This document provides examples of analyzing and graphing various functions based on their derivatives. It examines six examples: a radical function, polynomial function, trigonometric function. For each, it identifies the first and second derivatives, domain, x-intercepts, y-intercepts, asymptotes, critical numbers, possible points of inflection, and symmetry. It then analyzes the characteristics of each function graph based on the values and signs of the function, first derivative and second derivative over intervals of the domain.

1. 3.6 A Summary of Curve Sketching Intercepts, symmetry, domain & range, continuity, asymptotes, differentiability, extrema, concavity,infinite limits at infinity, WHEW!

2. Ex 4 p.211 Radical Function Analyze and graph 1 st Derivative: 2 nd Derivative: Domain: x-intercepts: y-intercept: Vertical asymptote(s): Horizontal asymptote(s): Critical number(s): Possible points of inflection: Symmetry: All reals (0, 0), (125/8, 0) (0, 0) None None x = 0, x = 8 x = 0, x = 1 None

3. Ex 4 continued f(x) f’(x) f”(x) Characteristics of Graph -∞ < x < 0 Pos Neg Increasing, concave down x = 0 0 0 Undef Relative maximum 0 < x < 1 Neg Neg Decreasing, concave down x = 1 -3 Neg 0 Point of inflection 1 < x < 8 Neg Pos Decreasing, concave up x = 8 -16 0 Pos Relative Minimum 8 < x < ∞ Pos Pos Increasing, concave up

5. Ex 5 p.213 Polynomial Function Analyze and graph 1 st Derivative: 2 nd Derivative: Domain: x-intercepts: y-intercept: Vertical asymptote(s): Horizontal asymptote(s): Critical number(s): Possible points of inflection: Symmetry: None x = 1, x = 4 At x = 2, x = 4 None None (0, 0) (0, 0), (4, 0) All real #’s

6. Ex 5 continued f(x) f’(x) f”(x) Characteristics of Graph -∞ < x < 1 Neg Pos Decreasing, concave up x = 1 -27 0 Pos Relative minimum 1 < x < 2 Pos Pos Increasing, concave up x = 2 -16 Pos 0 Point of inflection 2 < x < 4 Pos Neg Increasing, concave down x = 4 0 0 0 Point of Inflection 4 < x < ∞ Pos Pos Increasing, concave up

8. Ex 6 p.214 Trigonometric function Analyze and graph 1 st Derivative: 2 nd Derivative: Domain: x-intercepts: y-intercept: Vertical asymptote(s): Horizontal asymptote(s): Critical number(s): Possible points of inflection: Symmetry: x = - π /2, x = 3 π /2 None At x = π /2 None None (0, 1) ( π /2, 0 Period is 2 π . For convenience, choose (- π /2, 3 π /2) All real except x = (3+4n) π /2

9. Ex 5 continued f(x) f’(x) f”(x) Characteristics of Graph x = - π /2 Undef Undef Undef Vertical asymptote - π /2< x < π /2 Neg Pos Decreasing, concave up x = π /2 0 -1/2 0 Point of inflection π /2< x < 3 π /2 Neg Neg Decreasing, concave down x = 3 π /2 Undef Undef Undef Vertical asymptote

11. 3.6b p. 215/ 43, 45, 49-59 odd, 67-70