Maple 9 Learning Guide - Maplesoft

7.1 Introductory Calculus • 223enter , x six times to calculate the sixth derivative. Define the functionf6 to be that derivative.> f6 := unapply( %, x );f6 := x → −sin(x) e sin(x) + 16 cos(x) 2 e sin(x) − 15 sin(x) 2 e sin(x)+ 75 sin(x) cos(x) 2 e sin(x) − 20 cos(x) 4 e sin(x) − 15 sin(x) 3 e sin(x)+ 45 sin(x) 2 cos(x) 2 e sin(x) − 15 sin(x) cos(x) 4 e sin(x)+ cos(x) 6 e sin(x)The following is the error in the approximation.> err := 1/6! * f6(xi) * (x - a)^6;err := 1720 (−sin(ξ) esin(ξ) + 16 cos(ξ) 2 e sin(ξ) − 15 sin(ξ) 2 e sin(ξ)+ 75 sin(ξ) cos(ξ) 2 e sin(ξ) − 20 cos(ξ) 4 e sin(ξ) − 15 sin(ξ) 3 e sin(ξ)+ 45 sin(ξ) 2 cos(ξ) 2 e sin(ξ) − 15 sin(ξ) cos(ξ) 4 e sin(ξ)+ cos(ξ) 6 e sin(ξ) )(x − π) 6The previous plot indicates that the error is small (in absolute value)for x between 2 and 4.> plot3d( abs( err ), x=2..4, xi=2..4,> style=patch, axes=boxed );0.16022xi44xTo find the size of the error, you need a full analysis of the expressionerr for x between 2 and 4 and ξ between a and x, that is, on the twoclosed regions bounded by x = 2, x = 4, ξ = a, and ξ = x. The curvecommand from the plottools package can illustrate these two regions.