Logistic growth worksheet

A logistic equation grows almost exponentially at first and then levels off at the carrying capacity M. The fastest growth happens exactly halfway there, at M/2.

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Sigma Prep · sigmaprep.io/worksheets Name Logistic Growth Date Period Answer each question. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{650})$ , $P (0) = 130$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $\frac{d P}{d t} = 0.25 P (1 - \frac{P}{1000})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{200}{1 + 7 e^{- 0.25 t}}$ . Find $P (4)$ , to the nearest tenth. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{550})$ , $P (0) = 88$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $\frac{d P}{d t} = 0.2 P (1 - \frac{P}{800})$ , $P (0) = 20$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{800}{1 + 31 e^{- 0.5 t}}$ . Find $P (12)$ , to the nearest tenth. $\frac{d P}{d t} = 0.2 P (1 - \frac{P}{200})$ , $P (0) = 32$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{600})$ , $P (0) = 30$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{1000}{1 + 24 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth. $\frac{d P}{d t} = 0.5 P (1 - \frac{P}{700})$ , $P (0) = 252$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{950})$ , $P (0) = 5$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{700}{1 + 19 e^{- 0.3 t}}$ . Find $P (3)$ , to the nearest tenth. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{200})$ , $P (0) = 24$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $\frac{d P}{d t} = 0.3 P (1 - \frac{P}{550})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{900}{1 + 44 e^{- 0.1 t}}$ . Find $P (2)$ , to the nearest tenth. $\frac{d P}{d t} = 0.2 P (1 - \frac{P}{1000})$ , $P (0) = 280$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{900})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{150}{1 + e^{- 0.1 t}}$ . Find $P (7)$ , to the nearest tenth. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{300})$ , $P (0) = 96$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $\frac{d P}{d t} = 0.5 P (1 - \frac{P}{350})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Logistic Growth

Date Period

Answer each question.

$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{650})$ , $P (0) = 130$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest.
$\frac{d P}{d t} = 0.25 P (1 - \frac{P}{1000})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$
$P (t) = \frac{200}{1 + 7 e^{- 0.25 t}}$ . Find $P (4)$ , to the nearest tenth.
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{550})$ , $P (0) = 88$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest.
$\frac{d P}{d t} = 0.2 P (1 - \frac{P}{800})$ , $P (0) = 20$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$
$P (t) = \frac{800}{1 + 31 e^{- 0.5 t}}$ . Find $P (12)$ , to the nearest tenth.
$\frac{d P}{d t} = 0.2 P (1 - \frac{P}{200})$ , $P (0) = 32$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest.
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{600})$ , $P (0) = 30$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$
$P (t) = \frac{1000}{1 + 24 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth.
$\frac{d P}{d t} = 0.5 P (1 - \frac{P}{700})$ , $P (0) = 252$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest.
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{950})$ , $P (0) = 5$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$
$P (t) = \frac{700}{1 + 19 e^{- 0.3 t}}$ . Find $P (3)$ , to the nearest tenth.
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{200})$ , $P (0) = 24$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest.
$\frac{d P}{d t} = 0.3 P (1 - \frac{P}{550})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$
$P (t) = \frac{900}{1 + 44 e^{- 0.1 t}}$ . Find $P (2)$ , to the nearest tenth.
$\frac{d P}{d t} = 0.2 P (1 - \frac{P}{1000})$ , $P (0) = 280$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest.
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{900})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$
$P (t) = \frac{150}{1 + e^{- 0.1 t}}$ . Find $P (7)$ , to the nearest tenth.
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{300})$ , $P (0) = 96$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest.
$\frac{d P}{d t} = 0.5 P (1 - \frac{P}{350})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$



Sigma Prep · sigmaprep.io/worksheets Name Logistic GrowthAnswer keyVersion 1 Date Period Answer each question. $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{650})$ , $P (0) = 130$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $650$ ; $325$ $\frac{d P}{d t} = 0.25 P (1 - \frac{P}{1000})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{1000}{1 + 19 e^{- 0.25 t}}$ $P (t) = \frac{200}{1 + 7 e^{- 0.25 t}}$ . Find $P (4)$ , to the nearest tenth. $55.9$ $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{550})$ , $P (0) = 88$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $550$ ; $275$ $\frac{d P}{d t} = 0.2 P (1 - \frac{P}{800})$ , $P (0) = 20$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{800}{1 + 39 e^{- 0.2 t}}$ $P (t) = \frac{800}{1 + 31 e^{- 0.5 t}}$ . Find $P (12)$ , to the nearest tenth. $742.9$ $\frac{d P}{d t} = 0.2 P (1 - \frac{P}{200})$ , $P (0) = 32$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $200$ ; $100$ $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{600})$ , $P (0) = 30$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{600}{1 + 19 e^{- 0.1 t}}$ $P (t) = \frac{1000}{1 + 24 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth. $314.7$ $\frac{d P}{d t} = 0.5 P (1 - \frac{P}{700})$ , $P (0) = 252$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $700$ ; $350$ $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{950})$ , $P (0) = 5$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{950}{1 + 189 e^{- 0.1 t}}$ $P (t) = \frac{700}{1 + 19 e^{- 0.3 t}}$ . Find $P (3)$ , to the nearest tenth. $80.2$ $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{200})$ , $P (0) = 24$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $200$ ; $100$ $\frac{d P}{d t} = 0.3 P (1 - \frac{P}{550})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{550}{1 + 21 e^{- 0.3 t}}$ $P (t) = \frac{900}{1 + 44 e^{- 0.1 t}}$ . Find $P (2)$ , to the nearest tenth. $24.3$ $\frac{d P}{d t} = 0.2 P (1 - \frac{P}{1000})$ , $P (0) = 280$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $1000$ ; $500$ $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{900})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{900}{1 + 35 e^{- 0.1 t}}$ $P (t) = \frac{150}{1 + e^{- 0.1 t}}$ . Find $P (7)$ , to the nearest tenth. $100.2$ $\frac{d P}{d t} = 0.1 P (1 - \frac{P}{300})$ , $P (0) = 96$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $300$ ; $150$ $\frac{d P}{d t} = 0.5 P (1 - \frac{P}{350})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{350}{1 + 6 e^{- 0.5 t}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Logistic GrowthAnswer keyVersion 1

Date Period

Answer each question.

$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{650})$ , $P (0) = 130$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $650$ ; $325$
$\frac{d P}{d t} = 0.25 P (1 - \frac{P}{1000})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{1000}{1 + 19 e^{- 0.25 t}}$
$P (t) = \frac{200}{1 + 7 e^{- 0.25 t}}$ . Find $P (4)$ , to the nearest tenth. $55.9$
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{550})$ , $P (0) = 88$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $550$ ; $275$
$\frac{d P}{d t} = 0.2 P (1 - \frac{P}{800})$ , $P (0) = 20$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{800}{1 + 39 e^{- 0.2 t}}$
$P (t) = \frac{800}{1 + 31 e^{- 0.5 t}}$ . Find $P (12)$ , to the nearest tenth. $742.9$
$\frac{d P}{d t} = 0.2 P (1 - \frac{P}{200})$ , $P (0) = 32$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $200$ ; $100$
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{600})$ , $P (0) = 30$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{600}{1 + 19 e^{- 0.1 t}}$
$P (t) = \frac{1000}{1 + 24 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth. $314.7$
$\frac{d P}{d t} = 0.5 P (1 - \frac{P}{700})$ , $P (0) = 252$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $700$ ; $350$
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{950})$ , $P (0) = 5$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{950}{1 + 189 e^{- 0.1 t}}$
$P (t) = \frac{700}{1 + 19 e^{- 0.3 t}}$ . Find $P (3)$ , to the nearest tenth. $80.2$
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{200})$ , $P (0) = 24$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $200$ ; $100$
$\frac{d P}{d t} = 0.3 P (1 - \frac{P}{550})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{550}{1 + 21 e^{- 0.3 t}}$
$P (t) = \frac{900}{1 + 44 e^{- 0.1 t}}$ . Find $P (2)$ , to the nearest tenth. $24.3$
$\frac{d P}{d t} = 0.2 P (1 - \frac{P}{1000})$ , $P (0) = 280$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $1000$ ; $500$
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{900})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{900}{1 + 35 e^{- 0.1 t}}$
$P (t) = \frac{150}{1 + e^{- 0.1 t}}$ . Find $P (7)$ , to the nearest tenth. $100.2$
$\frac{d P}{d t} = 0.1 P (1 - \frac{P}{300})$ , $P (0) = 96$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. $300$ ; $150$
$\frac{d P}{d t} = 0.5 P (1 - \frac{P}{350})$ , $P (0) = 50$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ $P (t) = \frac{350}{1 + 6 e^{- 0.5 t}}$

How to do these

$\frac{d P}{d t} = 0.4 P (1 - \frac{P}{500})$ , $P (0) = 50$ . Find $\lim_{t \to \infty} P (t)$ and where $P$ grows fastest

The carrying capacity is the number under P, 500.
A population that starts between 0 and 500 climbs toward it.
Growth is fastest halfway, at 500/2.

The answer is $500$ ; $250$ .

Where students go wrong

Giving a time for the fastest growth. The question asks for the value of P, which is M/2 wherever P started.

Also called logistic differential equation, logistic model or carrying capacity.

What you can put on this worksheet

Carrying capacity and fastest growth: $\frac{d P}{d t} = 0.3 P (1 - \frac{P}{600})$ , $P (0) = 168$ . Find $\lim_{t \to \infty} P (t)$ and the value of $P$ where $P$ grows fastest. → $600$ ; $300$
Write the solution: $\frac{d P}{d t} = 0.4 P (1 - \frac{P}{900})$ , $P (0) = 25$ . Write $P (t)$ in the form $\frac{M}{1 + A e^{- k t}}$ → $P (t) = \frac{900}{1 + 35 e^{- 0.4 t}}$
Evaluate the solution: $P (t) = \frac{550}{1 + 21 e^{- 0.3 t}}$ . Find $P (15)$ , to the nearest tenth. → $446.0$

Questions about these worksheets

Yes. Every logistic growth sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the logistic growth questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: carrying capacity and fastest growth, write the solution and evaluate the solution. Tick as many as you want and set how many of each, or let it spread them evenly.

Giving a time for the fastest growth. The question asks for the value of P, which is M/2 wherever P started.

Logistic growth worksheet

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Logistic Growth

Logistic GrowthAnswer keyVersion 1

How to do these

Where students go wrong

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