Chapter 8: Q. 70 (page 681)

Let $f$ be a function with an nth-order derivative at a point $x_{o}$ and let $P_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}$ . Prove that $f^{(k)} (x_{0}) = P_{n}^{(k)} (x_{0})$ for every non-negative integer $k \leq n$ .

Short Answer

Expert verified

The equation $P_{n}^{(k)} (x_{0}) = f^{(k)} (x_{0})$ is true.

Step by step solution

Step 1. Given information

An function is given as $P_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}$

Step 2. Verification

First we have to find the derivative $P_{n} (x)$ ,

$P_{n}^{'} (x) = \frac{d}{d x} [\sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}] = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} \frac{d}{d x} {(x - x_{0})}^{k} = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} \times k {(x - x_{0})}^{k - 1} = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{(k - 1)!} {(x - x_{0})}^{k - 1}$

Similarly,

$P_{n}^{''} (x) = \frac{d}{d x} [\sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{(k - 1)!} {(x - x_{0})}^{k - 1}] = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{(k - 2)!} {(x - x_{0})}^{k - 2}$

Implies that $P_{n}^{(k)} (x_{0}) = f^{(k)} (x_{0})$

Hence proved.

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Let $f$ be a function with an nth-order derivative at a point $x_{o}$ and let $P_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}$ . Prove that $f^{(k)} (x_{0}) = P_{n}^{(k)} (x_{0})$ for every non-negative integer $k \leq n$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Verification

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Let fbe a function with an nth-order derivative at a point xoand let Pn(x)=∑k=0nf(k)x0k!x-x0k. Prove thatf(k)x0=Pn(k)x0 for every non-negative integerk≤n.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Let $f$ be a function with an nth-order derivative at a point $x_{o}$ and let $P_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}$ . Prove that $f^{(k)} (x_{0}) = P_{n}^{(k)} (x_{0})$ for every non-negative integer $k \leq n$ .