Chapter 3, Problem 12RE

Solution

To find: the y -intercept and the horizontal asymptote of the given function.

y -intercept − $\left(0,\frac{50}{7}\right)$

Horizontal asymptotes are $y=10$ and $y=0$ .

Given information:

The given function is $f\left(x\right)=\frac{50}{5+2e^{-0.04x}}$ .

Definition Used:

Logistic Growth Function:

Let a , b , c , and k be positive constants, with $b<1$ . A logistic growth function in x is a function that can be written in the form

  $f\left(x\right)=\frac{c}{1+a\cdot b^x}\text{ or }f\left(x\right)=\frac{c}{1+a\cdot e^{-kx}}$

Where the constant c is the limit to growth.

Explanation:

First, to find the y -intercept substitute $x=0$ in the given function.

  $\begin{array}{c}f\left(0\right)=\frac{50}{5+2e^{-0.04\left(0\right)}}\\ =\frac{50}{5+2\left(1\right)}\\ =\frac{50}{7}\end{array}$

So, y- intercept is $\left(0,\frac{50}{7}\right)$ .

Now, the given function can be rewritten in the following manner to convert it into general form of logistic growth model.

  $\begin{array}{c}f\left(x\right)=\frac{50}{5+2e^{-0.04x}}\\ =\frac{\frac{50}{5}}{\frac{5+2e^{-0.04x}}{5}}\\ =\frac{10}{\frac{5}{5}+\frac{2e^{-0.04x}}{5}}\\ =\frac{10}{1+0.4e^{-0.04x}}\end{array}$

By the above definition, the numerator of a logistic growth function is known as its limit to growth. Since, for any logistic growth function  f ( x ),

  $\underset{x\to \infty }{\lim }f\left(x\right)=c$

and thus  $y=c$  is a horizontal asymptote.

Here, the numerator of the given function $f\left(x\right)=\frac{10}{1+0.4e^{-0.04x}}$ is 10.

So, $y=10$  is a horizontal asymptote.

Also, every logistic growth function will approach zero as  x  tends to negative infinity, that is, for a logistic growth function  f ( x ),

  $\underset{x\to -\infty }{\lim }f\left(x\right)=0$

and thus  $y=0$ is a horizontal asymptote.