Estimate the maximum error made in approximating e^x by the polynomial 1 + x + {1}/{2}x^2 over the interval x of [-0.4,0.4].

E^x = 1 + x + x² / 2 + x³ / 3! + x^4 / 4! + ...

= (1 + x + x²/2) + x³ [ 1/6 + x / 4! + x² / 5! + ... ]

Error = e^x - (1 + x + x²) = x³ [ 1/6 + x / 4! + x² / 5! + ... ]

x / 4! < x / 6 x² / 5! < x² / 6 and so on

So if we replace all factorials by 1/6 ...

error < x² [ 1/6 + x/6 + x²/6 + ... ]

< x² / 6 [ 1 + x + x² ... ]

< x² / 6 * 1 / (1 - x) = x² / 6 (1-x) if x < 1

maximum error = x² / 6 (1-x) occurs at 0.4 or - 0.4 in the given interval.

= 0.0444444