Question

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].

Answer 1

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