Given $f(x)=17-x^2$, what is the average rate of change in $f(x)$ over the interval $[1, 5]$?
The average rate of change of a function \(f(x)\) over a given interval \([a, b]\) is calculated by finding the slope of the secant line that connects the points \((a, f(a))\) and \((b, f(b))\) on the graph of the function. The formula is given by:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
In this problem, the function is \(f(x) = 17 - x^2\) and the interval is \([a, b] = [1, 5]\).
First, we evaluate the function \(f(x)\) at the endpoints of the interval, \(x = 1\) and \(x = 5\):
For \(x = 1\):
\[ f(1) = 17 - (1)^2 = 17 - 1 = 16 \]
For \(x = 5\):
\[ f(5) = 17 - (5)^2 = 17 - 25 = -8 \]
Now, substitute these values into the average rate of change formula:
\[ \text{Average Rate of Change} = \frac{f(5) - f(1)}{5 - 1} \]
Substituting the calculated values:
\[ \text{Average Rate of Change} = \frac{-8 - 16}{4} = \frac{-24}{4} \]
Simplifying the fraction:
\[ \text{Average Rate of Change} = -6 \]
The average rate of change of the function \(f(x) = 17 - x^2\) over the interval \([1, 5]\) is \(-6\), which matches option b. Therefore, the final answer is \(\boxed{-6}\).
The concept of average rate of change is vital in many fields, including physics and economics. For example, if you are tracking the speed of a car on a highway, calculating the average rate of change of its position over a certain time interval tells you the average velocity during that trip. Similarly, a business might calculate the average rate of change in their daily revenue over a week to understand their growth or decline trend. Understanding this concept helps you interpret how quickly a quantity is changing on average relative to another.