Page 49 - Calc for the Clueless

You should be getting a lot better now!! Let us try a polynomial, to show you how easy it is.

Example 22—

+
Intercepts: (0,0), (1,0), (2,0), (3,0), and (5,0). No asymptotes. Watch how easy this is! f(5 ) is negative. (5 - x) 9
7
6
is an odd power, so cross. (x- 3) is an odd power, so cross. (x- 2) is an even power, no cross. (x- 1) is an odd
5
31
power, so cross, x is an even power, no cross. The leading term, -X , dominates when x is big (say x = 100) or
4
31
when x is small (say x = -100). -(100) negative—right end goes to minus infinity. -(-100) positive—left end
31
goes to plus infinity. Briefly, right to left, looking at the exponents only—cross, cross, no cross, cross, no cross,
and the sketch looks like...

Example 23—

Intercept (4,0). Vertical asymptote x = 0 (y axis). No oblique or horizontal asymptote since degree of top is 5
5
more than the bottom. (100) is positive—right end goes to plus infinity. (-100) is negative—left end goes to
5
minus infinity. The sketch is... ta-da!...
The next area of curve sketching involves maximum points, minimum points, inflection points, and cusps. Since
much of this involves derivatives and factoring, more care and time is needed.

Definitions