Insight into Antiderivatives and Numerical Methods
Explore the challenges and solutions in dealing with antiderivatives, turning corners with Euler's method, and enhancing numerical approximations for accuracy in calculus.
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Presentation Transcript
Looking for insight in the special case of antiderivatives
Turning Corners (or Not!!!) Euler s method is very bad at turning corners. Think about a solution curve like this one . . .
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.
Point of View f f t t ( ) ( ) ( ) = = = Area f t t slope y t f t t dy dt= ( ) g t When our differential equation is of the form Euler s method is a generalization of the left end-point Riemann sum!
Midpoint Approximations We use this insight to improve on Euler s method. f t t The midpoint Riemann sum is much more accurate. t
Improved Eulers Method We don t know the value of the function at the midpoint. We only know the value of the function at the left endpoint. The idea obviously has merit. There s only one problem . . . But we can approximate the value of the function at the midpoint using the ordinary Euler approximation! t t
