Job 16 Taylor Formula

Source: Internet
Author: User
Tags cos


4.

(1) Solution: According to Taylor unfold
\[
\cos x =1-\frac{x^2}{2} + O (x^3)
\qquad \mbox{and} \qquad
\LN (1+x) = x-\frac{1}{2} x^2 +o (x^2),
\]
So
\[
\lim_{x\to 0} \frac{\cos x \ln (1+x)-x}{x^2}
= \lim_{x\to 0} \frac{(1+o (x)) (X-\frac12 x^2 +o (x^2))-X}{x^2}
=\lim_{x\to 0} \frac{x-\frac12 x^2-x+o (x^2)}{x^2}=-\frac12.
\]

Note: Because the denominator is $x ^2$, we only need to expand the molecule to $x ^2$, and other high-order items can be written as $o (x^2) $. Here $\cos x$ only used the previous expansion, because $\cos x$ 's second expansion is $x ^2$ and $\ln (1+X) $ The first expansion is $x $, so at this time to multiply is $x ^3=o (x^2) $, so we do not need to consider. Similarly, because the first item of $\cos x$ is $1$, $\ln (1+x) $ needs to be expanded to $x ^2$ item.



(2) $\frac{7}{360}$, courseware example



5. Proof: According to the book on the 144-page theorem 3.3.1, because the function $f (x) $ at $a $ point exists in the second derivative, then
\[
F (x) =f (a) +f ' (a) (x-a) +\frac{f "(A)}{2} (X-a) ^2+o ((x-a) ^2).
\]
To make $x =a+h$ and $x =a-h$
\[
F (a+h) =f (a) +f ' (a) h+\frac{f ' (a)}{2}h^2+o (h^2)
\]
And
\[
F (a-h) =f (a)-F ' (a) h+\frac{f ' (a)}{2}h^2+o (h^2).
\]
So
\[
\begin{aligned}
\lim_{h \to 0} \frac{f (a+h) +f (a-h) -2f (a)}{h^2}
&= \lim_{h \to 0} \frac{f (a) +f ' (a) h+\frac{f ' (a)}{2}h^2+f (a)-F ' (a) h+\frac{f ' (a)}{2}h^2-2f (a) +o (h^2)}{h^2}
\\
&=\lim_{h\to 0} \frac{f "(a) h^2 +o (h^2)}{h^2}
\\
&=f ' (a).
\end{aligned}
\]


Another method of proof:

Job 16 Taylor Formula

Contact Us

The content source of this page is from Internet, which doesn't represent Alibaba Cloud's opinion; products and services mentioned on that page don't have any relationship with Alibaba Cloud. If the content of the page makes you feel confusing, please write us an email, we will handle the problem within 5 days after receiving your email.

If you find any instances of plagiarism from the community, please send an email to: info-contact@alibabacloud.com and provide relevant evidence. A staff member will contact you within 5 working days.

A Free Trial That Lets You Build Big!

Start building with 50+ products and up to 12 months usage for Elastic Compute Service

  • Sales Support

    1 on 1 presale consultation

  • After-Sales Support

    24/7 Technical Support 6 Free Tickets per Quarter Faster Response

  • Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.