How do I know if Euler's method overestimates?

Check the concavity of the solution. Concave down means the estimate is too high, and concave up means it's too low.

When is logistic growth the fastest?

When the population is at half of the carrying capacity, P = L/2. That's the inflection point of the logistic curve.

Are Euler's method and logistic growth only on BC?

Yes. Both are in the BC course only, and AP Calculus AB doesn't include them.

Euler's method and logistic growth for AP Calculus BC

Updated November 9, 2026

Euler's method and logistic growth are the BC-only parts of the differential equations unit. Both are very learnable, and both tend to reward a tidy setup.

This guide walks through Euler's method step by step and shows what a logistic equation tells you without solving it.

Euler's method

Euler's method estimates a solution by taking short straight steps along the slope. Starting from a known point, each new value is the old value plus the slope times the step size: $y_{\text{new}} = y_{\text{old}} + \frac{dy}{dx} \cdot \Delta x$ .

Three steps

Let

\frac{dy}{dx} = x - y

with

y(0) = 3

. Use Euler's method with three equal steps of size

0.5

to approximate

y(1.5)

1At $(0, 3)$ the slope is $-3$ , so $y(0.5) \approx 3 + 0.5(-3) = 1.5$ .
2At $(0.5, 1.5)$ the slope is $-1$ , so $y(1) \approx 1.5 + 0.5(-1) = 1$ .
3At $(1, 1)$ the slope is $0$ , so $y(1.5) \approx 1 + 0.5(0) = 1$ .

Answer: $y(1.5) \approx 1$

Overestimate or underestimate?

Each step follows a tangent line. If the solution is concave up, tangent lines lie below it, so Euler's method underestimates. If it's concave down, Euler's method overestimates. To find concavity, take the derivative of $\frac{dy}{dx}$ and substitute.

Check the concavity

For the equation above,

\frac{dy}{dx} = x - y

with

y(0) = 3

, the exact solution is

y = x - 1 + 4e^{-x}

. Is the Euler estimate of

y(1.5)

too high or too low?

$\frac{d^2y}{dx^2} = 1 - \frac{dy}{dx} = 4e^{-x}$ , which is always positive, so the solution is concave up and the estimate is too low. The exact value is $y(1.5) = 0.5 + 4e^{-1.5} \approx 1.393$ .

y = x - 1 + 4e^{-x}

After the starting point, the Euler points fall below the exact solution.

Answer: Too low

Logistic growth

A logistic model starts out growing almost exponentially, speeds up until it reaches half of its limit, then levels off. Its standard form is $\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$ , where $L$ is the carrying capacity. The solution has the form $P = \frac{L}{1 + Ae^{-kt}}$ .

The long-term limit is $L$ , for any positive starting value.
The population is growing fastest when $P = \frac{L}{2}$ , which is the inflection point of the curve.
You can read both from the equation, with no need to solve it.

Read the equation, then solve

A population satisfies

\frac{dP}{dt} = 0.4P\left(1 - \frac{P}{200}\right)

with

P(0) = 20

. Find

\lim_{t \to \infty} P(t)

, the population when it's growing fastest, and the time when that happens.

The limit is $L = 200$ , and growth is fastest at $P = 100$ .

From $P(0) = 20$ , $\frac{200}{1 + A} = 20$ gives $A = 9$ , so $P = \frac{200}{1 + 9e^{-0.4t}}$ . Setting $P = 100$ gives $9e^{-0.4t} = 1$ , so $t = \frac{\ln 9}{0.4} \approx 5.493$ .

P = \frac{200}{1 + 9e^{-0.4t}}

Answer: Limit $200$ , fastest at $P = 100$ , when $t \approx 5.493$

Keep practicing

Questions on these topics show up in both multiple choice and free response. See how to earn free-response points for what to write down.

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Questions

How do I know if Euler's method overestimates?: Check the concavity of the solution. Concave down means the estimate is too high, and concave up means it's too low.
When is logistic growth the fastest?: When the population is at half of the carrying capacity, P = L/2. That's the inflection point of the logistic curve.
Are Euler's method and logistic growth only on BC?: Yes. Both are in the BC course only, and AP Calculus AB doesn't include them.