- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 3
Slope fields and Euler’s method practice question 11
After an injection, a drug is removed by an enzyme that saturates, so its concentration , in mg/L after hours, satisfies
Euler’s method with is used to estimate . Without computing the steps, decide whether the estimate is above or below the true value, and why.
Euler’s method steps along tangent lines. Which derivative of decides whether the curve lies above or below its tangents?
That is concavity, the sign of . Differentiate with the chain rule: .
Find by the quotient rule. While , both and are negative: what does that make , and which side of its tangents does the curve lie on?
Theory UBC Math 100 §12.3Key results
Worked solution
Strategy
Each Euler step runs along a tangent line, and a concave-up curve lies above its tangents, so the estimate runs low when the solution is concave up. Find the sign of C″ from the equation itself, with the chain rule.
Working
Quotient rule.
Chain rule, since C depends on t. The concentration falls towards 0 but stays positive, so C″ > 0 throughout.
A concave-up curve lies above its tangent lines, so the first Euler step lands below the true curve. Each later step starts lower and again runs along a tangent that lies below the solution through its starting point, so the estimate of C(1) is too low.
Answer
The answer is Below the true value, because the solution is concave up.
Check
Two Euler steps give 3, then 2.25, then about 1.56, and the true C(1) ≈ 1.62 is higher, as the concavity predicts.
Where this goes wrong
Deciding from the sign of C′. Whether the curve rises or falls says nothing about which side of its tangent lines it lies on; that is concavity, the sign of C″.
Leans on
Concavity and the second derivative, from Curve sketching
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