Approximating Solutions Using Euler’s Method
Build Euler estimates step by step from the current point and distinguish an approximation from an exact solution.
College Board Topic 7.5 · CED p. 136 (PDF p. 141) · CED effective Fall 2020; project course outline 2026–27.
BC extension: Euler’s method turns a slope field into a sequence of short tangent-line steps. Start from (x₀,y₀), choose step h, and repeat xₙ₊₁=xₙ+h, yₙ₊₁=yₙ+hF(xₙ,yₙ). Evaluate the slope at the current point, not the point you have not reached yet. Keep a table of x, estimated y, slope, and increment. A negative h steps backward.
The estimate follows tangent segments rather than the exact curve. If the exact solution is concave up over a step, its tangent line lies below it; concave down puts the tangent above it. For repeated steps, support any overall error claim using the equation or a comparison—local tangent reasoning alone is not a universal error theorem for every differential equation. Smaller steps often improve accuracy in smooth stable examples, but a rounded table can still accumulate error. Keep full precision until the final answer.
Let's work through it
- For y′=y, y(0)=1, h=0.5, the first slope is 1 and y₁=1+0.5(1)=1.5. Reason: Use the current point’s slope.
- At x=0.5, the estimated slope is 1.5, so y₂=1.5+0.5(1.5)=2.25. Reason: Recalculate the slope at the updated estimate.
- The exact y(1)=e≈2.718, so this estimate is low. Reason: The positive exponential is concave up, and this recurrence can be compared with its exact multiplier.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Where is the slope evaluated in an Euler step?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Where is the slope evaluated in an Euler step?
Worked answer and reason
At the current estimated point.
The forward recurrence uses available data.
Retrieval 2
What does h=−0.1 mean?
Worked answer and reason
Step left by 0.1 in the independent variable.
The same slope formula is multiplied by a negative horizontal change.
Catch the tempting mistake
- Euler’s method integrates the derivative exactly.
- It uses tangent-line approximations. Label the result as an estimate and retain the step size.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 7 problem 25 · No calculator · analytic
- Unit 7 problem 26 · No calculator · analytic
- Unit 7 problem 27 · No calculator · analytic
For the teacher
- Require the x,y,slope,increment table before allowing automation. Compare two step sizes and an exact solution where available. Shared AB slope-field reasoning remains a prerequisite lesson here.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Slope field
A slope field displays short line segments with slopes specified by a first-order differential equation. For y′=F(x,y), the segment at (x,y) has slope F(x,y). For example, y′=y gives horizontal segments along y=0 and positive slopes above that axis. Solution curves follow the local slopes, and an initial point helps choose the relevant curve.
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Concavity
Concavity describes how a graph’s slope changes as the input increases. A graph is concave up where its derivative increases and concave down where its derivative decreases. For example, x² is concave up because its derivative 2x increases. A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
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Tangent line
A tangent line at a differentiable point passes through the point and has slope equal to the derivative there. It describes the function’s local linear behavior. For example, y=x² has tangent line y=4+4(x−2) at x=2. A tangent can cross the curve, so touching at exactly one point is not its general definition.