Chapter 6.4, Problem 86E

The Logistic Function and Grant Spending

a. Show that the logistic function $f\left(x\right)=\frac{N}{1+A{e}^{-kx}}$ can be written in the form

$f\left(x\right)=\frac{N{e}^{kx}}{A+e^{kx}}$

[HINT: See the note after Example 5 in Section 6.2.]

b. Use the result of part (a) and a suitable substitution to show that

${\displaystyle \int \frac{N}{1+A{e}^{-kx}}dx=\frac{N\ln \left(A+e^{kx}\right)}{k}+C}$

c. The rate of graduation of private high school students in the United States for the period 1993–2003 was approximately

$s\left(t\right)=11+\frac{20}{1+1,{800}^{-0.9t}}\text{ billion dollars per year }\left(3\le t\le 13\right),$

Where t years since 1994.61 Use the result of part (b) to estimate the total number of private high school graduates over the period 1998–2003.