Calculating upper bound on interpolation error

[aguharay]

A table of f(x) = (e^x)sin(x), is to be published for x = 0 to x = 1 with a tabular interval h, where h
is of the form 1/n where n is an integer. If x is not a tabular value but lies between 0 and 1, then
f(x) is to be determined by quadratic interpolation with an error less than 0.5 × 10^−5 in absolute
value. What value of h would you recommend?

Any ideas??

 

[Count Iblis]

A table of f(x) = (e^x)sin(x), is to be published for x = 0 to x = 1 with a tabular interval h, where h
is of the form 1/n where n is an integer. If x is not a tabular value but lies between 0 and 1, then
f(x) is to be determined by quadratic interpolation with an error less than 0.5 × 10^−5 in absolute
value. What value of h would you recommend?

Any ideas??

The interpolating polynomial will approximately be the second order Taylor approximation. The error term is thus given by the remainder term, (which is the the third order term evaluated at some point p in the interval, but because the interval is small, you can take p to be any point in the interval). This is the case for each interval, so the maximum error is given by the maximum of h^3/6 f'''(x).