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Longhand

Chapter 3: Applications of derivatives

Curve sketching

The first derivative tells you where a graph rises and falls, the second tells you how it bends, and limits at infinity tell you what it settles towards. Put together they let you draw a curve you have never seen, and read one you have.

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Practice questions

Every question below is original material written for Longhand. Open a solution to read the full working, or answer it yourself to have it counted towards your progress.

  • Introductory2 marks · 3 min

    Find the largest open interval on which f(x)=exx+2f(x) = \dfrac{e^{x}}{x + 2} is increasing.

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  • Introductory1 mark · 2 min

    Find the xx-coordinate of the inflection point of f(x)=x3−6x2+xf(x) = x^{3} - 6x^{2} + x.

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  • Introductory1 mark · 2 min

    Which of the following functions is even?

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  • Standard2 marks · 4 min

    On which interval is f(x)=x4−6x2f(x) = x^{4} - 6x^{2} concave down?

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  • Standard2 marks · 4 min

    Find the local maximum value of f(x)=x3−3x2−9x+5f(x) = x^{3} - 3x^{2} - 9x + 5.

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  • Standard2 marks · 4 min

    The function below has exactly one vertical asymptote. Find its xx-value.

    f(x)=x2−4x2−5x+6f(x) = \frac{x^{2} - 4}{x^{2} - 5x + 6}
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  • Standard3 marks · 4 min

    Find the xx-coordinate of the local maximum of f(x)=xe−2xf(x) = xe^{-2x}.

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  • Challenging3 marks · 5 min

    A function ff, defined for all real xx, has derivative f′(x)=(3−x)(x+1)e−xf'(x) = (3 - x)(x + 1)e^{-x}. You are given f′f', not ff. At which value of xx does ff have a local maximum?

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  • Challenging4 marks · 6 min

    Let f(x)=x2+1x−1f(x) = \dfrac{x^{2} + 1}{x - 1}. Find the xx-coordinate of its local minimum.

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  • Capstone6 marks · 12 min

    Let f(x)=x2x−1f(x) = \dfrac{x^{2}}{x - 1}. Find its domain and asymptotes (vertical and slant), the intervals on which it increases and decreases, its local extrema, and the intervals on which it is concave up and concave down. Then sketch the graph.

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