sketch the graph of y = x^3 / (x^2 - 4), keep in mind the information provided: 1. x and y intercept: (0, 0) 2. Vertical asymptotes: x = 2, x = -2 3. There is no horizontal asymptote. 4. Discontinuity at x = -2 and x = 2. 5. First derivative: y’ = (x^4 - 12x^2) / (x^2 - 4)^2 6. Critical points: x = 0, x = 2√3, x = -2√3 7. Intervals of increase: (-∞, -2√3) U (2√3, ∞) 8. Intervals of decrease: (-2√3, -2) U (-2, 0) 9. Local maxima: (-3.46, -5.20) and (3.46, 5.20) 10. Second derivative: y’’ = (8x^3 + 96x) / (x^2 - 4)^3 11. Critical numbers: (0, 0), (√12, 3√3), (-√12, -3√3) 12. Point of inflection: (0, 0) 13. Concave upward: (-2, 0) U (2, ∞) 14. Concave downward: (-∞, -2) U (0, 2) A sketch of the graph (by hand) - Labeling of vertical asymptotes on the sketch - Labeling of horizontal asymptotes on the sketch - Labeling of x and y intercepts on the sketch - Labeling of maximum and minimum points on the sketch - Labeling of points of inflection on the sketch A sketch of the graph on Desmos with all key points and features labeled
sketch the graph of y = x^3 / (x^2 - 4), keep in mind the information provided: 1. x and y intercept: (0, 0) 2. Vertical asymptotes: x = 2, x = -2 3. There is no horizontal asymptote. 4. Discontinuity at x = -2 and x = 2. 5. First derivative: y’ = (x^4 - 12x^2) / (x^2 - 4)^2 6. Critical points: x = 0, x = 2√3, x = -2√3 7. Intervals of increase: (-∞, -2√3) U (2√3, ∞) 8. Intervals of decrease: (-2√3, -2) U (-2, 0) 9. Local maxima: (-3.46, -5.20) and (3.46, 5.20) 10. Second derivative: y’’ = (8x^3 + 96x) / (x^2 - 4)^3 11. Critical numbers: (0, 0), (√12, 3√3), (-√12, -3√3) 12. Point of inflection: (0, 0) 13. Concave upward: (-2, 0) U (2, ∞) 14. Concave downward: (-∞, -2) U (0, 2) A sketch of the graph (by hand) - Labeling of vertical asymptotes on the sketch - Labeling of horizontal asymptotes on the sketch - Labeling of x and y intercepts on the sketch - Labeling of maximum and minimum points on the sketch - Labeling of points of inflection on the sketch A sketch of the graph on Desmos with all key points and features labeled
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
sketch the graph of y = x^3 / (x^2 - 4), keep in mind the information provided:
1. x and y intercept: (0, 0)
2. Vertical asymptotes: x = 2, x = -2
3. There is no horizontal asymptote.
4. Discontinuity at x = -2 and x = 2.
5. First derivative: y’ = (x^4 - 12x^2) / (x^2 - 4)^2
6. Critical points: x = 0, x = 2√3, x = -2√3
7. Intervals of increase: (-∞, -2√3) U (2√3, ∞)
8. Intervals of decrease: (-2√3, -2) U (-2, 0)
9.
10. Second derivative: y’’ = (8x^3 + 96x) / (x^2 - 4)^3
11. Critical numbers: (0, 0), (√12, 3√3), (-√12, -3√3)
12. Point of inflection: (0, 0)
13. Concave upward: (-2, 0) U (2, ∞)
14. Concave downward: (-∞, -2) U (0, 2)
A sketch of the graph (by hand)
- - Labeling of vertical asymptotes on the sketch
- - Labeling of horizontal asymptotes on the sketch
- - Labeling of x and y intercepts on the sketch
- - Labeling of maximum and minimum points on the sketch
- - Labeling of points of inflection on the sketch
A sketch of the graph on Desmos with all key points and features labeled
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