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Euler's Method

A slope field shows the direction everywhere, but it never hands you a number. Euler's method walks the field — small step by small step — to build an actual numerical answer. It's the bridge from seeing a solution to computing one.

Stand at your starting point, read the slope the equation gives you, and take a short straight step in that direction. Now you're at a new point — read the slope there and step again. Chain those steps together and the path approximates the true solution. Make the steps smaller and the approximation gets better. That's the whole method: follow the tangent, a little at a time.

Where the formula comes from

You don't have to memorize the rule — it falls right out of the tangent line you already know from differential calculus.

Walk the field, watch the error

The grey curve is the true solution; the pink staircase is Euler's approximation, each step a straight tangent segment. Drag the step size down and watch the staircase snap onto the curve; drag it up and watch it drift, with the error at each step marked. Hit walk it to take the steps one at a time.

true solution Euler approximation error at each step

step size h = 0.50

Pick an equation

Run the table

This is the calculation the exam asks for. In Show me mode, reveal the steps one at a time; switch to Test me to type each value yourself and get checked — with the worked step shown if you miss.

Step	The arithmetic

The ideas everything else builds on

The update rule

From a point, take the slope the equation gives, multiply by the step, and add it on. Then move x forward by h and repeat.

It's just tangent-line stepping

Each step is one use of local linear approximation — the same tangent-line idea from derivatives, applied over and over.

Smaller steps, less error

Halving the step size roughly halves the error — but doubles the number of steps. Accuracy and work pull in opposite directions.

It's only an approximation

Euler rides the tangents, so the way the curve bends decides the miss: concave up it falls short, concave down it overshoots.

Your turn

Six checks on the method — running the steps, reading the error, and the concavity reasoning the exam loves to ask about. Try each before revealing the answer.

Quick quiz

Six fast multiple-choice checks across the whole lesson. Pick an answer to see whether it's right and why — then work through all six.

Question 1 of 6

Score: 0 / 0

Common mistakes & exam tips

Using the wrong point for the slope

Each step uses the slope at the start of the interval, f(xₙ, yₙ) — not f(xₙ₊₁, yₙ₊₁). Plug in the values you're standing on, then step.

Forgetting to advance x

Every row moves x forward by h as well as updating y. Skip the x update and the next slope is evaluated at the wrong place.

Arithmetic and sign slips

The table is short but unforgiving — one wrong slope or a dropped minus sign corrupts every row after it. Carry numbers carefully, row to row.

Thinking smaller h is free

Smaller steps cut the error, but they also mean more steps, more arithmetic, and more rounding. There's always a tradeoff.

Confusing the estimate with the truth

Euler gives an approximation. Unless told otherwise, don't treat the final yₙ as the exact value of the solution.

Guessing over vs under

It's not random: concave-up solutions are underestimated, concave-down are overestimated. Check the concavity (often the sign of y″) to decide.

On the AP exam

“Use Euler's method with two steps to approximate y(1)”: build the table and show every row — the points are in the work, not just the final number.
A frequent follow-up: “is this an over- or under-estimate?” Answer from concavity — concave up means Euler underestimates, concave down means it overestimates.
Read the step size and start point carefully, and keep x and y updates in lockstep on every line.

Euler approximates a solution by walking it. The next lesson solves one exactly: Separation of Variables pulls the x's and y's to opposite sides and integrates. Or head back to the Unit 7 Guide.