Problem 5 - 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. $$ h(x)=x^{2}-4 x+2 ;[-2,2] $$

Step-by-Step Solution

Answer

The average rate of change is calculated as the slope of the secant line passing through the points corresponding to the function values at the interval endpoints. The instantaneous rates of change at these points, obtained from the derivative, are generally different in nonlinear functions.

Step 1: Compute Function Values at the Endpoints

Firstly, evaluate \(h(x)\) at the endpoints of the interval \([-2,2]\). This means to plug in x = -2 and x = 2 into the function.

Step 2: Calculate the Average Rate of Change

The average rate of change is the difference of these function values divided by the difference in x. This means to subtract \(h(-2)\) from \(h(2)\) and divide the result by \(2 - (-2)\).

Step 3: Find the Derivative

Calculate the derivative of \(h(x)\), which represents the rate of change of the function at any point x.

Step 4: Calculate the Instantaneous Rates of Change

Evaluate the derivative at -2 and 2 to find the instantaneous rates of change at the endpoints of the interval.

Step 5: Compare the Rates of Change

Lastly, compare the average rate of change with the instantaneous rates of change at the endpoints. If they are equal, then the function is linear; otherwise, it is nonlinear.