Question in image Compute P'(0) and use it to approximate the population after six months. \ A colony of penguins grows according to the logistic model and its population is given by P(t)penguins after t years where7500P(t) =1+4e-t/4'with an initial population of P(0) = 1500 penguins.

Given that, the logistic function is Pt=75001+4e-t4. Differentiate the function with respect to x as…

Answered: Compute P'(0) and use it to approximate…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section: Chapter Questions

Problem 64RE: What is the carrying capacity for a population modeled by the logistic equation...

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Compute P'(0) and use it to approximate the population after six months. \

A colony of penguins grows according to the logistic model and its population is given by P(t)
penguins after t years where
7500
P(t) =
1+4e-t/4'
with an initial population of P(0) = 1500 penguins.

Expert Solution

Step 1

Given that, the logistic function is $P (t) = \frac{7500}{1 + 4 e^{- \frac{t}{4}}}$ .

Differentiate the function with respect to x as follows,

$\begin{array}{rcl} P' (t) & = & \frac{d}{d t} (\frac{7500}{1 + 4 e^{- \frac{t}{4}}}) \\ = & 7500 \frac{d}{d t} (\frac{1}{1 + 4 e^{- \frac{t}{4}}}) \\ = & \frac{7500 e^{- \frac{x}{4}}}{{(1 + 4 e^{- \frac{x}{4}})}^{2}} \end{array}$

Step 2

Substitute t=0 in $\begin{array}{rcl} P' (t) & = & \frac{7500 e^{- \frac{x}{4}}}{{(1 + 4 e^{- \frac{x}{4}})}^{2}} \end{array}$ .

$\begin{array}{rcl} P' (0) & = & \frac{7500 e^{- \frac{(0)}{4}}}{{(1 + 4 e^{- \frac{(0)}{4}})}^{2}} \\ = & \frac{7500 e^{0}}{{(1 + 4 e^{0})}^{2}} \\ = & \frac{7500}{{(1 + 4)}^{2}} \\ = & \frac{7500}{25} \\ = & 300 \end{array}$

Therefore, the value of $P' (0) = 300$ .

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