Question

Math

Posted about 2 months ago

```
\begin{tabular}{|c||c|c|c|c|}
\hline$x$ & $f(x)$ & $f^{\prime}(x)$ & $f^{\prime \prime}(x)$ & $f^{\prime \prime \prime}(x)$ \\
\hline 0 & 3 & -2 & 1 & 4 \\
\hline 1 & 2 & -3 & 3 & -2 \\
\hline 2 & -1 & 1 & 4 & 5 \\
\hline
\end{tabular}
5. Selected values of a function $f$ and its first three derivatives are indicated in the table above. What is the third-degree Taylor polynomial for $f$ about $x=1$ ?
(A) $2-3 x+\frac{3}{2} x^{2}-\frac{1}{3} x^{3}$
(B) $2-3(x-1)+\frac{3}{2}(x-1)^{2}-\frac{1}{3}(x-1)^{3}$
(C) $2-3(x-1)+\frac{3}{2}(x-1)^{2}-\frac{2}{3}(x-1)^{3}$
(D) $2-3(x-1)+3(x-1)^{2}-2(x-1)^{3}$
```

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Answer from Sia

Posted about 2 months ago

Solution by Steps

step 2

From the table, we have:
$f(1) = 2, \quad f'(1) = -3, \quad f''(1) = 3, \quad f'''(1) = -2$

step 3

Substituting these values into the Taylor polynomial formula, we get:
$P_3(x) = 2 - 3(x-1) + \frac{3}{2}(x-1)^2 - \frac{2}{6}(x-1)^3$
$P_3(x) = 2 - 3(x-1) + \frac{3}{2}(x-1)^2 - \frac{1}{3}(x-1)^3$

B

Key Concept

Taylor Polynomial

Explanation

The Taylor polynomial of degree $n$ for a function $f$ about $x = a$ is given by $P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$.

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