QuestionFind the average rate of change of $f(x)=\frac{1}{x+2}$ on $[8,8+h]$ . Express your answer in terms of $h$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\frac{1}{x+}$ . We are finding the average rate of change on the interval $[8,8+h]$

STEP 2

The average rate of change of a function $f(x)$ over the interval $[a,b]$ is given by the formula $\frac{f(b)-f(a)}{b-a}$

STEP 3

In our case, $a=8$ and $b=8+h$ . So, we need to find the values of $f(a)$ and $f(b)$ .

STEP 4

First, let's find $f(a)$ by substituting $a=8$ into the function $f(x)$ $f(a) = f(8) = \frac{1}{8+2}$

STEP 5

Calculate the value of $f(a)$ $f(a) = \frac{1}{10}$

STEP 6

Now, let's find $f(b)$ by substituting $b=8+h$ into the function $f(x)$ $f(b) = f(8+h) = \frac{1}{(8+h)+2}$

STEP 7

implify the expression for $f(b)$ $f(b) = \frac{1}{10+h}$

STEP 8

Now that we have $f(a)$ and $f(b)$ , we can substitute these into the formula for the average rate of change $\frac{f(b)-f(a)}{b-a} = \frac{\frac{1}{10+h} - \frac{1}{10}}{(8+h)-8}$

STEP 9

implify the denominator of the fraction $\frac{f(b)-f(a)}{b-a} = \frac{\frac{}{+h} - \frac{}{}}{h}$

STEP 10

To simplify the numerator, find a common denominator for the two fractions $\frac{}{10+h} - \frac{}{10} = \frac{10 - (10+h)}{10(10+h)} = \frac{-h}{10(10+h)}$

STEP 11

Substitute this back into the formula for the average rate of change $\frac{f(b)-f(a)}{b-a} = \frac{\frac{-h}{10(10+h)}}{h}$

STEP 12

implify the expression by cancelling out $h$ in the numerator and denominator $\frac{f(b)-f(a)}{b-a} = \frac{-}{10(10+h)}$ So, the average rate of change of $f(x)$ on the interval $[8,8+h]$ is $\frac{-}{10(10+h)}$ .

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