Question

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=\frac{1}{\sqrt{x}} ;[1,4] $$

Short answer

After performing the calculations, you will find the average rate of change over the interval and the instantaneous rates of change at the end points of the interval. Comparing these figures will give an idea about the function's behavior on the given interval.

Step-by-step solution

Express the Function

The function is already provided: \(f(x)=\frac{1}{\sqrt{x}}\). Our target interval is from \(x=1\) to \(x=4\). In order to use the graphing utility, you will need to familiarize yourself with the graphing tool's functionality and input this function into it.

Calculate Average Rate of Change

The average rate of change of a function over an interval [a, b] is given by \( \frac{f(b) - f(a)}{b - a}\). By plugging the provided interval values into the function, the average rate becomes \( \frac{f(4) - f(1)}{4 - 1}\). Solve this equation to find the average rate of change.

Calculate Instantaneous Rates of Change

The instantaneous rate of change, also known as the derivative, can be found by applying the power rule for differentiation to \(f(x) = x^{-1/2}\). The derivative is \(f'(x) = -\frac{1}{2}x^{-3/2}\). By substituting \(x = 1\) and \(x = 4\), the instantaneous rates of change at the endpoints of the interval can be obtained.

Comparison

Now, compare the values obtained in step 2 and step 3 to analyze the behavior of the function at these intervals. Consider how the instantaneous rates of change at the endpoints approximate the average rate of change over the whole interval.