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, how do you estimate the value of f'(1.9)?

Answer 1

Depending on the final details in your approximation method, you could either say `f'(1.9) approx -4.612` (more accurate) or `f'(1.9) approx -4.4` (less accurate).

Explanation:

We have

`f(x) approx T_{4}(x)=7-3(x-2)+7(x-2)^2-6(x-3)^3+8(x-2)^4` for `x approx 2`.

Therefore,

`f'(x) approx T_{4}'(x)=-3+14(x-2)-18(x-2)^2+32(x-2)^3` for `x approx 2`.

To get the more accurate approximation, plug `x=1.9` into this last equation:

`f'(1.9) approx T_{4}'(1.9)=-3+14*(-.1)-18*(-.1)^2+32*(-.1)^3 approx -4.612`.

To get a less accurate approximation, we can get a linear approximation to `T_{4}'(x)` itself as `T_{4}'(x) approx -3+14(x-2)`. Then `f'(1.9) approx T_{4}'(1.9) approx -3+14*(-.1)=-4.4`.