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Tangent Lines and Normal Lines Practice Problems

Find the point (a, f(a)), find the slope f′(a), then write the line. Type the right-hand side of y = ... for each line.

  1. 1.

    Find the tangent line to at . Type the right-hand side of y = ..., for example 6x - 9.

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    The point is and the slope is . So , which is .

  2. 2.

    Find the tangent line to at .

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    The point is and the slope is . So , which is .

  3. 3.

    Find the slope of the tangent line to at .

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    , so at the slope is .

  4. 4.

    Find the tangent line to at .

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    The point is . , so the slope is . Then , which is .

  5. 5.

    Find the tangent line to at .

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    The point is . , so the slope is . Then , which is .

  6. 6.

    Find the tangent line to at .

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    The point is and the slope is , so the tangent line is .

  7. 7.

    Find the tangent line to at . Type e for the number e.

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    The point is and the slope is . Then , so . It passes through the origin.

  8. 8.

    Find the normal line to at .

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    The point is and the tangent slope is , so the normal slope is . Then , which is .

  9. 9.

    Find the normal line to at .

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    The point is and the tangent slope is , so the normal slope is . Then , which is .

  10. 10.

    At what value of is the tangent line to parallel to the line ?

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    Parallel means slope 3. gives .

  11. 11.

    Find the tangent line to the circle at .

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    Implicit differentiation gives . Then , which is .

  12. 12.

    Find the tangent line to at .

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    The point is . The product rule gives , so the slope is . Then , which is .
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MPhil in Mathematics, Founder of Mathovia
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Tangent Lines and Normal Lines Practice Problems

These problems follow the tangent lines and normal lines lesson. They start with tangent lines to polynomials and build up to normal lines, a given slope, a circle and a product.

The method

  1. The point: plug into the function to get .
  2. The slope: differentiate and plug in to get . For a normal line, use .
  3. The line: , then solve for .

Common mistakes to watch for

  • Leaving in the slope instead of plugging in .
  • Taking the y-coordinate from instead of from .
  • Using for the normal slope instead of the negative reciprocal .

Frequently Asked Questions

How should I type my answer?+

Type only the right-hand side of y = ..., in terms of x. For example, for y = 6x − 9 type 6x - 9. Fractions like x/4 + 1 and constants like e and pi are fine.

Do I have to write the line in slope-intercept form?+

No. Any equivalent expression is accepted, so 2 + (x - 4)/4 and x/4 + 1 are both marked correct.

More practice problems

Power Rule Practice Problems Implicit Differentiation Practice Problems Limit Definition of the Derivative Practice Problems