Applied Homework #1, Due Wed Jan 30, 2002

1.) f(x) =

A. Determine the domain of f(x).

^B. Find its intercepts.

^C. Check the function’s symmetry. (i.e. Is the function even (symmetric about the y-axis) or is the function odd (symmetric about the origin)? Is the function a periodic function?)

^D. Does the function have any asymptotes?

^· The line y = L is a horizontal asymptote if either or .

^· The line x = a is a vertical asymptote if at least one of the statements is true:

E. Compute(x) and find the intervals on which (x) is positive (f is increasing) and the intervals on which (x) is negative (f is decreasing).

F. Find the critical numbers of f (the numbers c where (c) = 0 or (c) does not exist. Then say if the critical number is a local minimum or a local maximum.

G. Determine the concavity of f(x) using the second derivative.

H. Sketch the curve. Using the information in A-G, draw the graph. Draw in the asymptotes as broken lines. Plot the intercepts, maximum and minimum points and inflection points. Then make the curve pass through these points, rising and falling according to E, with concavity according to G, and approaching the asymptotes.

2.) Discuss g(x) using A-H in #1.

g(x) =

3.) A painting in an art gallery has height AB=h and is hung so that its lower edge is a distance BC=d above the eye E of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle t subtended at his eye E by the painting?)

4.) Evaluate by interpreting it as an area and integrating with respect to y instead of x.