Find the average rate of change of the function $f(x)=\frac{1}{x-7}$ as $x$ changes from $x=-2$ to $x=1$.
Answer
Final Answer: The average rate of change of the function \(f(x)=\frac{1}{x-7}\) as \(x\) changes from \(x=-2\) to \(x=1\) is \(\boxed{-0.0185}\).
Step 1 ：We are given the function \(f(x)=\frac{1}{x-7}\) and we need to find the average rate of change as \(x\) changes from \(x=-2\) to \(x=1\).
Step 2 ：The formula for the average rate of change of a function \(f(x)\) over the interval \([a, b]\) is \(\frac{f(b) - f(a)}{b - a}\).
Step 3 ：Substitute \(a=-2\) and \(b=1\) into the formula, we get \(\frac{f(1) - f(-2)}{1 - (-2)}\).
Step 4 ：Substitute \(f(1)\) and \(f(-2)\) into the formula, we get \(\frac{\frac{1}{1-7} - \frac{1}{-2-7}}{1 - (-2)}\).
Step 5 ：Simplify the above expression to get the average rate of change, which is approximately -0.0185.
Step 6 ：Final Answer: The average rate of change of the function \(f(x)=\frac{1}{x-7}\) as \(x\) changes from \(x=-2\) to \(x=1\) is \(\boxed{-0.0185}\).
