verify that the graph of has a vertical asymptote,

give the equation of that asymptote, and determine the behavior of the graph

on each side of the asymptote as x approaches the asymptote from each side.

b) Find the equation of the horizontal asymptote for the graph of

.

c) If , find y' and then simplify the expression obtained to

the extent possible.

2. A point, P, on a wheel of radius 4 inches has coordinates (3, ) relative to the

center of the wheel. Find the coordinates of this point relative to the center of the

wheel after the wheel has rolled 13 inches to the left. (Show quadrant diagrams

to justify the evalaution of any trigonometric functions.)

3. a) Sketch 1 forward cycle and 1/2 backward cycle of the graph of y = cos ().

Be sure to label the x- value of the starting point and of all the quarter period

points on the X-axis and the value of the amplitude on the Y-axis. Include a

small sketch of the basic curve y = cos x.

b) Expand and simplify sin (3x-). Show a special right triangle in a quadrant

diagram to justify the evaluation of trigonometric functions.

4. a) Sketch the graph of r = cos 4 from .

b) Find the area bounded by the curve r = 1 where and

the ray =. Show graphs of a basic trigonometric function to justify the

evaluation of trigonometric functions.

5. a) Sketch 1 cycle to the left of the Y-axis and 1 cycle to the right of the Y-axis of

the curve whose equation is y = sec ½ x. Show the asymptotes and give their

equations.

b) Represent the function y = tan 3x as a quotient of two trigonometric functions

and use this new representation and the Quotient Rule for Derivatives to

compute a formula for y', the first derivative. Express the formula in simplest

terms.

c) Find the equation of the tangent line to y = csc ½ x at the point where the

argument of the function is. Express your answer in y = mx + b form. Show

a special right triangle in a quadrant diagram to justify the evaluation of

trigonometric functions. [d/dx csc ax = -a csc ax cot ax.]

6. a) Sketch the hyperbola . Indicate the coordinates of the

center, the coordinates of the vertices, the coordinates of the focal points, and

the equations of the transverse and coordinate axes. Show the asymptotes, as

well, in your sketch.

b) Find the equations of the asymptotes of the hyperbola in Part "a".

c) Convert the equation to the standard form of the

equation of a hyperbola.

7. Find the values of A and that will allow the function y =

to be written in the form y = A sin (2x + ) where A is the amplitude of the function and is an

inverse tangent function. [Hint: Be sure to determine the quadrant is in.]

8. a) Solve for b: .

b) Evaluate .

9. a) Sketch the triangle, AOB, whose vertices are A (5, 160°), B (2, 280°) and

O (0,0). Then use the Law of Cosines to compute the length of side AB.

b) If A = , perform the

indicated operations and combine similar terms to obtain the exact value of A

in simplest terms.

10. a) Complete the evaluation of the integral that leads to .

b) Evaluate the integral by letting = sin and then making

the appropriate substitutions for "x" and "dx" so that the integration can be

performed in terms of "". After obtaining an answer in terms of "",

convert the answer back to an answer in terms of "x".

[Note: cos a d = 1a sin a .]

Extra Credit: If , , give a mathematical justification for

the fact that cos =the absolutr value of cos .

5.