Chapter 2, Problem 1GYR

To determine

Calculate the average rate of change of the function over the interval and provide the relation of the function to a secant line.

Answer to Problem 1GYR

The average rate of change of the function is $\underset{\_}{\frac{g\left(b\right)-g\left(a\right)}{b-a}}$.

Explanation of Solution

Given information:

The function is $g\left(t\right)$.

The interval from $t=a$ to $t=b$.

Calculation:

Calculate the average rate of change of the function $g\left(t\right)$ over the interval as shown below.

$\begin{array}{c}\text{Average}\text{rate}\text{of}\text{change}=\frac{\Delta g\left(t\right)}{\Delta t}\\ =\frac{g\left(b\right)-g\left(a\right)}{b-a}\end{array}$

Hence, the average rate of change of the function is $\underset{\_}{\frac{g\left(b\right)-g\left(a\right)}{b-a}}$.

The slope of the line through the points $\left(a,g\left(a\right)\right)$ and $\left(b,g\left(b\right)\right)$ is the rate of change of the function over the interval $\left(a,b\right)$.

A line joining the points $\left(a,g\left(a\right)\right)$ and $\left(b,g\left(b\right)\right)$ is known to be the secant line.

That is the slope of secant line is identical to the average rate of change of the function.