AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/7/7-5)
Unit 7 · Topic 7.5
BC only7.5 Approximating Solutions Using Euler's Method
BC only. Euler's method approximates a solution to a differential equation one small step at a time. From a known point, you follow the tangent line for a step of size Δx, land on a new point, and repeat.
Key terms
- Euler's method
- step size (Δx)
- tangent line approximation
- initial condition
- approximation
The idea
This topic is tested only on the BC exam.
Suppose you know a solution passes through (x₀, y₀) and you know dy/dx = f(x, y). Near that point the curve is close to its tangent line. So take a short step along the tangent line instead of the curve. That puts you at a new point, which is probably a little off the true curve but close. Find the slope there from the equation and take another step. Each step is a tangent line approximation (4.6) built on the last one.
The procedure
A table keeps you organized. Use columns for x, y, slope dy/dx and change Δy = slope × h.
- Start at (x₀, y₀) with step size h = Δx.
- Compute the slope at the current point: m = f(xₙ, yₙ).
- Update: xₙ₊₁ = xₙ + h and yₙ₊₁ = yₙ + m·h. In words: new y = old y + slope × step.
- Repeat until you reach the target x-value.
- To step to the left, use a negative h.
How accurate is it?
Smaller steps usually give a better estimate, because you follow each tangent line for a shorter distance before correcting. The error comes from the curve bending away from the tangent line.
Concavity tells you the direction of the error for each step. If the solution is concave up along the way, every tangent line lies below the curve, so Euler's method underestimates. If it's concave down, Euler's method overestimates. To know which, find d²y/dx² from the differential equation (7.4). If the concavity changes during the steps, you can't easily say which way the error goes.
Euler's method and Riemann sums
Each Euler step adds slope × width to y. If the slope depends only on x, as in dy/dx = f(x), the total change Euler's method adds up is f(x₀)h + f(x₁)h + …, which is exactly a left Riemann sum for ∫ f(x) dx. So Euler's method is the same idea as estimating accumulated change with rectangles (6.2), except that in general the slope also depends on the y-values you've estimated along the way.
On the exam
Euler's method questions are usually two steps by hand, with friendly numbers. You'll be asked to show the work and sometimes to say whether the approximation is too high or too low. You don't need to memorize anything beyond new y = old y + slope × Δx.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Two steps to the right
Let y = f(x) be the solution to dy/dx = x + y with f(0) = 1. Use Euler's method with two steps of size 0.5 to approximate f(1). Is it an over- or underestimate?
Show the solutionHide the solution
- Step 1: Step 1: at (0, 1), slope = 0 + 1 = 1. New y = 1 + 1(0.5) = 1.5. Now at (0.5, 1.5).
- Step 2: Step 2: at (0.5, 1.5), slope = 0.5 + 1.5 = 2. New y = 1.5 + 2(0.5) = 2.5. Now at (1, 2.5).
- Step 3: Concavity: d²y/dx² = 1 + dy/dx = 1 + x + y. Along the path, x and y are positive, so d²y/dx² > 0 and the solution is concave up.
- Step 4: Concave up means tangent lines lie below the curve, so Euler's method gives an underestimate.
- Step 5: (For reference, the exact solution is y = 2eˣ − x − 1, so f(1) = 2e − 2 ≈ 3.437. The estimate 2.5 is indeed low.)
Answer: f(1) ≈ 2.5, an underestimate.
- Example 2
Trap: stepping to the left
Let y = g(x) be the solution to dy/dx = xy with g(2) = 1. Use Euler's method with two steps of size 0.1 to approximate g(1.8).
Show the solutionHide the solution
- Step 1: Moving left means h = −0.1.
- Step 2: Step 1: at (2, 1), slope = 2(1) = 2. New y = 1 + 2(−0.1) = 0.8. Now at (1.9, 0.8).
- Step 3: Step 2: at (1.9, 0.8), slope = 1.9(0.8) = 1.52. New y = 0.8 + 1.52(−0.1) = 0.648. Now at (1.8, 0.648).
- Step 4: Common wrong move: using h = +0.1 and getting 1.2 after the first step, which heads the wrong way.
Answer: g(1.8) ≈ 0.648
Common mistakes
- Using the slope from the starting point for every step. Recompute the slope at each new point.
- Forgetting to multiply the slope by the step size.
- Using a positive step when moving to the left.
- Rounding heavily in the middle of the steps. Keep exact decimals until the end.
On the exam
- Show each step's computation, like “f(0.5) ≈ 1 + 1(0.5) = 1.5.” A final number alone won't earn full credit.
- Euler's method often shares a free-response question with a slope field, a tangent line and a separable solution, so the same differential equation appears in several parts.
Connected topics
Videos
Check yourself
4 questions on 7.5 Approximating Solutions Using Euler's Method. Pick an answer to see if you got it, and why.
Let y = f(x) be the solution to the differential equation dy/dx = x + y with f(0) = 1. What is the approximation for f(1) found by using Euler's method with two steps of equal size, starting at x = 0?
Let y = f(x) be the solution to the differential equation dy/dx = 1 + y² with f(0) = 1. Euler's method with two steps of size 0.1 is used to approximate f(0.2). Which of the following gives the approximation and correctly describes it?
Let y = f(x) be the solution to the differential equation dy/dx = sin x + √y with f(1) = 4. What is the approximation for f(2) found by using Euler's method with two steps of equal size, starting at x = 1?
Let y = f(x) be the solution to the differential equation dy/dx = 2x − y with f(1) = 3. What is the approximation for f(2) found by using Euler's method with two steps of equal size, starting at x = 1?
0 of 4 answered