Mathematics

Problem

The table gives values of the derivatives of a differentiable function

f

x=2

. $\begin{aligned} f(2)&=3, & f'(2)&=-1, & f''(2)&=4, & f^{(3)}(2)&=0, & f^{(4)}(2)&=-8. \end{aligned}$ Using the fourth-degree Taylor polynomial for

f

centered at

x=2

, which of the following is the best approximation of

f(1.9)

? a)

3.118

3.120

3.100

3.140

Intuitive explanation

The key is building a Taylor polynomial: it's a polynomial centered at $x=2$ that matches $f$ and all its derivatives at that point. Each derivative term tells you how much the function curves or twists near $x=2$ , and you weight each by powers of the distance $(x-2)$ . Plug in $x=1.9$ to get your approximation.

Intuitive explanation

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