The general procedure for curve sketching is based on the material learned in the last few sections. Essentially, the first and second derivatives are used to determine which of the following pieces will be used where to form the graph:
Then, intercepts and asymptotes are found to refine the graph and make it more accurate.
Example: Sketch a graph of
f (x) =
.
Step 1: Determine the domain of
f
.
In this case, f is undefined only at x = 3 and x = - 3 , so the domain includes all x except those two points. These two points will turn out to be important, because places where the graph is undefined could potentially be vertical asymptotes or places where the function changes concavity or direction.
Step 2 Use the first derivative to find the critical points and determine the
direction of the graph.
A) Find the critical points
f'(x) = = | |||
f'(x) = 0 when x = 0 |
f'(x) is undefined at x = - 3 and x = + 3 , but these points are not part of the domain of f , so these are not considered critical points.