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Unit1– Derivative Graphs Section 11.2 – Second Derivative Graphs

Unit1– Derivative Graphs Section 11.2 – Second Derivative Graphs. Second Derivative Concavity of a Function. f “ (a) gives us information on the concavity of the graph of f(x). If f ”(a) > 0, the graph of f(x) is concave up. If f ”(a) < 0, the graph of f(x) is concave down.

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Unit1– Derivative Graphs Section 11.2 – Second Derivative Graphs

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  1. Unit1– Derivative Graphs Section 11.2 – Second Derivative Graphs Second Derivative Concavity of a Function

  2. f “ (a) gives us information on the concavity of the graph of f(x). If f ”(a) > 0, the graph of f(x) is concave up If f ”(a) < 0, the graph of f(x) is concave down

  3. If f ” (a) = 0, a point of inflection MAY exist on f. A point of inflection occurs when concavity changes

  4. Given f(x), find the x-coordinates of the point(s) of inflection on f 2 B. Given f ‘ (x) is graphed, find the x-coordinates of the point(s) of inflection on f. 1, 3 C. If f(x) is graphed, for what intervals is f ‘ ‘ (x) negative? (-1, 2) D. If f ’ (x) is graphed, for what intervals is f “ (x) positive? (-1, 1) and (3, 5) The graph is on the interval (-1, 5) E. If f “ (x) is graphed, for what intervals is f “ (x) negative? (-1, 0)

  5. Given f(x), at what values of x are the points of inflection? Given f ‘ (x), at what values of x are the points of inflection? Given f(x), for what intervals is f “ (x) > 0? Given f ‘ (x), for what intervals if f “ (x) positive? Given f(x), when do f “ (x) and f ‘ (x) have opposite signs? -1.5, -0.5, 0.5, 1.5 -1, 0, 1 (-1.5, -0.5), (0.5, 1.5) (-1, 0), (1, 2) The graph is on the interval [-2, 2] (-1.5, -1), (-0.5, 0) (0.5, 1), (1.5, 2)

  6. -1, 1 (-1, 1) (-1, 0), (1, 3) None Given f(x), where is f ‘ (x) undefined? Given f(x), write using interval notation the interval(s) on which the graph is concave down. Given f ‘ (x), write using interval notation the interval(s) on which the graph is concave up. Given f ‘’ (x), write using interval notation the interval(s) on which the graph is concave down. The graph is defined on (-3, 3)

  7. Given f(x), where is f ‘ (x) = 0? Given f ‘ (x) write using interval notation the interval(s) on which the graph is increasing. Given f ‘ (x), write using interval notation the interval(s) on which f “ (x) is negative. Given f “ (x) for what value(s) of x would there be a point of inflection? -0.5, 0.5 (-1, 0), (1, 2] (-0.5, 0.5) –1, 0 1 The graph is defined on [-2, 2]

  8. Given f ‘ (x), where is f ‘ (x) = 0? Given f(x), write using interval notation the interval(s) on which f ‘ ‘ (x) is positive. Given f ‘ (x), write using interval notation the interval(s) on which f ‘ ‘ (x) is negative. Given f(x), at approximately what value of x would f ‘ (x) = 1? 1, 3 (-2, 5) (-2, 2) 2.5 The graph is defined on [-2, 5]

  9. Given f(x), where is f ‘ (x) undefined? Given f ‘ (x), where is f “ (x) undefined? Given f “ (x), write using interval notation the interval(s) on which the graph would be concave up. Given f “ (x), for what values of x is there a point of inflection? 0 0 (-1, 3] -1 The graph is defined on [-10, 3]

  10. Which of the following is/are true about the function f if its derivative is defined by increasing I) f is decreasing for all x < 4 II) f has a local maximum at x = 1 III) f is concave up for all 1 < x < 3 NO TRUE A) I only B) II only C) III only D) II and III only E) I, II, and III

  11. The graph of the second derivative of a function f is shown below. Which of the following are true about the original function f? I) The graph of f has an inflection point at x = -2 II) The graph of f has an inflection point at x = 3 III) The graph of f is concave down on the interval (0, 4) A) I only B) II only C) III only D) I and II only E) I, II and III NO YES NO

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