Problem 9 - College Algebra and Calculus: An Applied Approach

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. $$ f(x)=\frac{1}{x} ;[1,4] $$

Step-by-Step Solution

Answer

The average rate of change of the function on the interval [1,4] is -0.5. The instantaneous rates of change at the endpoints of the interval are -1 and -1/16 respectively. Thus, the average rate of change isn't same as the instantaneous rates of change at these endpoints.

Step 1: Calculate the Average Rate of Change

First, find the average rate of change by plugging the x-values from the interval into the function and subtracting the resulting y-values. This gives you the change in y. Divide this by the change in x, or the length of the interval, to find the average rate of change. \[Average\ Rate\ of\ Change = \frac{f(4) - f(1)}{4-1} = \frac{\frac{1}{4} - \frac{1}{1}}{4-1}\]

Step 2: Find the Instantaneous Rate of Change

To find the instantaneous rates of change, take the derivative of the function. The derivative of \(f(x)=\frac{1}{x}\) is \(f'(x)=-\frac{1}{x^2}\). Then, substitute the x-values from the ends of the interval into the derivative to find the instantaneous rates. \[f'(1)= -\frac{1}{1^2} = -1, f'(4) = -\frac{1}{4^2} = -\frac{1}{16}\]

Step 3: Compare the Rates

Now, compare the average rate of change to the instantaneous rates of change at the endpoints. The average rate of change is might not same as the instantaneous rates at the endpoints, due to the nature of the function and its curve within the interval [1,4].