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Math Practice

Curve Sketching Practice Problems

Use f′ for rising and falling and the local extrema, f″ for concavity and inflection points, and limits for asymptotes. Give intervals in interval notation, such as (-inf, -1) U (1, inf), and lists of x-values separated by commas.

  1. 1.

    Find the intervals where is increasing.

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    , which is positive outside . So is increasing on .

  2. 2.

    Find the x-value of the local maximum of .

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    changes from + to − at and from − to + at . The local maximum is at , where .

  3. 3.

    Find the interval where is decreasing.

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    , which is negative between its zeros. So is decreasing on .

  4. 4.

    Find the y-coordinate of the inflection point of .

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    changes sign at . Then , so the inflection point is .

  5. 5.

    Find the x-values of the local minima of .

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    . The signs are − on , + on , − on , + on . So there are local minima at and a local maximum at .

  6. 6.

    Find the interval where is concave down.

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    , which is negative between and . So is concave down on .

  7. 7.

    Find the x-values of all local extrema of .

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    . Critical points are and , but is negative on both sides of , so is not an extremum (it's a flat inflection point). The only local extremum is a minimum at .

  8. 8.

    Find the vertical asymptote of .

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    The denominator is zero at , and the numerator is there, so is a vertical asymptote.

  9. 9.

    Find the horizontal asymptote of .

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    The degrees are equal, so is the ratio of leading coefficients, . The horizontal asymptote is .

  10. 10.

    Find the x-value of the inflection point of .

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    and . Since , changes sign at .

  11. 11.

    The derivative of a function is . Find the x-values of all local extrema of .

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    , so the sign of follows : − before , + after. has a local minimum at . At , touches zero without changing sign, so there's no extremum.

  12. 12.

    The derivative of a function is . Find the x-values of the inflection points of .

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    , which changes sign at and at . Equivalently, the graph of has a local maximum at and a local minimum at .
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MPhil in Mathematics, Founder of Mathovia
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Curve Sketching Practice Problems

These problems follow the curve sketching lesson. Each one asks for a single feature you’d need to draw the graph: an interval, an extremum, an inflection point or an asymptote.

The checklist

  1. Domain, intercepts and symmetry.
  2. Asymptotes, from limits.
  3. Sign chart for : increasing, decreasing, local extrema.
  4. Sign chart for : concavity and inflection points.
  5. Plot the key points and connect them with the matching shapes.

Common mistakes to watch for

  • Calling a critical point an extremum when doesn’t change sign.
  • Calling a zero of an inflection point when doesn’t change sign.
  • Mixing up and when reading a derivative graph.

Frequently Asked Questions

How do I type an interval or a union?+

Use parentheses and inf for infinity, and U for union: (-inf, -1) U (1, inf). Inequalities such as x < -1 or x > 1 are accepted too.

Why isn't x = 0 an answer in problem 7?+

The derivative is zero there, but it doesn't change sign, so the graph keeps falling through a flat spot. A critical point is only a local maximum or minimum if f′ changes sign.

More practice problems

First Derivative Test Practice Problems Concavity and Inflection Points Practice Problems Critical Points Practice Problems