Problem 7 - 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. $$ f(x)=3 x^{4 / 3} ;[1,8] $$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x)=3 x^{4/3}\) over the interval [1,8] is 20.14. The instantaneous rate of change at the endpoints of the interval are 4 and 16, respectively. The average rate of change is higher than the instantaneous rates at the endpoints.

Step 1: Drawing the Graph

The function \(f(x)=3 x^{4/3}\) is graphed over the interval [1,8]. Graphing is usually done with the help of a graphing calculator or software.

Step 2: Calculate the Average Rate of Change

Next, calculate the average rate of change over the interval [1,8]. This is found by using the formula \(\frac{f(b)-f(a)}{b-a}\) where a and b are the endpoints of the interval. Here, \(f(a) = 3(1)^{4/3} = 3\) and \(f(b) = 3(8)^{4/3} = 144\). The average rate of change therefore is \(\frac{144 - 3}{8 - 1} = 20.14\).

Step 3: Calculate the Instantaneous Rates of Change

The instantaneous rate of change at a point is the derivative of the function at that point. The derivative of \(f(x) = 3x^{4/3}\) is \(f'(x) = 4x^{1/3}\). So, the instantaneous rate at x=1 is \(4(1)^{1/3} = 4\) and at x=8 is \(4(8)^{1/3} =16\).

Step 4: Comparison of Rates

Comparing the rates of change, it can be observed that the average rate of change (20.14) over the interval is larger than both of the instantaneous rates of change at the endpoints (4 and 16).