math_ Taylor formula

Differential Taylor formula

• Use an easy calculation / A simple function can approximately express a complex function , This approximate expression is mathematically called Close to

• Taylor's formula uses polynomials $P$(polynominal) To approximate a given function $f(x)$;

• We use it $P_i$ To describe approximation $f(x)$ The process of :

  • First order approximation :

    • $$P_1=f(x_0)+f^{\prime }(x_0)(x-x_0)=a_0+a_1(x-x_0)$$

  • Second order approximation :

    • $$P_2=a_0+a_1(x-x_0)+a_2(x-x_0{)}^2$$

  • …( Higher precision approximation function )

• The problem is , How to determine the coefficient $a_i$

$irankforcednearLetterCount{P}_i$

• $$\begin{gathered} irankforcednearLetterCount{P}_{i}surfaceinbymanytermtype: \\ {P}_i=a_0+\sum _{k=1}^n{a}_k(x-x_0{)}^k; \\ =f(x_0)+\sum _{k=1}^n{a}_k(x-x_0{)}^k; \end{gathered}$$

  $$\begin{gathered} andMaletype\bcancel{\sum _{k=0}^n{a}_k(x-x_0{)}^k} \\ Meetingstayx=x_{0}OutHairrawdifferentoften,Outpresent{0}^{0}Ofterm,becausethissinglestateDemolitionbranchOutCome\; on{a}_1,sendhave\; tovarioustermOffingerCounttoLessby1 \\ butyes,canWithWritedo:(notesIt\; meanstiredAddNextworldk=0)\{\begin{array}{l}\sum _{k=0}^n{a}_k(x-x_0{)}^k;x\ne x_0\\ f(x_0);x=x_0\end{array} \end{gathered}$$

• $$\begin{gathered} LetterCount{P}_{i}andf(x)OfguideCountAndbetweenYesSuch\; asNextaboutbeam: \\ nrankOfforcednearLetterCount{P}_{n}andByforcednearLetterCountf(x)stayspot{x}_{0}Of(irank)guideCountvaluephaseSame\; as(i\in [0,N];i\in N^{\ast })) \end{gathered}$$

  $${P}_n^{(i)}(x_0)=f^{(i)}(x_0)\text{\hspace{1em}(cf (constraint family(i))}$$

  $$\begin{gathered} thisintakei\in N^{\ast }OfvariousindividualivaluebeltEnter\; intoOntype(cf),canWithhave\; toToOnesystemColumnOfaboutbeametc.type, \\ Every\; timeindividualetc.typecanWithseekExplainOutOneindividualterm{a}_i,a_{i}yesTurn\; offOn{f}^{(i)}Ofsurfacereachtype \end{gathered}$$

about $(x+a{)}^n$ Higher derivative of form

$$\begin{gathered} (x^n{)}^{(n)}=n!; \\ (x^n{)}^{(n+1)}=0 \\ canWithhave\; toTo \\ ((x+a{)}^n{)}^{(n)}=(\sum _{i=0}^n{x}^i{a}^{n-i}){)}^{(n)}=1\cdot (x^n{)}^{(n)}=n! \end{gathered}$$

$$\begin{gathered} moreOnelikeOf,IPeoplecanWithPUSHguide: \\ remembery=(x+a{)}^n \\ ((x+a{)}^n{)}^{(k)}=y^{(k)} \\ 1⩽k⩽n;k,n\in N^{+}when, \\ ((x+a{)}^n{)}^{(k)}=\frac{n!}{(n-k)!}(x+a{)}^{n-k}=P_n^k{(x+a)}^{n-k} \end{gathered}$$

$$\begin{gathered} ,otherOf,Whenk=nwhen,(oftenCountaOfvaluestaythiswhennothingTurn\; offtightwant)thenhave\; to: \\ ((x+a{)}^n{)}^{(n)}=n! \end{gathered}$$

$P_{i}OfkrankguideCount$

$$P_i(x)=a_0+\sum _{k=1}^n{a}_k(x-x_0{)}^k;$$

$$\begin{gathered} YesOnirankforcednearLetterCount{P}_i,YesItsseekkrankguideCount; \\ {P}_i^{(k)}(x_0)=0+\sum 0+a_{k}k!+\sum 0=a_{k}k! \\ rootAccording\; to\; theaboutbeamstripPieces\; of \\ =f^{(k)}(x_0) \\ fromandhave\; toTo{a}_k=\frac{f^{(k)}(x_0)}{k!} \end{gathered}$$

Taylor polynomial $P_n$$LetterCountf(x)stayspotx=x_{0}It\text{'}s\; aboutOfnTime\mathbb{T}aylormanytermtype$

$a_k:forcednearLetterCount{P}_{i}varioustermsystemCount{a}_{k}Ofvalue$

$$\begin{gathered} f(x)Ofstayx=x_{0}It\text{'}s\; about(namelyf(x_0))OfirankforcednearLetterCount{P}_{i}surfaceinbymanytermtype: \\ f(x)\approx P_n(x) \\ {P}_n(x,x_0)=a_0+\sum _{k=1}^n{a}_k(x-x_0{)}^k; \end{gathered}$$

$$\begin{gathered} ThetypeSoncallby:LetterCountf(x)stayspotx=x_{0}It\text{'}s\; aboutOfnTime\mathbb{T}aylormanytermtype, \\ orpersoncallbyf(x)Press(x-x_0)OfpowerexhibitionopenOfnTimeThaiLemanytermtype \end{gathered}$$

$$a_k=\frac{f^{(k)}(x_0)}{k!}$$

Coefficient characteristics

• so ,$a_k$$yesTurn\; offOnf(x)stay{x}_{0}It\text{'}s\; aboutOfkrankguideCount{f}^{(k)}(x_0)zWithAndk!Ofsurfacereachtype$

• Items in polynomials coefficient Have the same characteristics : There are two parts

  • k Step derivation
  • k Factorial

General

• $mostafter,systemCountAgainrideWithnucleushearttype(x-x_0)OfkTimepower(x-x_0{)}^k$, Get a complete general term of Taylor polynomial :

  • $$a_k(x-x_0{)}^{(k)}=\frac{f^{(k)}(x_0)}{k!}\times (x-x_0{)}^k$$

  • general term formula
    $$\begin{gathered} T(x_0,k,\xi )=a_k(x-x_0{)}^{(k)}=\frac{f^{(k)}(x_0)}{k!}\times (x-x_0{)}^k \\ {R}_n(x)=T(x_0,n+1,\xi )=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1} \end{gathered}$$

$R_n(x)$: Remainder ( Error description function )

• be aware ,$more\; thanterm{R}_n(x)alsoyesLetterCount,theWithcanWithWant\; tolike,forcednearLetterCount{P}_n(x)stayEstimatecountf(x)OfNoSame\; asspotLetterCountvalue,canYesNoSame\; asOffineindeedchengdegree$

Type of remainder

There are two kinds of remainder :

• Lagrange (Lagrange) Type remainder

• The remainder and taylor The coefficients of polynomials have the same characteristics : It consists of three parts

  • k Step derivation ,

  • k The next power ,

  • k Factorial

  • $$\begin{gathered} R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1} \\ more\; thantermOfsystemCountMinistrybranch\frac{f^{(n+1)}(\xi )}{(n+1)!} \\ AnduniversalthroughsystemCountphaseThan: \\ \{\begin{array}{l}{x}_0\to \xi \\ n\to (n+1)\end{array} \end{gathered}$$

• Peano (Peano) Type remainder

  • Largrange Simplified description of type ( High order infinitesimal )

  • $${R}_n(x)=o((x-x_0{)}^n)$$

Formula macro definition ( Some editors do not support , The formula cannot be rendered ,typora Support )
\def\ltzero{\lim_{x\rightarrow 0}} \def\ltxzero#1{\lim_{x\rightarrow x_0}} \def\ltx#1{\lim_{x\rightarrow #1}} \def\ltxi#1{\lim_{x\rightarrow x_{#1}}} \def\limtoxi#1{\lim_{x\rightarrow x_{#1}}} \def\Rn#1{R_n^{(#1)}(x)}

remember :
$$R_n(x)=f(x)-P_n(x)$$

$$\begin{gathered} {P}_n^{(i)}(x_0)=f^{(i)}(x_0) \\ namely: \\ {f}^{(i)}(x_0)-p_n^{(i)}(x_0)=0\text{\hspace{1em}(cf (constraint family))} \end{gathered}$$

$$\begin{gathered} R_n^{(i)}(x_0)=f^{(i)}(x)-P_n^{(i)}(x)=0 \\ namely,R_n(x_0)=R_n^{{}^{\prime }}(x_0)=R_n^{{}^{\prime \prime }}(x_0)=\cdots =R_n^{(n)}(x_0) \end{gathered}$$

Relation between remainder and infinitesimal

$${R}_n^{(n)}(x_0)=o((x-x_0{)}^n)$$

$$\begin{gathered} \because (backcomplexsenduseLuohave\; toreachLawbe)\underset{x\to x_0}{\lim }\frac{R_n(x)}{(x-x_0{)}^n}=\underset{x\to x_0}{\lim }\frac{R_n^{{}^{\prime }}(x)}{n(x-x_0{)}^{n-1}} \\ =\underset{x\to x_0}{\lim }\frac{R^{{}^{\prime \prime }}(x)}{n(n-1)(x-x_0{)}^{n-2}}=\cdots \\ =\underset{x\to x_0}{\lim }\frac{R_n^{(n-1)}(x)}{n!(x-x_0)}=\frac{1}{n!}\underset{x\to x_0}{\lim }\frac{R_n^{(n-1)}(x)-R_n^{(n-1)}(x_0)}{x-x_0}=\frac{1}{n!}{R}_n^{(n)}(x_0)=0 \end{gathered}$$

• among , The process of converting limits into derivatives :

$$\begin{gathered} rememberg(x)=R_n^{(n-1)}(x);fromOnfrontNoodlesOfOnStatementcanknow{R}_n^{(i)}(x_0)=0,be,g(x_0)=0 \\ fromguideCountOfsetThe\; righteous{g}^{\prime }(x_0)=\underset{x\to x_0}{\lim }\frac{g(x)-g(x_0)}{x-x_0} \\ be{R}_n^{(n)}(x_0)=g^{\prime }(x_0) \end{gathered}$$

$R_n^{(i)}(x)$ Derivative order i	1	2	3	…	n	(n+1)
Corresponding i Of the first derivative ($(x-x_0{)}^n{)}^{(i)}$ Multiplicative multiplication Coefficient factor	n	n-(1)	n-(2)	…	n-(n-1)=1	n-n

Taylor formula

$$\begin{gathered} f(x)=P_n(x)+R_n(x) \\ fromchengorderset\; upmeterOfhorndegree,by了strongtransfer{x}_{0}YesMaletypeOfshadowring,canWithWritedo \\ *f(x)=g(x,x_0,\xi )=P_n(x,x_0)+R_n(x,x_0,\xi );(constant\xi \in (x_0,x)) \end{gathered}$$

$$\begin{gathered} f(x)=g(x,x_0,\xi )=f(x_0)+\sum _{k=1}^n{a}_k(x-x_0{)}^k \\ +\frac{f^{(n+1)}(\xi )}{n!}(x-x_0{)}^{n+1} \\ =f(x_0)+\sum _{k=1}^{k=n}\frac{f^{(k)}(x_0)}{k!}{(x-x_0)}^k+\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1} \end{gathered}$$

$$\begin{gathered} ByforcednearLetterCount=forcednearLetterCount+By\; mistakeBad \\ ByforcednearLetterCountcanWithuseforcednearLetterCount{P}_n(x,x_0)Come\; onEstimatemeter \\ By\; mistakeBadcanWithuse{R}_n(x,x_0)Come\; onEstimatemeter \end{gathered}$$

McLaughlin (Maclaurin) The formula

When Taylor formula
$$\begin{gathered} Make{x}_0=0; \\ f(x)=g(x,0,\xi )=f(x_0)+\sum _{k=1}^n{a}_k(x-0{)}^k \\ +\frac{f^{(n+1)}(\xi )}{n!}(x-0{)}^{n+1} \\ {x}_0\to \xi \end{gathered}$$

$$\begin{gathered} Itsin:a_k=\frac{f^{(k)}(x_0)}{k!},x_0=0; \\ {a}_{k_{x_0=0}}=\frac{f^{(k)}(0)}{k!} \end{gathered}$$

namely :
$$\begin{gathered} \underset{maclaurin}{f(x)}=g(x,0,\xi ) \\ =f(x_0)+\sum _{k=1}^n\frac{f^{(k)}(0)}{k!}{x}^k+R_n(x) \\ Itsin,R_n(x)canWithyeslagrangetypeOfmore\; thanterm,alsocanWithyespeanotypemore\; thanterm \end{gathered}$$

$$\begin{gathered} staymaclaurinMaletypeNext,take{R}_n(x)bylagrangetypemore\; thantermwhen, \\ fromOn\xi \in (x_0,x),canWithMake\xi =\theta x;(\theta \in (0,1)) \\ be,R_n(x)=\frac{f^{(n+1)}(\theta x)x^{n+1}}{(n+1)!} \end{gathered}$$

Taylor formula ( McLaughlin's formula ) Application

$sin(x)beltYeslargrangetypemore\; thantermOfnrankmaclaurinMaletype(exhibitionopen)$

• According to the general maclaurin The formula :

• $$\begin{gathered} \underset{maclaurin}{f(x)}=f(0)+\sum _{k=1}^{k=n}\frac{f^{(k)}(0)}{k!}\cdot x^k+R_n(x) \\ {R}_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}\cdot x^{(n+1)}=\frac{f^{(n+1)}(\theta x)}{(n+1)!}\cdot x^{(n+1)} \end{gathered}$$

• $$\begin{gathered} f^{(k)}(x)=sin(x+k\cdot \frac{\pi }{2}) \\ {f}^{(k)}(0)=sin(k\frac{\pi }{2});(k=1,2,3,\cdots ) \\ =1,0,-1,0,1,0,\cdots (rootAccording\; to\; thesinglepositionround,NodifficultHairpresentItsZhouperiodgaugeLaw) \\ f(0)=sin(0)=0 \\ f(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots \end{gathered}$$

  k	1	2	3	4	5	6	…	n=2m
  $f^{(k)}(0)$	1	0	-1	0	1	0
  $\frac{f^{(k)}(0)}{k!}$	$\frac{1}{1!}$	$\frac{0}{2!}=0$	$\frac{-1}{3!}$	$\frac{0}{4!}=0$	$\frac{1}{5!}$	$\frac{0}{6!}$
  $\frac{f^{(k)}(0)}{k!}{x}^k$	$x^1$	0	$-\frac{1}{3!}{x}^3$	0	$+\frac{1}{5!}{x}^5$	0

  The first m term	1	2	3	4	5	6	…	m	(m+1) Remainder
  2m	2	4	6	8	10	12
  m-1	0	1	2	3	4	5
  2m-1	1	3	5	7	9	11
  2m+1	3	5	7	9	11	13
  $(-1{)}^{m-1}$( As the symbol of the item )	1	-1	1	-1	1	-1		$(-1{)}^{m-1}$	$(-1{)}^m$
  $(-1{)}^m$	-1	1	-1	1	-1	1
  $\frac{f^{(k)}(0)}{k!}{x}^k$	$x^1$	0	$-\frac{1}{3!}{x}^3$	0	$+\frac{1}{5!}{x}^5$	0		$(-1{)}^{m-1}\frac{x^{2m-1}}{(2m-1)!}$	$(-1{)}^m\frac{cos(\theta x)}{(2m+1)!}{x}^{2m+1}$

  $$\begin{gathered} f(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots +(-1{)}^{(m-1)}\frac{x^{2m-1}}{(2m-1)!}+R_{2m} \\ f(x)=\sum _{k=1}^{k=m}(-1{)}^{(m-1)}\frac{x^{2m-1}}{(2m-1)!}+R_{2m} \\ {R}_{2m}(x)=\frac{sin(\theta x+(2m+1)\frac{\pi }{2})}{(2m+1)!}{x}^{(2m+1)}=(-1{)}^m\frac{cos(\theta x)}{(2m+1)!}{x}^{2m+1} \end{gathered}$$

  If you take m=1; obtain
  $$sinx\approx x$$

  $thiswhenuseP(x)=xEstimatecountf(x)=sin(x)productionrawOfBy\; mistakeBadby$:
  $$|R_2|=|-\frac{cos(\theta x)}{3!}{x}^3|⩽\frac{|x^3|}{6}$$

  so , When it comes to estimating f(x),x When the value is small , Use P(x)=x Estimate f(x)=sin(x) The error of poor students is very limited

  When x When the value is large , The upper limit of error will increase , The estimation effect may be very unreliable , At this time , Consider using higher-order approximation functions

  m=2; obtain
  $$sinx\approx x-\frac{1}{3!}{x}^3$$

  m=3; obtain
  $$sinx\approx x-\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5$$
  Their errors will not exceed $\frac{1}{5!}|x^5|,\frac{1}{7!}{x}^7$; And the factorial growth is faster than the index , Therefore, it is believed that higher-order approximation can better control the error in a small enough range

The application of Taylor expansion $(x_0\ne 0)$

$$f(x)=f(x_0)+\sum _{k=1}^{k=n}\frac{f^{(k)}(x_0)}{k!}{(x-x_0)}^k+R_n(x)$$

$$f(x)=x^3+3x^2-2x+4OfPress(x+1)Oflpowerexhibitionopen$$

• $indeedset{x}_0=-1$
• Calculation f(x) At each derivative , And bring in $x=x_0,have\; toTo{f}^{(k)}(x_0)$
• Calculation $R_n(x)$
• Bring in the formula

$$\begin{gathered} x_0=-1; \\ f(-1)=8; \\ f(x)stayx=x_0=-1It\text{'}s\; aboutOfvariousrankguideCountvalue:f^{\prime }(x)=3x^2+6x-2;f^{\prime }(-1)=-5 \\ {f}^{\prime \prime }(x)=6x+6;f^{\prime \prime }(-1)=0 \\ {f}^{\prime \prime \prime }(x)=6;f^{\prime \prime \prime }(-1)=6 \\ {f}^{(k)}(x)=0;(k⩾4)theWithR=R_4(x)=0 \\ \therefore f(x)=f(-1)+\frac{f^{\prime }(-1)}{1!}(x+1)+\frac{f^{\prime \prime }(-1)}{2!}(x+1)+\frac{f^{\prime \prime \prime }(-1)}{3!}x+1+R_4(x) \\ =8-5(x+1)+(x+1{)}^3 \end{gathered}$$