# Suppose T_4(x) = 7-3(x-2)+7(x-2)^2-6(x-2)^3+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(2.1)?

Oct 8, 2016

$6.7$

#### Explanation:

${T}_{4} \left(2\right) = 7$ and

${T}_{4} \left(2 + \delta\right) \approx {T}_{4} \left(2\right) + T {'}_{4} \left(2\right) \delta$

with

$T {'}_{4} \left(x\right) = - 3 + 2 \times 7 \left(x - 2\right) - 3 \times 6 {\left(x - 2\right)}^{2} + 4 \times 8 {\left(x - 2\right)}^{3}$

so

$T {'}_{4} \left(2\right) = - 3$

Finally

${T}_{4} \left(2 + 0.1\right) \approx 7 - 3 \times 0.1 = 6.7$