Problem 8 - 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. $$ f(x)=x^{3 / 2} ;[1,4] $$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x)=x^{3 / 2}\) on the interval [1,4] is computed using the formula \((f(b) - f(a)) / (b - a)\). The instantaneous rates of change at the interval's endpoints are given by the derivative of the function, evaluated at these points which is \((3/2) \sqrt{x}\). Comparing these values provides insight into the function's behavior over this interval.

Step 1: Compute Average Rate of Change

The average rate of change of a function on an interval [a, b] is given by the formula \((f(b) - f(a))/(b - a)\). Substitute \(a = 1\) and \(b = 4\) into the formula. Thus, the average rate of change of \(f\) on [1,4] is \((f(4) - f(1))/(4 - 1) = ((4^{3/2} - 1^{3/2})/ (4 - 1))\). Simplify this to get the answer.

Step 2: Compute Instantaneous Rate of Change at the Endpoints

The instantaneous rate of change at a point is the derivative of the function evaluated at that point. The derivative of \(f(x)\) is given by \(f'(x) = (3/2) \sqrt{x}\). Compute \(f'(1)\) and \(f'(4)\) to find the instantaneous rates of change at the endpoints of the interval [1, 4].

Step 3: Compare the Instantaneous Rates of Change with the Average Rate of Change

Compare the result from Step 1 with the results from Step 2. This will provide insight into how the function behaves on the interval [1,4], specifically how the average rate of change relates to the instantaneous rates of change at the endpoints of the interval.