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IB Mathematics: Applications and Interpretation SL · 12.9: Differentiation · 12.9.2

Name Date

From f to f′ Problem set A

Mark the turning points of f first: they are where f′ meets the x-axis. Then read the sign of f′ from whether f rises or falls.

Difficulty: first skills applying them stretch

Skills

  1. The graph shows the straight line y = f(x) through (0, 4) and (2, 0).

    −2−11234−4−2246xyy = f(x)
    1. (a) Write down f′(x).
    2. (b) Does the graph of f′ cross the x-axis? Give a reason.

  2. The graph shows y = f(x), a parabola with its vertex at (1, 4). The graphs P, Q and R are shown with it. One of them is the graph of f′.

    −2−11234−6−4−2246xyfPQR(1, 4)
    1. (a) Which of P, Q, R is the graph of f′?
    2. (b) State the x-value where the graph of f′ meets the x-axis.

  3. The graph shows y = f(x), a cubic with its turning points marked. The graphs P, Q and R are shown with it. One of them is the graph of f′.

    −2−112−6−4−2246810xyfPQR(−1, 2)(1, −2)
    1. (a) Which of P, Q, R is the graph of f′?
    2. (b) Write down the values of x where f′(x) = 0.

  4. The graph shows y = f(x), a quartic with its three turning points marked.

    −2−11234−10−551015xyy = f(x)(−1, −9)(1, 7)(3, −9)
    1. (a) Write down the values of x where the graph of f′ meets the x-axis.
    2. (b) Write down the values of x for which f′(x) > 0.

  5. The graph shows y = f(x) for 0 ≤ x ≤ 360, where x is in degrees. Its maxima and minima are marked.

    90180270360−22x (degrees)yy = f(x)(90, −2)(270, 2)
    1. (a) Write down the values of x where f′(x) = 0.
    2. (b) Write down the values of x for which f′(x) > 0.

  6. The table shows f(x) = xᵃ and its gradient function for a = 1 to 4. Use the pattern to write f′(x) for a = 5.

    a1234f(x)xx²x³x⁴f′(x)12x3x²4x³

In context

  1. A ball is thrown upwards. The graph shows its height h metres after t seconds, for 0 ≤ t ≤ 4. The graphs P, Q and R are shown with it. One of them shows the rate of change of h (the vertical speed).

    0.511.522.533.54−20−101020t (s)h (m)hPQR(2, 20)
    1. (a) Which of the three graphs shows the rate of change?
    2. (b) State the value of t where the rate of change is zero and say what it means.

Stretch

  1. The graph of y = f(x) is a cubic with a local minimum at (−2, −20) and a local maximum at (1, 7). It falls, rises, then falls. Describe the graph of f′: its shape, where it meets the x-axis and where it is above the axis.

    −4−3−2−1123−20−10102030xyy = f(x)(−2, −20)(1, 7)

  2. The graph of y = f(x) for 0 ≤ x ≤ 360 is a wave with maxima at (0, 4) and (360, 4) and a minimum at (180, −2). Describe the graph of f′: where it is zero, where it is negative and where it has its smallest and largest values.

    90180270360−224x (degrees)yy = f(x)(0, 4)(180, −2)(360, 4)

jmaths.xyz · AI SL · 12.9.2 From the graph of f to the graph of f′