MATH 121  EXAM III  FALL 1996

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1. (13 points) Let f(x) = \frac{x+13}{x^{2}−7x+10}. Identify the horizontal and vertical asymptotes for y = f(x). Find the x and y intercepts. Draw a sketch of the graph.

2. (12 points) What does f(x) (as in 1) approach as:

a) x → −2^{−}

b) x → +2^{+}

c) x → −∞

d) x → +∞

3. (12 points) Find a, b and d so that the rational function f(x) = \frac{ax+b}{x+d} has:

horizontal asymptote y=3, vertical asymptote x=5 an intercept with the y axis at y = 8.

4. (13 points) Use long division of polynomials to determine the end behaviour asymptote of y = f(x) = \frac{5x^{3}+2x^{2}+x+2}{x^{2}−1}.

5. (10 points) Simplify the trig expression tan(arccos(\frac{x}{5})). Assume that the angles involved are in the first quadrant.

6. (10 points) Prove the identity:

\frac{cos^{2} x − 1}{sin x} = − tan x sin x.

MATH 121  EXAM III  FALL 1996

7. (15 points)

a) Find all (not just some) of the vertical asymptotes of the graph y = tan(2x − 1) + 5.

b) Describe in words how the graph of part a) is obtained from the graph of y = tan(x). (A sketch of the graph is NOT required.)

c) What is the period of f(x) = tan(2x − 1) + 5?

8. (15 points) A ball thrown at angle θ and initial velocity v_{o} (in feet/second) travels a distance in feet of d = (\frac{1}{32})v_{o}^{2} sin(2θ). If Alfred E. Neumann throws a ball a distance of 400 feet, with initial velocity of 125 feet/sec., what was the throwing angle θ (in degrees, to 2 decimal places)?