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Math Practice

Difference Quotient Practice Problems

Find f(x + h), subtract f(x), divide by h and simplify until the h in the denominator cancels. The problems move from lines to rational and square root functions.

  1. 1.

    Find the difference quotient for .

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    , and .

  2. 2.

    Find the difference quotient for .

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    , so the quotient is , the slope.

  3. 3.

    Find the difference quotient for .

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    ; dividing by gives .

  4. 4.

    Find the difference quotient for .

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    The constant 4 cancels: over is .

  5. 5.

    Find the difference quotient for .

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    ; divide by to get .

  6. 6.

    Evaluate the difference quotient for and simplify the result.

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    , so the quotient is .

  7. 7.

    Find the difference quotient for .

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    ; dividing by gives .

  8. 8.

    Find the difference quotient for .

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    ; dividing by gives .

  9. 9.

    Find the difference quotient for .

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    ; dividing by gives .

  10. 10.

    Find the difference quotient for .

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    , so the quotient is .

  11. 11.

    Find the difference quotient for and rationalize the numerator. Type square roots as sqrt( ).

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    Multiply top and bottom by : the numerator becomes , which cancels, leaving .

  12. 12.

    For , evaluate the difference quotient at with .

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    and , so . The formula agrees: .
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MPhil in Mathematics, Founder of Mathovia
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Difference Quotient Practice Problems

These problems follow the difference quotient lesson. Use brackets when you substitute , and distribute the minus sign when you subtract .

The formula

Common mistakes to watch for

  • Not substituting everywhere. Every in becomes : gives , not .
  • Dropping the subtraction brackets. subtracts every term of ; bracket it.
  • Cancelling too early. Expand and simplify the numerator first; only factor out and cancel at the end.
  • Leaving an in the denominator. If the denominator still has , the simplification is not finished.

Frequently Asked Questions

What is the difference quotient?+

The expression (f(x + h) − f(x)) / h, the slope of the secant line between x and x + h.

How do I know my simplification is right?+

After subtracting f(x), every remaining term should contain h. If a term without h is left over, recheck the expansion.

How do I type my answers?+

Type numbers without units, fractions like 5/3, and expressions like 2x + h. For intervals use parentheses or brackets and oo for infinity, for example (2, oo).

Calculators for this topic

Difference Quotient Calculator

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