Maple 9 Learning Guide - Maplesoft

236 • Chapter 7: Solving Calculus ProblemsAn Approximation of the Integral off(x) = 1/2+sin(x)on the Interval [0, 10]Using a Left-endpoint Riemann SumApproximate Value: 6.8390715291.510.50–0.52 4 6 8 10x–1Area: 6.954499888F(x)f(x)By specifying method = right or method = midpoint when usingthe ApproximateInt command, you can draw and sum boxes evaluatedat the right-hand side or at the midpoint of the box.Mixed Partial DerivativesThis section describes the D operator for derivatives and gives an exampleof a function whose mixed partial derivatives are different.Consider the following function.> f := (x,y) -> x * y * (x^2-y^2) / (x^2+y^2);f := (x, y) → x y (x2 − y 2 )x 2 + y 2The function f is not defined at (0, 0).> f(0,0);Error, (in f) numeric exception: division by zeroAt (x, y) = (r cos(θ), r sin(θ)) the function value is> f( r*cos(theta), r*sin(theta) );r 2 cos(θ) sin(θ) (r 2 cos(θ) 2 − r 2 sin(θ) 2 )r 2 cos(θ) 2 + r 2 sin(θ) 2As r tends to zero so does the function value.> Limit( %, r=0 );