Given $$ f(x)=\frac{1}{x+11} $$, find the average rate of change of $$ f(x) $$ on the interval $$ [3,3+h] $$. Your answer will be an expression involving $$ h $$. Submit Question

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UpStudy AI · Accepted Answer

The average rate of change of a function on an interval $$[a, b]$$ is given by

$$
\frac{f(b)-f(a)}{b-a}.
$$

For the function $$ f(x)=\frac{1}{x+11} $$ on the interval $$[3, 3+h]$$,

1. Compute $$ f(3) $$:
   $$
   f(3)=\frac{1}{3+11}=\frac{1}{14}.
   $$

2. Compute $$ f(3+h) $$:
   $$
   f(3+h)=\frac{1}{(3+h)+11}=\frac{1}{14+h}.
   $$

3. Form the difference and divide by $$ h $$:
   $$
   \frac{f(3+h)-f(3)}{h}=\frac{\frac{1}{14+h}-\frac{1}{14}}{h}.
   $$

4. To combine the fractions in the numerator, use a common denominator:
   $$
   \frac{1}{14+h}-\frac{1}{14}=\frac{14- (14+h)}{14(14+h)}=\frac{-h}{14(14+h)}.
   $$

5. Now, divide by $$ h $$:
   $$
   \frac{-h}{h \cdot 14(14+h)}=-\frac{1}{14(14+h)}.
   $$

Thus, the average rate of change of $$ f(x) $$ on the interval $$[3, 3+h]$$ is

$$
-\frac{1}{14(14+h)}.
$$
The average rate of change of $$ f(x) $$ on the interval $$[3, 3+h]$$ is $$ -\frac{1}{14(14+h)} $$.

Given , find the average rate of change of on the interval . Your answer
will be an expression involving .
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Given , find the average rate of change of on the interval . Your answer will be an expression involving . Submit Question