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Linear Approximation Practice Problems

Pick a nearby point a where f is easy to evaluate, find f(a) and f′(a), and use L(x) = f(a) + f′(a)(x − a). Give estimates exactly as L gives them.

  1. 1.

    Find the linearization of at . Type it in terms of x, for example 6x - 9.

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    and , so .

  2. 2.

    Find the linearization of at .

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    and , so .

  3. 3.

    Use the linearization of at to estimate .

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    and , so . The true value is .

  4. 4.

    Use the linearization of at to estimate .

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    , so .

  5. 5.

    Use the linearization of at to estimate .

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    , so . The true value is , so this is an underestimate, as expected since is concave up.

  6. 6.

    Use a linear approximation at to estimate . Give at least 4 decimal places, or an exact fraction.

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    , and . So .

  7. 7.

    Use to estimate .

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    With and : . The true value is .

  8. 8.

    Use the linearization of at to estimate .

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    , so . The true value is .

  9. 9.

    Find the linearization of at .

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    and , so . Then .

  10. 10.

    For , find the differential when and .

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    . At : .

  11. 11.

    A sphere's radius is measured as 5 cm with a possible error of 0.1 cm. Use differentials to estimate the possible error in its volume, in cm³. You can type pi.

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    cm³.

  12. 12.

    Find the linearization of at . You can type pi.

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    and , so . Then .
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MPhil in Mathematics, Founder of Mathovia
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Linear Approximation Practice Problems

These problems follow the linear approximation lesson. They start with finding linearizations and build up to estimating roots and powers and using differentials to estimate error.

The method

  1. Choose close to the value you want, where is easy to find exactly.
  2. Compute and .
  3. Write and evaluate it.
  4. For differentials, use .

Common mistakes to watch for

  • Choosing far from the value you want, which makes the estimate poor.
  • Using degrees in trig approximations instead of radians.
  • Using in place of the number .

Frequently Asked Questions

How should I type my answer?+

For a linearization, type L(x) in terms of x, such as 6x - 9. For an estimate, type the number the linearization gives, such as 4.025. Exact forms like 10pi and 271/90 are accepted.

Why is my estimate marked wrong when it is closer to the true value?+

These problems ask for the value the tangent line gives, not the true value. For example, the linear estimate of ln 1.05 is 0.05, even though ln 1.05 is really 0.04879.

More practice problems

Tangent Lines and Normal Lines Practice Problems Higher-Order Derivatives Practice Problems Related Rates Practice Problems