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

Name Date

From f to f′ Problem set B

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.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The graph shows the straight line y = f(x) through (0, −3) and (3, 3).

    −2−112345−6−4−224xyy = 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, −3). The graphs P, Q and R are shown with it. One of them is the graph of f′.

    −4−3−2−112−6−4−2246xyfPQR(−1, −3)
    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. Here is y = f(x), a cubic. Its turning points are marked, and three more graphs P, Q and R are drawn. One of P, Q, R is the graph of f′.

    −1−0.50.511.522.53−6−4−22468xyfPQR(0, 0)(2, 4)
    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. Here is y = f(x), an M-shaped quartic. Its three turning points are marked.

    −3−2−112−30−20−10102030xyy = f(x)(−2, 32)(0, 0)(1, 5)
    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. Here is y = f(x) over one full turn (0 to 360 degrees). Its highest and lowest points are marked.

    90180270360−6−4−2246x (degrees)yy = f(x)(0, 5)(180, −5)(360, 5)
    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 g(x) = xᵇ and its gradient function for b = 2 to 5. Use the pattern to write g′(x) for b = 6.

    b2345g(x)x²x³x⁴x⁵g′(x)2x3x²4x³5x⁴

In context

  1. The graph shows the profit P thousand pounds of a company when it makes x hundred items, for 0 ≤ x ≤ 8. The graphs J, K and L are shown with it. One of them shows the rate of change of the profit.

    12345678−20−101020x (hundreds)P (thousand pounds)profitJKL(4, 8)
    1. (a) Which of the three graphs shows the rate of change?
    2. (b) State the value of x 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 maximum at (0, 0) and a local minimum at (2, −4). It rises, falls, then rises. Describe the graph of f′: its shape, where it meets the x-axis and where it is below the axis.

    −2−11234−10−5510xyy = f(x)(0, 0)(2, −4)

  2. The graph of y = f(x) for 0 ≤ x ≤ 360 is a wave with a maximum at (90, 1) and a minimum at (270, −3). Describe the graph of f′: where it is zero, where it is positive and where it has its largest and smallest values.

    90180270360−4−3−2−112x (degrees)yy = f(x)(90, 1)(270, −3)

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