Answer :
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
= (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
