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Published byKallie Bardell Modified over 2 years ago

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Looking for insight in the special case of antiderivatives

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Turning Corners (or Not!!!) Euler’s method is very bad at turning corners. Think about a solution curve like this one...

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Turning Corners (or Not!!!) Euler’s method is very bad at turning corners. When the curve nears a maximum, Euler’s method “overshoots.” Likewise, when the curve nears a minimum, Euler’s method drops too far.

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Point of View tt tt When our differential equation is of the form Euler’s method is a generalization of the left end-point Riemann sum! y = slope t = f’(t) tArea = f’(t) t

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Midpoint Approximations tt The midpoint Riemann sum is much more accurate. tt tt We use this insight to improve on Euler’s method.

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Improved Euler’s Method The idea obviously has merit. There’s only one problem... We don’t know the value of the function at the midpoint. We only know the value of the function at the left endpoint. But we can approximate the value of the function at the midpoint using the ordinary Euler approximation! tt tt

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Here it is! Old t and Old y temp t = Old t + 0.5( t) Temp y = Old y + 0.5( t) y’(Old t, Old y) New t = Old t + t New y = Old y + y’(Temp t, Temp y) t

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Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

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