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Calc 3.6b
1. 3.6b 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
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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
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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
