Taylor Definition and 849 Threads

Homework Statement For example cosh(x) = 1+x2/2!+x4/4!+x6/6!+... Homework EquationsThe Attempt at a Solution So plugging in x=0 you get that coshx = 1 at the origin. The approximate graph for the coshx function up to the second order looks like a 1+x2/2! graph, but what about graphing coshx...

Hey! :o Let $f :\rightarrow \mathbb{R}$, $f(x) := tan(x)$. I want to find a $N\in \mathbb{N}$ such that for the $N$-th Taylor polynomial $P_N$ at $0$, that is defined as follows $P_N(x)=\sum_{n=0}^N\frac{f^{(n)}(0)}{n!}x^n$, it holds that $$\left |f(x)-P_N(x)\right |\leq 10^{-5}, \ \ x\in...

Homework Statement Same as title. Homework Equations Taylor expansion. The Attempt at a Solution Okay - what?! I don't even know where to begin. I taylor expanded the function and pretended like n was just some number and that doesn't help. I've never learned this. How? Can you point me in...

Homework Statement The coefficient of the term (x−π)2 in the Taylor expansion for f(x)=cos(x) about x=π is: Homework Equations ##cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!}...## The Attempt at a Solution Unless my taylor series for cosine is incorrect, I'm...

I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field, L=½(∂φ)^2 - m^2 φ^2 in the equation, S[φ]=∫ d4x L[φ] ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2) Particularly, it is in the Taylor series...

Homework Statement Hello, I solved others but not 6.9: Find the equation of the path joining the origin O to point P(1,1) in the xy plane that makes the integral ∫(y'2 +yy' + y2) dx stationary. ∫ from O to P. y' = dy/dx Homework Equations I need use ∂f/∂y = d/dx (∂f/∂y') (euler-lagrange...

Homework Statement Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression: $$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$ Homework Equations 3. The Attempt at a Solution [/B] ##L=\lim_{x \rightarrow 0}...

Homework Statement NOTE - When I post the thread my embedded images aren't showing up on my web browser, but they do show up when I bring it up to edit, so I don't know if other users can see the pictures or not... If not, they're here: Problem outline: http://tinypic.com/r/34jeihj/9 Solution...

Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...

Homework Statement and the solution (just to check my work) Homework Equations None specifically. There seems to be many ways to solve these problems, but the one used in class seemed to be partial fractions and Taylor series. The Attempt at a Solution The first step seems to be expanding...

$\textsf{a. Find the first three nonzero terms of the Taylor series $a=\frac{3\pi}{4}$}$ \begin{align} \displaystyle f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\sin{x} \\ f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\ f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=\frac{\sqrt{2}}{2} \\...

$\tiny{206.11.3.39}$ $\textsf{a. Find the first four nonzero terms of the Taylor series $a=0$}$ \begin{align} \displaystyle f^0(x)&=(1+x)^{-2} &\therefore \ \ f^0(a)&= 1 \\ f^1(x)&=\frac{-2}{(x+1)^3} &\therefore \ \ f^1(a)&= -2 \\ f^2(x)&=\frac{6}{(x+1)^4} &\therefore \ \ f^2(a)&= 6 \\...

$\tiny{242.13.3}$ $\textsf{1. Using the known series expansion of $\displaystyle e^x = \sum_{n=0}^{\infty}$, find the series representation of}\\$ $\textsf{a. $e^{-3x}$}\\$ $\textsf{b. $e^{x^3}$}$

Homework Statement Using Taylor series, Find a polynomial p(x) of minimal degree that will approximate F(x) throughout the given interval with an error of magnitude less than 10-4 F(x) = ∫0x sin(t^2)dt Homework Equations Rn = f(n+1)(z)|x-a|(n+1)/(n+1)![/B] The Attempt at a Solution I am...

$\tiny {11. 1.33-T} $ $\textsf{Find the nth order Taylor polynomials of the given function centered at a=100, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=\sqrt{x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n} \frac{f^{(k)}\left(a\right)}{k!}(x-a)^k$$ $\textsf{n=0}\\$...

$\tiny{206.11.1.16-T}$ $\textsf{Find the nth-order Taylor polynomials centered at 0, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=e^{-4x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$ $\textsf{n=0}\\$ \begin{align} f^0(x)&\approx...

$\tiny{206.11.1.15-T}$ $\textsf{Find the nth-order Taylor polynomials centered at 0, for $n=0, 1, 2.$}$ \\ $$\displaystyle f(x)=cos(3x)$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$ $\textsf{n=0}\\$...

Homework Statement Determine the Taylor series for the function below at x = 0 by computing P5(x) f(x) = cos(3x2) Homework Equations Maclaurin Series for degree 5 f(0) + f1(0)x + f2(0)x2/2! + f3(0)x3/3! + f4(0)x4/4! + f5(0)x5/5! The Attempt at a Solution I know how to do this but attempting...

I was studying the derivation for taylor series in ℝ##^n## on my book and I have some trouble understanding a passage; it's the very beginning actually: ##f : A## ⊆ ℝ##^n## → ℝ ##f ## ∈ ##C^2(A)## ##x_0## ∈ ##A## "be ##g_{(t)} = f_{(x_0 + vt)}## where v is a generic versor, then we have...

Homework Statement two man,each of equal mass m,are standing at one end of a stationary railroad flatcar with frictionless wheel and mass mcar.Find the car's speed if the two men run to the other end of the car and jump off simultaneously with the same speed u(relative to the car) Homework...

Homework Statement two man,each of equal mass m,are standing at one end of a stationary railroad flatcar with frictionless wheel and mass mcar.Find the car's speed if the two men run to the other end of the car and jump off simultaneously with the same speed u(relative to the car) Homework...

Homework Statement Perform a Taylor Series expansion for γ in powers of β^2, keeping only the third terms (ie. powers up to β^4). We are assuming at β < 1. Homework Equations γ = (1-β^2)^(-1/2) The Attempt at a Solution I have no background in math so I do not know how to do Taylor expansion...

Homework Statement What's the first order term in the expansion ln(x) about x = 1? Homework Equations Taylor series formula The Attempt at a Solution The question is multiple choice, and the choices are x, 2x, or (1/2)x. However, when I calculate the first order term in the expansion of ln(x)...

[Note from mentor: this thread was originally posted in a non-homework forum, therefore it does not use the homework template.] I have been given an equation for the relativistic doppler effect but I'm struggling to see this as a function and then give a first order Taylor expansion. Any help...

I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector. Does that mean I will have to treat it as a Taylor expansion about two variables...

Homework Statement Find a power series that represents $$ \frac{x}{(1+4x)^2}$$ Homework Equations $$ \sum c_n (x-a)^n $$ The Attempt at a Solution $$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$ since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} $$ x*\frac{d}{dx}\frac{1}{(1+4x)^2}...

I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is...

i watched a lot of videos and read a lot on how to choose it, but i what i can't find anywhere is, what's the physical significance of the a, if we were to draw the series, how will the choice of a affect it?

Homework Statement Find the Taylor Series for f(x)=1/x about a center of 3. Homework EquationsThe Attempt at a Solution f'(x)=-x^-2 f''(x)=2x^-3 f'''(x)=-6x^-4 f''''(x)=24x^-5 ... f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n I'm not sure where I went wrong...

I am a second year physics and I want to study CM in more depth than that of the general textbooks

hi, when I dug up something about metric tensors, I found a equation in my attached file. Could you provide me with how the derivation of this ensured? What is the logic of that expansion in terms of metric tensor? I really need your valuable responses. I really wonder it. Thanks in advance...

Hello, Can someone explain this to me? In the above case ct=yt-gt I tried to solve it as a three variable taylor approximation but got a few extra terms that weren't included in the above. So I am a little confused now. I only need to understand how the first line was derived because I get...

Hello friends, I need to compute the taylor expansion of $$\frac{x^4 e^x}{(e^x-1)^2}, $$ for ##x<<1##, to find $$ x^2 + \frac{x^4}{12}.$$ Can someone explain this to me? Thanks!

Homework Statement Page 35 of Jackson's Electrodynamics (3rd ed), it gives the following equation (basically trying to prove your standard 1/r potential is a solution to Poisson equation): \nabla^2 \Phi_a = \frac{ -1 }{ \epsilon_0 } \int \frac{ a^2 }{( r^2 + a^2)^{5/2} } \rho( \boldsymbol{x'}...

Homework Statement To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0## where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)## Homework Equations I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...

hi, first of all in this image there is a fact that we have parallel transported vector, and covariant derivative is zero along the "pr"path as you can see at the top of the image. I consider that p, and r is a point and in the GREEN box we try to make a taylor expansion of the contravariant...

Hello, I can't find solution for Maclaurin (Taylor a=0) polynom of function: f(x)=1/(√1-e3x). Could you help me please? Thank you so much for help Andrea

Hi everyone, I've been trying to buy a copy of the first edition of the textbook "Spacetime Physics" by Taylor and Wheeler in my country, but I haven't been able to get my hands on a copy of it. Moreover, the e-books available online are poorly scanned with a bad font. I was able to download...

Hi all, I am very confused about how one can find the upper bound for a Taylor series.. I know its general expression, which always tells me to find the (n+1)th derivative of a certain function and use the equation f(n+1)(c) (x-a)n+1/(n+1)! for c belongs to [a,x] However, there are...

Homework Statement Hi everybody! In the middle of an exercise, our teacher suddenly wrote: sin(\frac{x}{y} sin y) = \frac{x}{y} sin y - \frac{1}{2} sin θ (\frac{x}{y} sin y)^2 I don't get where does that come from? The closest I've managed to reach is: sin(\frac{x}{y} sin y) =...

Homework Statement Consider the position vector of a mass m at height h above the Earth's surface to be \underline{r}=(R+h)\underline{e}_z where R is the radius of the Earth. Make a Tylor expansion in h/R <<1 of the gravitational field...

Let f(x) = (1+x)-4 Find the Taylor Series of f centered at x=1 and its interval of convergence. \sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!} is general Taylor series form My attempt I found the first 4 derivatives of f(x) and their values at fn(1). Yet from here I do not know how to find the...

Homework Statement http://imgur.com/1aOFPI7 PART 2 Homework Equations Taylor series form The Attempt at a Solution My thought process is that the answer is 3 because using the geometric series equation (1st term)/(1-R) then you can get the sum. In this case R would be x+2 where x is -2 so 0...

I have been studying Hulse Taylor PSR 1913+16 calculation of period shift which is regarded as indirect proof for gravitational waves, but I don't understand one thing. If you look on the graph of Cumulative period shift, around every 10 years the shift doubles...

1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...

My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero. I have a function ##f[x,r]## I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0## This is impossible algebraically for me, but may be possible if I can...

Homework Statement Use zero- through third-order Taylor series expansion f(x) = 6x3 − 3x2 + 4x + 5 Using x0=1 and h =1. Once I found that the Taylor Series value is 49. I want to be able to check the value. On the board our teacher plugged in a value into the equation to show that the answer...

Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...

well, i have an calculus exam tomorrow and I'm 100% gona fail. I've neglected calculus so i could study for other subjects and left only 2 days to study taylor's polynomial aproximation, series and function series, the latter two are way more complicated than i expected. good thing is i can...

Consider the potential ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##, where ##\phi## is a scalar field and the mass dimensions of the couplings are: ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##. Expanding the field ##\phi## about the point...