What Kind of Error?

Calculus Level 2

For which of the following differential equations will Euler's method always produce an overestimate of the true value?

d y d x = x 2 \frac{dy}{dx}=x^2 d y d x = e x \frac{dy}{dx}=e^x d y d x = sin ⁡ x \frac{dy}{dx}=\sin x d y d x = log ⁡ x \frac{dy}{dx}=\log x

Samir Khan by Samir Khan , 23, USA

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1 solution

Samir Khan
Jun 11, 2016

Since Euler's method uses tangent lines repeatedly, Euler's method will overestimate the true value when the tangent lines lie above the graph, or when the graph is concave down. This will happen when f ′ ′ ( x ) ≤ 0 f''(x)\leq 0 for all x x , and we can check that this only happens for log ⁡ x \log x .

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