AP Calculus BC 7.5: Euler’s Method

Practice AP Calculus BC questions on using equal-step Euler approximations to estimate points on a differential-equation solution curve.

[Maximum number: 2]

Let y=f(x) be the particular solution to the differential equation $\frac{d y}{d x}=(3-x) y^{2}$ with initial

condition f(1)=-1.

Use Euler's method, starting at x=1 with two steps of equal size, to approximate f(1.4).

Show the work that leads to your answer.

D Use Euler's method, starting at x=1 with two steps of equal size, to approximate f(1.4). Show the

work that leads to your answer.

$\begin{aligned} & f(1.2) \approx f(1)+\left.(1.2-1) \cdot \frac{d y}{d x}\right\|_{(1,-1)} & =-1+0.2(2)=-0.6 \end{aligned}$
$\begin{aligned} & f(1.4) \approx f(1.2)+\left.(1.4-1.2) \cdot \frac{d y}{d x}\right\|_{(1.2,-0.6)} & \approx-0.6+0.2(3-1.2)(-0.6)^{2}=-0.4704 \end{aligned}$

An approximation of f(1.4) is -0.47.

Scoring Notes for Part D

- To earn P8, a response must demonstrate the first step of Euler's method, with the correct initial

condition, correct step size, and correct (or imported) expression for the derivative.

Note: Any subsequent error in simplification or rounding will not affect the scoring for P8.

- The two steps of Euler's method may be explicit expressions or may be presented in a table. For

example:

x	y	$\frac{d y}{d x} \cdot \Delta x\left(\right.$ or $\left.\frac{d y}{d x} \cdot 0.2\right)$
1	-1	0.4
1.2	-0.6	0.1296
1.4	-0.4704

Note: In the presence of a correct answer, a table does not need to be labeled to earn both P8 and P9.

In the presence of no answer or an incorrect answer, such a table must be correctly labeled to earn

P8.

- A response of $-0.6+0.2(3-1.2)(-0.6)^{2}$ earns P9, regardless of any subsequent errors in

simplification or rounding.

- A response that imports an incorrect value for $f^{\prime}(1)$ from part A or part B is eligible to earn P9 with

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FUN-7.C—Estimate solutions to differential equations—Topic 7.5 question 1