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IB Mathematics: Applications and Interpretation HL · 12.9: Differentiation · 12.9.4

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Sketching f and f' from graphs Problem set A

Sketch each graph, marking turning points and roots. Use the calculator to check turning points and intervals.

Difficulty: first skills applying them stretch

Skills

  1. The graph shows \(y=f(x)\), a parabola with roots at \(x=-1\) and \(x=5\). Sketch \(y=f'(x)\).

    −2246−6−4−2246810xyy = f(x)

  2. The graph shows \(y=f(x)\), a parabola that touches the x-axis at \(x=3\). Sketch \(y=f'(x)\).

    −11234567−224681012xyy = f(x)

  3. The graph shows \(y=f(x)\), a cubic with roots at \(x=-2\), \(0\) and \(3\). Sketch \(y=f'(x)\), giving the x-values where it crosses the x-axis.

    −3−2−11234−10−5510xyy = f(x)

  4. The graph shows \(y=f(x)\), a cubic that touches the x-axis at \(x=-1\) and crosses it at \(x=2\). Sketch \(y=f'(x)\).

    −3−2−11234−8−6−4−22468xyy = f(x)

  5. The graph shows \(y=f(x)\), an increasing curve with the horizontal asymptote \(y=2\). It crosses the x-axis at \(x=0.405\). Sketch \(y=f'(x)\).

    −2−11234−8−6−4−224xyy = f(x)

  6. Sketch a possible graph of a function f that satisfies each condition.

    1. (a) \(f(x)>0\) for all x, and \(f'(x)>0\) for all x.
    2. (b) f is always increasing, and \(f'\) is always decreasing.

In context

  1. The graph shows \(g'(t)\), the rate of change of the height of a plant in cm per day, for \(0\le t\le 6\). It is a parabola with roots at \(t=0\) and \(t=6\). Sketch two possible graphs of the height \(g(t)\).

    −11234567−224681012xyy = g'(t)

  2. The graph shows \(y=f(x)=3\sin x\) for \(0\le x\le 2\pi\).

    123456−4−3−2−11234xyy = f(x)
    1. (a) Sketch \(y=f'(x)\).
    2. (b) Taking this same graph as \(y=f'(x)\), sketch a possible \(y=f(x)\).
    3. (c) Explain why your answer to part (b) is not the only one.

Stretch

  1. Let \(f(x)=x^2(x^2-4)\).

    1. (a) Use technology to sketch \(y=f(x)\).
    2. (b) Find the values of x for which \(f(x)>0\).
    3. (c) Find the values of x for which \(f'(x)<0\).

  2. The graph shows four curves, A, B, C and D. They form two pairs: in each pair, one curve is the derivative of the other. Write down the pairs and say which curve is the derivative.

    −2−112−3−2−1123xyABCD

jmaths.xyz · AI HL · 12.9.4 Sketching f and f' from graphs