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Question

a. Show that the logistic function $f(x)=\frac{N}{1+A b^{-x}}$ can be written in the form $f(x)=\frac{N b^{x}}{A+b^{x}}$. b. Use the result of part (a) and a suitable substitution to show that $\int \frac{N}{1+A b^{-x}} d x=\frac{N \ln \left(A+b^{x}\right)}{\ln b}+C$. The rate of graduation of private high school students in the United States for the period 1994-2008 was approximately $r(t)=220+\frac{110}{1+3.8(1.27)^{-t}}$ thousand students per year $(0 \leq t \leq 14)$ t years since 1994. Use the result of part (b) to estimate the total number of private high school graduates over the period 2000-2008.

Solution

VerifiedAnswered 2 years ago

Answered 2 years ago

Step 1

1 of 10$\textbf{a.}$

Note that the $b^{-x}$ in the denominator can be written as

$b^{-x} = \dfrac{1}{b^x}$

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