The Lagrange remainder R, after n terms, for some c in the interval |x|
2, is
Since R is greatest when c = 2, n needs to satisfy the inequality
Using a calculator to evaluate successively at various integral
values of x gives y(8) > 0.01, y(9) > 0.002, y(10) < 3.8 × 10 −4 < 0.0004.
Thus we achieve the desired accuracy with a Taylor polynomial at 0 of
degree at least 10.
(D) On your calculator, graph one arch of the cycloid for t in [0,2π] and
(x,y) in [0,7] × [−1,3]. Use disks; then the desired volume is
(C) In the first quadrant, both x and y must be positive; x(t) = et is
positive for all t, but y(t) = 1 − t 2 is positive only for −1 < t < 1. The arc
