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How to evaluate the difference quotient?

Understand the Problem

The question is asking how to compute the difference quotient, which is a fundamental concept in calculus that involves finding the average rate of change of a function over a specific interval. The difference quotient is typically expressed as (f(x+h) - f(x)) / h.

Answer

The difference quotient for the function $f(x) = x^2$ is $2x + h$.
Answer for screen readers

The difference quotient for the function $f(x) = x^2$ is given by $2x + h$.

Steps to Solve

  1. Identify the Function

    First, you need to know what the function $f(x)$ is for which you want to compute the difference quotient. For example, let’s say $f(x) = x^2$.

  2. Calculate $f(x+h)$

    Now, calculate $f(x+h)$. For our function, this means substituting $x+h$ into the function:

    $$ f(x+h) = (x+h)^2 $$

    Expanding this gives:

    $$ f(x+h) = x^2 + 2xh + h^2 $$

  3. Compute the Difference Quotient

    Now, use the difference quotient formula $(f(x+h) - f(x)) / h$. Substitute $f(x+h)$ and $f(x)$ into the formula:

    $$ \frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2) - x^2}{h} $$

    Simplifying the numerator:

    $$ \frac{2xh + h^2}{h} $$

    Now divide each term by $h$ (assuming $h \neq 0$):

    $$ 2x + h $$

  4. Final Expression

    The final expression for the difference quotient is:

    $$ 2x + h $$

The difference quotient for the function $f(x) = x^2$ is given by $2x + h$.

More Information

The difference quotient is a crucial part of calculus as it lays the groundwork for understanding derivatives, which measure how a function changes at a specific point. When $h$ approaches 0, this expression approaches the derivative of the function.

Tips

  • Forgetting to simplify the expression before dividing by $h$ can lead to incorrect results. Always simplify $f(x+h) - f(x)$ fully before performing the division.
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