Chapter 7.7, Problem 35PPE

To determine

Toshow :Example of exponential function $y=a{b}^x$ which is exponential growth as well as exponential decay function.

Explanation of Solution

Given information:exponential function of the form $y=a{b}^x$

Solution:

The original exponential function is $y=a{b}^x$

The exponential growth function can be expressed as $y=a{\left(1+r\right)}^x$ and the exponential decay function is written as $y=a{\left(1-r\right)}^x$ .

Here the growth rate (r) is determined as $b=1+r$ and the decay rate (r) is $b=1-r$

Where $a$ is the initial value before measuring growth or decay

For example let the population of a home town in 2020 was 35000 with an annual rate of increase of 2.5 percent.

Therefore after one year the growth rate of the town is $\begin{array}{l}35000\left(1+0.024\right)\\ \Rightarrow 35000\left(1.024\right)\end{array}$

The growth factor is $\left(1.024\right)$

Hence the equation is given by $y=a{b}^x=35000{\left(1.024\right)}^x$

And before one year the decay rate of the town is $\begin{array}{l}35000\left(1-0.024\right)\\ \Rightarrow 35000\left(0.976\right)\end{array}$

Hence the equation is given by $y=a{b}^x=35000{\left(0.976\right)}^x$