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Calculus I

Calculus I. Section 4.3 Graphs and Derivatives. Homework Changed. Read (pp 296-303) and (pp 316-329) Do (pp 304) 1, 5, 7, 10, 13, 17, 23, 25, 27, 31, 33, 43, 45 Do (pp 324) 1, 17, 29, 33, 41. Increasing/Decreasing Test.

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Calculus I

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  1. Calculus I Section 4.3 Graphs and Derivatives

  2. Homework Changed • Read (pp 296-303) and (pp 316-329) • Do (pp 304) 1, 5, 7, 10, 13, 17, 23, 25, 27, 31, 33, 43, 45 • Do (pp 324) 1, 17, 29, 33, 41

  3. Increasing/Decreasing Test • If f’(x) > 0 on an interval, then f is increasing on that interval • If f’(x) < 0 on an interval, then f is decreasing on that interval • Example: Use the I/D test to find whereis increasing/decreasing

  4. First Derivative Test • Suppose that c is a critical number of a continuous function f • If f’ changes from + to – at c, then f has a local maximum at c. • If f’ changes from – to + at c, then f has a local minimum at c. • If f’ doesn’t change sign at c, then there is no local max/min at c.

  5. Example • Find the intervals on which f is increasing or decreasing. • Find the local maximum and minimum values of f.

  6. Given the Graph, Plot f’ and f’’

  7. Concavity Test • If f”(x) > 0 for all x in interval I, then the graph of f is concave upward on I. • If f”(x) < 0 for all x in interval I, then the graph of f is concave downward on I. • The point where a continuous function changes its concavity is called an inflection point.

  8. Example • Find the intervals of concavity and the inflection points of

  9. Example • Sketch the graph that satisfies these conditions

  10. Second Derivative Test • Suppose f” is continuous near c. • If f’(c) = 0 and f”(c) > 0, then f has a local minimum at c. • If f’(c) = 0 and f”(c) < 0, then f has a local maximum at c. • If f”(c) = 0, then the test fails and you need to use other methods.

  11. Example • Page 305: 49

  12. Section 4.5 How to Graph Using Calculus

  13. Guidelines • Domain: Start by determining the domain of the function algebraically. • Intercepts: You can find the y-intercepts by setting x to 0, that is find f(0). The x-intercepts are found by solving f(x) = 0. • Symmetry: (i) If f(–x) = f(x) for all x then the curve is symmetric to the y-axis and is an even function.

  14. Guidelines • Symmetry: (i) If f(–x) = -f(x) for all x then the curve is symmetric about the origin (rotate 180°) and is an odd function. • Asymptotes: If then f has horizontal asymptote at the line y = L. If f(x) goes to as x approaches a (from left or right side), then x = a is a vertical asymptote.

  15. Guidelines • Increasing/Decreasing: Use the I/D test. If f’ is positive, then f is increasing. If f’ is negative, then f is decreasing. • Max/Min: Find the critical values, f’(x) = 0 or is undefined, and check signs on either side. The second derivative test can also be used. • Inflection Points:Find where f”(x) = 0, concave up where f”(x) > 0 and concave down where f”(x) < 0. • Sketch: Plot all this information and sketch the curve.

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