Problem 3 - 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. $$ f(t)=3 t+5 ;[1,2] $$

Step-by-Step Solution

Answer

The instantaneous rate of change of a linear function \(f(t)=3t+5\) at any point, including at the endpoints of the interval [1,2], is equal to the average rate of change over the interval, which is 3.

Step 1: Graphing the Function

Use a graphing program or a calculator to plot the function \(f(t)=3t+5\). The linear graph should clearly show a positive slope.

Step 2: Calculating the Average Rate of Change

The average rate of change of the function on the interval [1,2] is defined as the change in the function value divided by the change in the time interval, i.e., \(\frac{f(2)-f(1)}{2-1}\). Fill in \(f(2)\) and \(f(1)\) with their respective values from the function \(f(t)\). After substitution, it will be found that the average rate of change of the given function on interval [1,2] is 3.

Step 3: Calculating Instantaneous Rates of Change

For a linear function like \(f(t)=3t+5\), the instantaneous rate of change at any point is simply the slope of the line which is the constant 3.

Step 4: Comparing the Average Rate and Instantaneous Rates of Change

Now, have a look on the average rate of change and the instantaneous rates of change. You will find they are the same, which is 3 in this case. For linear functions, this is always the case; the average rate of change over any interval is equal to the instantaneous rate of change at any point.