Curve sketching  Derivative Applications
Curve sketching
In this section we will expand our knowledge on the connection between derivatives and the shape of a graph. By following the “5Steps Approach”, we will quantify the characteristics of the function with application of derivatives, which will enable us to sketch the graph of a function.
Lessons
Notes:
Guidelines for Curve Sketching
a) domain
b) Intercepts
yintercept: set x=0 and evaluate y.
xintercept: set y=0 and solve for x. (skip this step if the equation is difficult to solve)
c) Asymptotes
vertical asymptotes:
for rational functions, vertical asymptotes can be located by equating the denominator to 0 after canceling any common factors.
horizontal asymptotes:
evaluate $lim_{x \to \infty } f(x)$ to determine the rightend behavior;
evaluate $lim_{x \to \infty } f(x)$ to determine the leftend behavior.
slant asymptotes:
for rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.
d) Compute$f' (x)$
find the critical numbers:
• use the First Derivative Test to find: intervals of increase/decrease and local extrema.
e) Compute$f'' (x)$
• inflection points occur where the direction of concavity changes.
find possible inflection points by equating the$f'' (x)$ to 0.
•Concavity Test:
•inflection points occur where the direction of concavity changes.

1.
Use the guidelines to sketch the graph of:
$f(x)=\frac{x^38}{x^3+8}$