- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Slope fields and Euler’s method practice question 14
A student uses Euler’s method with two steps of to estimate for the problem below. Find the first line that is wrong.
Check each line against the rule of Euler’s method: every step uses the slope at the point it starts from, with that point’s and its .
Write down the point each step starts from, and then . Is every slope in the working taken at one of those points?
Theory UBC Math 100 §12.3Key results
Corrected working
Strategy
Each Euler step uses the slope at the point it leaves, with both the current t and the current y.
Working
First step, from (0, 1).
The second step starts at t₁ = 0.5, with y₁ = 1.5.
So y(1) ≈ 2.5.
For comparison: this formula solves the problem (substitute to check). The solution is concave up, so Euler runs low.
Answer
The answer is about 2.5.
Check
Four steps of 0.25 give about 2.88, between 2.5 and the true value 2e − 2 ≈ 3.44, so a smaller step moves the estimate up towards the truth.
Where this goes wrong
Taking the slope at the end of the step, (1, 1.5), a point that pairs the new time with the old y. Euler’s method uses only what is known at the start of each step.
Leans on
Linear approximation, from Linear approximation
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