Question

Suppose `T_4(x) = 7-3(x-2)+7(x-2)^2-6(x-2)^2+8(x-2)^4` is the degree 4 Taylor polynomial centered at x=2 for some function f, what is the value of f(2)?

Answer 1

`f(2)=7`

Explanation:

The Taylor Series of `f(x)` centred about `x=a` is given by
`f(x) = f(a) + (f'(a))/(1!)(x-a) + (f''(a))/(2!)(x-a)^2 + (f'''(a))/(3!)(x-a)^3 ...`

So if we substitute `a=2` and compare the constant coefficients we get:

`f(a) = 7 => f(2)=7`