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)?

Nov 15, 2016

$f \left(2\right) = 7$

Explanation:

The Taylor Series of $f \left(x\right)$ 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:

$f \left(a\right) = 7 \implies f \left(2\right) = 7$