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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 B

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

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 \(y=f(x)\), a parabola with roots at \(x=-4\) and \(x=2\). Sketch \(y=f'(x)\).

    −6−4−224−10−55xyy = f(x)

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

    −5−4−3−2−1123−12−10−8−6−4−22xyy = f(x)

  3. The graph shows \(y=f(x)\), a cubic with roots at \(x=-3\), \(1\) and \(4\). Sketch \(y=f'(x)\), giving the x-values where it crosses the x-axis.

    −4−224−30−20−1010203040xyy = f(x)

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

    −2−11234−4−2246810xyy = f(x)

  5. The graph shows \(y=f(x)\), a decreasing curve with the horizontal asymptote \(y=-1\). It crosses the x-axis at \(x=1.39\). Sketch \(y=f'(x)\).

    −2−11234−2246810xyy = 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 decreasing, and \(f'\) is always increasing.

In context

  1. The graph shows \(T'(t)\), the rate of change of the temperature of a room in °C per hour, for \(0\le t\le 6\). It is a parabola with roots at \(t=1\) and \(t=5\). Sketch two possible graphs of the temperature \(T(t)\).

    −11234567−6−4−22468xyy = T'(t)

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

    123456−3−2−1123xyy = 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 \(f(x)=x^2(x^2-6)\).

    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, P, Q, R and S. 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.

    −2−1.5−1−0.50.511.52−3−2−1123xyPQRS

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