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IB Mathematics: Applications and Interpretation HL · 12.9: Differentiation · 12.9.3

Name Date

From f′ to f Problem set A

f is increasing where the graph of f′ is above the x-axis. Find where f′ = 0 first.

Difficulty: first skills applying them stretch

Skills

  1. The graph shows y = f′(x), the straight line through (0, 6) and (2, 0). Write the values of x for which f is:

    −2−112345−8−6−4−22468xyy = f′(x)
    1. (a) increasing;
    2. (b) decreasing.

  2. The graph shows y = f′(x), a parabola meeting the x-axis at x = −1 and x = 3.

    −3−2−112345−6−4−2246xyy = f′(x)
    1. (a) Write the values of x for which f is increasing.
    2. (b) Write the values of x where f has a stationary point.

  3. The graph shows y = f′(x). It has asymptotes the dashed lines x = 2 and y = −1 and meets the x-axis at x = 5.

    −22468−4−2246xyy = f′(x)
    1. (a) Write the values of x for which f is increasing.
    2. (b) Write the x-value of the stationary point of f.

  4. The graph shows y = f′(x), a cubic that meets the x-axis at x = −1, 2 and 4.

    −2−112345−10−5510xyy = f′(x)
    1. (a) Write the values of x for which f is increasing.
    2. (b) Write the values of x where f has a stationary point.

  5. The graph shows y = f′(x), the line through (0, 10) and (5, 0). A different function g has g′(x) = 3 for all x.

    −22468−5510xyy = f′(x)
    1. (a) Find the x-coordinate of the stationary point of f.
    2. (b) Is it a maximum or a minimum?
    3. (c) Describe the graph of g.

  6. The graph shows y = f′(x) for x > 0. It is a decreasing curve that gets closer to the x-axis.

    123456782468xyy = f′(x)
    1. (a) Find the gradient of f at x = 3.
    2. (b) Is f increasing or decreasing? Does its gradient get larger or smaller as x increases?

In context

  1. The graph shows the rate of change of a population of bacteria, P′(t) thousand per hour, for 0 ≤ t ≤ 6, where P is the population in thousands and t is the time in hours. P′ meets the t-axis at t = 1 and t = 5.

    123456−6−4−2246t (hours)P′P′(t)
    1. (a) For which values of t is the population decreasing?
    2. (b) At what time is the population at a local maximum?

Stretch

  1. The graph of y = f′(x) is the line through (0, 8) and (4, 0). Describe the graph of y = f(x): where it has a stationary point, whether that is a maximum or a minimum, and its overall shape.

    −2−1123456−4−2246810xyy = f′(x)

  2. For x > 0 the graph of y = f′(x) is the decreasing curve through (1, 6), (2, 3) and (6, 1), getting closer to the x-axis. The graph of y = f(x) passes through (1, 2). Describe the graph of f for x > 0.

    123456782468xyy = f′(x)

jmaths.xyz · AI HL · 12.9.3 From the graph of f′ back to f