Euler's Method

Part 1 of 7 — The Algorithm

The Idea

Given rac{dy}{dx} = f(x, y) and initial condition $(x_0, y_0)$, approximate the solution by stepping along tangent lines.

Euler's Method

$x_{n+1} = x_n + Delta x$ $y_{n+1} = y_n + f(x_n, y_n)cdot Delta x$

Each step: new $y$ = old $y$ + slope $imes$ step size

Euler Basics 🎯

Key Takeaways — Part 1

$y_{n+1} = y_n + f(x_n, y_n)\Delta x$. One step at a time.

Euler's Method

Part 2 of 7 — Multi-Step Computations

Example: $dy/dx = 2x$, $y(1) = 3$, $Delta x = 0.5$

Exact: $y = x^2 + 2$, $y(2) = 6$. Euler gives 5.5 (underestimate for concave up).

Multi-Step 🎯

Key Takeaways — Part 2

Organize in a table. Smaller $\Delta x$ gives better approximations.

Euler's Method

Part 3 of 7 — Over- and Under-Estimates

When Does Euler Over/Under-Estimate?

This is because tangent lines on a concave-up curve lie below the curve, and vice versa.

Over/Under 🎯

Key Takeaways — Part 3

Concave up → underestimate. Concave down → overestimate.

Euler's Method

Part 4 of 7 — Step Size & Accuracy

Effect of Step Size

Smaller $Delta x$ → more steps → better approximation.

Exact: $e approx 2.718$

Step Size 🎯

Key Takeaways — Part 4

Error $\propto \Delta x$. Halve the step → halve the error (roughly).

Euler's Method

Part 5 of 7 — AP Exam Contexts

What the AP Exam Asks

1. Compute: Perform 2-3 steps of Euler
2. Interpret: Is result an over/underestimate?
3. Justify: Explain using concavity

Template Answer

"Using Euler's method with step size $\Delta x$: $y_1 = y_0 + f(x_0, y_0)\Delta x = ...$

Since $f''(x) > 0$ (concave up), the tangent lines lie below the curve, so Euler's method produces an underestimate."

AP Context 🎯

Key Takeaways — Part 5

Compute, then justify over/underestimate using concavity.

Euler's Method

Part 6 of 7 — Practice Workshop

Workshop 🎯

Workshop Complete!

Euler's Method — Review

Part 7 of 7 — Final Assessment

Final 🎯

Euler's Method — Complete! ✅