Given f(x)=1/x, find the number of terms needed to approximate f(0.5) accurate to 4 decimal places.

Answer 1

First we need the general term for the derivatives

f(x) = x^(-1)

f'(x) = -x^(-2)

f''(x) = 2 x^(-3)

f'''(x) = -6x^(-4)

f_4(x) = 24*x^(-5)

...

f_n(x) = n! * x^ -(n+1) = n! / (x^(n+1))

The Taylor polynomial has general term:

f_n(x0)*(x-x0)^n / n!

Assuming we are expanding about x0=0 with x=0.5;

f_n(0) * (0.5 -0) / n! =

n!* (0)^-(n+1)(0.5)^n / n!

= 0.5^n

We want 0.5^n <= 0.00001 to be the error bound.

0.5^n <= 10^(-5)

THis happens are n=16.616

you will need 17 terms