Chapter 2, Problem 1GYR

To determine

Define the average rate of change of function g(t) over the interval $t=a$ to $t=b$. Also define how its related to secant line.

Explanation of Solution

Calculation:

The average rate of change of function g(t) over the interval $t=a$ to $t=b$.

The average rate is defined as if one function is changing with respect to other function.

For the above case Average rate of change is equal to change in $g(t)$ over the time interval.

$\frac{g(b)-g(a)}{(b-a)}$.

If $\left(a,g(a)\right)$ and $\left(b,g(b)\right)$ are considered as two points on the graph.

Now curve $y=g(t)$, then slope of secant line joining these two points are given by

$slopem=\frac{y_2-y_1}{x_2-x_1}\mathrm{..................}(2)$

And $\frac{g(b)-g(a)}{(b-a)}\mathrm{...............}(2)$

From equation (1) and (2) it is concluded that the rate of change is the slope of the secant line through the point $P\left(a,g(a)\right),Q\left(b,g(b)\right)$.