![When you are sketching a graph, either by hand or calculator You normally can’t show the entire graph. You will have to choose what you show. Graph f(x) = x 3 – 25x 2 + 74 x – 20 OR X : [-2,5] Y: [-10,40] X : [-10,30] Y: [-1200,200]](https://image.slidesharecdn.com/calc3-6a-111031212836-phpapp02/85/Calc-3-6a-2-320.jpg)
Calc 3.6a
- 1. 3.6a A Summary of Curve Sketching Intercepts, symmetry, domain & range, continuity, asymptotes, differentiability, extrema, concavity,infinite limits at infinity, WHEW!
- 2. When you are sketching a graph, either by hand or calculator You normally can’t show the entire graph. You will have to choose what you show. Graph f(x) = x 3 – 25x 2 + 74 x – 20 OR X : [-2,5] Y: [-10,40] X : [-10,30] Y: [-1200,200]
- 3. Calculus can help you know highlights of graph
- 4. Ex 1 p. 210 Rational Function Analyze and sketch graph of Vertical asymptotes: Domain: x-intercepts: all reals except x≠-2, 2 1 st derivative: 2 nd derivative: (-3, 0), (3, 0) y-intercept: (0, 9/2) Horizontal asymptote Possible pts of inflection: x = -2, x = 2 y = 2 Critical number(s): x = 0 (2 & -2 not critical, domain problems) None Symmetry: w/ respect to y-axis
- 5. Ex 1 continued . . . And yes, I expect this much calculus work to be shown! f(x) f’(x) f”(x) Characteristics of Graph -∞ < x < -2 Neg Neg Decreasing, concave down x = -2 Undef Undef Undef Vertical asymptote -2 < x < 0 Neg Pos Decreasing, concave up x = 0 9/2 0 Pos Relative min 0 < x < 2 Pos Pos Increasing, concave up x = 2 Undef Undef Undef Vertical asymptote 2 < x < ∞ Pos Neg Increasing, concave down
- 6. Draw an excellent graph, label all asymptotes, important points like x- and y-intercepts, and extrema
- 7. Ex 2 p.211 Rational 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 real numbers except x ≠ 2 none (0, -2) x = 2 None x = 0, 4 None None
- 8. Ex 2 continued f(x) f’(x) f”(x) Characteristics of Graph -∞ < x < 0 Pos Neg Increasing, concave down x = 0 -2 0 Neg Relative maximum 0 < x < 2 Neg Neg Decreasing, concave down x = 2 Undef Undef Undef Vertical asymptote 2 < x < 4 Neg Pos Decreasing, concave up x = 4 6 0 Pos Relative Minimum 4 < x < ∞ Pos Pos Increasing, concave up
- 9. This doesn’t have a horizontal asymptote, but since the degree in numerator is one more than that of denominator, it has a slant asymptote. Do division to see what the equation of the slanted line is. Rewriting it after division, So the graph approaches the slant asymptote of y = x as x approaches +∞ or -∞
- 10. Ex 3 p.212 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 real #’s (0, 0) Same (0, 0) None y = 1 (right) and y = -1 (left) None At x = 0 With respect to origin
- 11. f(x) f’(x) f”(x) Characteristics of Graph -∞ < x < 0 Pos Pos Increasing, concave up x = 0 0 0 Point of inflection 0 < x < ∞ Pos Neg Increasing, concave down
- 12. 3.6a p. 215/ 3-33 mult 3, 39