Taylor expansion Definition and 166 Threads

Hi. I borrowed many multivariable Calculus book so that I can choose one for the next semester. The one I liked most is Multivariable Calculus by Ron Larson. It is full of graphics and colours, somthing that is essential to understand functions of two variables. The only thing is that it does...

Homework Statement From step 1 to step 2, what do they mean by "Taking the weighted sum of the two squares " ? I tried and expanded everything in step 2 and it ends up as the same as step 1 (as expected), The Attempt at a Solution I tried looking up "weighted sum" and "...

Homework Statement Folks, how is the following expansion obtained for the following function ##F(x,u,u')## where x is the independent variable. The change ##\epsilon v## in ##u## where ##\epsilon## is a constant and ##v## is a function is called the variation of ##u## and denoted by...

Homework Statement I need to use Taylor Expansion to show that: (1+x)^n = 1 + nx + n(n-1)(x^2)/2! + ...Homework Equations y(x0 + dx) = y(x0) + dx(dy/dx) + [(dx)^2/2!](d^2y/dx^2) + ... The Attempt at a Solution I've only just begun Taylor Expansion, according to my textbook I need the above...

Homework Statement arctan x = ∫du/(1+u2), from 0 to x Homework Equations The Attempt at a Solution I noticed that 1/(1+u2) = 1/(1+u2)1/2 × 1/(1+u2)1/2. I decided to take the taylor series expansion of 1/(1+u2)1/2, square the result and then integrate. I got 1/(1+u2)1/2 =...

Something came up while I was trying to solve a problem. This is a Taylor Expansion of a function: x+(x^2)/2+(x^3)/3+(x^4)/4+(x^5)/5+(...) What's the function associated with it?

I have the next function: z^3-2xz+y=0 and I want to find taylor expansion of z(x,y) at the point (1,1,1), obviously I need to define F(x,y,z) as above and use the implicit function theorem to calculate the derivatives of z(x,y), but I want mathematica to compute this to me. I tried the Series...

Hi. I just want to ask: how can I realize that I need to do the 4th order taylor's expansion for solving a precise limit? e.g. [SIZE="4"]\mathop {\lim }\limits_{x\to 0} \frac{{{e^x}-1-\frac{{{x^2}}}{2}+\sin x-2x}}{{1-\cos x-\frac{{{x^2}}}{2}}} We need the 4th order of the expansion but how can...

In thermodynamic perturbation theory (chapter 32 in Landau's Statistical Physics) for the Gibbs (= canonical) distribution, we have E = E_0 + V, where V is the perturbation of our energy. When we want to calculate the free energy, we have: e^{-F/T} = \int e^{-(E+V)/T} \mathrm{d}\Gamma We can...

Can someone please explain the Taylor expansion for f(x,y) about (x0,y0) ? Would really appreciate some sort of step by step answer :) thankyou

I wasn't sure where to put this, so I put this here! In the photo, you see there's written 'Taylor expanding for small delta-r2, we find' ... I really don't get the two steps in the next line. Any help would be greatly appreciated.

Homework Statement Hi Say I want to Taylor-expand f(\omega + m\sin(\Omega t)) where ω and Ω are frequencies, m is some constant and t denotes time. Then I would get f(\omega + m\sin(\Omega t)) = f(\omega) + (m\sin(\Omega t)\frac{dI}{d\omega} + \ldots Is it necessary to make any...

Homework Statement Find the first two non-zero terms in the Taylor expansion of \frac{x}{\sqrt{x^2-a^2}} where a is a real constantHomework Equations f(x)=f(x_0)+f^{\prime}(x_0)(x-x_0)+\frac{f^{\prime\prime}(x_0)}{2!}(x-x_0)^2+...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n The Attempt at a Solution If...

i am very confuse how my profs always use taylor expansion in physics which somehow doesn't follow the general equation of f(x) = f(a) + f'(a)(x-a) + 1/2! f''(a)(x-a)2 and so on... like for example, what is the taylor expansion of x - kx where k is small it was given as something like...

Not sure under which forum this should have gone under, anyway can someone who really understands it explain it to me in as simple terms as they can, from what I'm getting its approximates something for a function or something? No idea.

I am attempting to complete a problem for a problem set and am having difficulty simplifying an expression; any help would be greatly appreciated! The question is a physics question which attempts to derive an equation for the temperature within a planet as a function of depth assuming...

Homework Statement Show that if F is twice continuously differentiable on (a,b), then one can write F(x+h) = F(x) + h F'(x) + \frac{h^2}{2} F''(x) + h^2 \varphi(h), where \varphi(h) \to 0 as h\to 0. Homework Equations The Attempt at a Solution I'm posting this here...

Homework Statement Find the critical point(s) of this function and determine if the function has a maxi- mum/minimum/neither at the critical point(s) (semi colons start a new row in the matrix) f(x,y,z) = 1/2 [ x y z ] [3 1 0; 1 4 -1; 0 -1 2] [x;y;z] Homework Equations The...

Hi, I have a following question... Can it be that there is given some Lagrangian and instead of considering whole Lagrangian one makes its series expansion and considers only some orders of expansion? Can you bring some examples or why and when does this happen... ? Thank you

Homework Statement The following is a modification of Newton's method: xn+1 = xn - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn) Homework Equations We are supposed to use the following method: let En = xn + p where p = root → xn = p + En Moreover, f(xn) = f(p + En) = f(p) +...

what does it mean to say taylor expansion of ex centered at 0? does it mean that the sum of the expansion will give me the value that the function ex will take when x = 0 ? so its e0 = 1? also, how do we know what value to center on? because as i encounter taylor series in my...

Hello all, I understand that the taylor expansion for a multidimensional function can be written as f(\overline{X} + \overline{P}) = f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P}) where t is on (0,1). Although I haven't seen that form before, it makes sense...

Homework Statement This shouldn't be so hard to do I guess, but I just cannot figure it out. The problem statement: Prove that the special form of the discrete Laplacian operator in radial coordinates acting on a grid function u_{l,m} at the central grid point l=0, m=0, given by...

For retarded scalar potential of arbigtrary source around origin: V(\vec r, t) = \frac 1 {4\pi\epsilon_0}\int \frac { \rho(\vec r\;',t-\frac {\eta}{c}) }{\eta} d\;\tau' \;\hbox { where }\;\eta =\sqrt{r^2 + r'^2 - 2 \vec r \cdot \vec r\;' } Where \;\vec r \; point to the field point where V...

Dear all, This question is close to the post "Laplace transform of a Taylor series expansion" in PhysicsForums.com, dated Jul06-09. This is my problem: Consider the Laplace transform F(s) = 1 / ( s - K(s) ) , where K(s) = -1/2 + i/(2*Pi) * ln[ ( Lambda - (b+i*s) )/( b + i*s...

On p35 of Jackson's Classical Electrodynamics 3rd Edition, the author gives the expansion of the charge density \rho(\mathbf{x'}) around \mathbf{x'}=\mathbf{x} as \rho(\mathbf{x'}) = \rho(\mathbf{x}) + \frac{r^2}{6}\nabla^2\rho + ... where r = |\mathbf{x} - \mathbf{x'}| My question is...

I have this translation operator T(a) that acts on a function y(x) and causes the transformation T(a)y(x) = y(x+a). I am supposed to be "expanding y(x+a) as a taylor series in a" to show that T(a)=eipa, where p is the operator p = -i.d/dx] So, I've started out with the general equation for the...

What does it mean to have a taylor expansion of a gradient (vector) about the position x? I.e. taylor expansion of g(x + d) where g is the gradient and d is the small neighborhood.

Homework Statement In my never ending quest to suck and never be able to do Taylor Expansions, I have another one. I hope one day I'll be able to do these. I have an unknown material and a scanning tunnel microscope. A layer of hydrogen atoms of radius R are added to the surface. This of...

Well - I am a 16 year old student, whos really interested in math. I do a lot of studying on my own, because I am a bit bored with the present math in school. Right now I am reading about solving differential equations with power series. I can do this, and i do understand the recurrence...

I want to show this taylor expansion: \frac{1}{\sqrt{1+{x}^{2}}} \rightarrow x^2 what I keep getting is something to the x^3 could some one please help me with this simple expansion?

ok .lets say the expression we have is ex the taylor expansion becomes 1+x+x2/2+... integrating becomes x+x2/2+x3/6+...+c so how do we know that c = 1? for it to become back to ex becos it is said that integral of ex = ex do we just let x be 0 to find c = 1? does it work for all...

Homework Statement With a Taylor series expansion of the well-behaved \rho ({\bf{x'}}) around {\bf{x'}} = {\bf{x}}, one finds the Taylor expansion of the charge density to be, \rho ({\bf{x'}}) = \rho ({\bf{x}}) + {\textstyle{1 \over 6}}{r^2}{\nabla ^2}\rho + ... Homework Equations...

Homework Statement I have the function (a(1+z)3 + b)-1/2 and i need to taylor expand it around z=0 to the first order, a and b are constants, there sum is equal to one. I have the answer: 1 - (1+q)z where q = a/2 - b This is in my physics book but it does not explain the...

hello, please help to calculate the taylor polynomial for http://latex.codecogs.com/gif.latex?f(x)=x^{x}-1 around the point a=1 i thought to write it as g(x)=x^x and then f(x)=g(x)-1 and then find the polynomial for g(x) as lng(x)=xln(x) but it seems incorrect.

I am wroking through an electrodynamics textbook and there is this Taylor expansion to do later a multipole expansion. But I can't figure out how the author does it. Please any help? the expansion: \frac{1}{|\vec{r}-\vec{r'}|} = \frac{1}{r} - \sum^3_{i=1} x'_i \frac{\partial}{\partial...

Homework Statement a) Using a geometric series, find the Taylor expansion of the function f(x) = x/(1+x^2) b) Use the series found in (a) to obtain the Taylor expansion of ln(1 + x^2) Homework Equations The Attempt at a Solution I really don't know where to start; I can't find...

Hi everyone. The problem I have to face is to perform a taylor series expansion of the integral \int_{-\infty}^{\infty}\frac{e^{-\sum_{i}\frac{x_{i}^{2}}{2\epsilon}}}{\sqrt{2\pi\epsilon}^{N}}\cdot e^{f(\{x\})}dx_{i}\ldots dx_{N} with respect to variance \epsilon. I find some difficulties...

Homework Statement Use the taylor's expansion of f(x)= x1/4 about x= 16 to estimate (16.1)1/4 Homework Equations Taylors formula: f(a) + f'(a) (x-a) + (f''(a)/2!) (x-a)2+...The Attempt at a Solution Ok I have calculate the taylor expansion to be: 2 + (1/32) (x-16)-(3/320) (x-16)2+ (7/262144)...

Homework Statement Find the taylor expansion of the following formula in the case where r > > d to the first order in \epsilon = \frac{d}{r} \frac{1}{r_{+}} = \frac{1}{\sqrt{r^{2} + (\frac{d}{2})^{2} - rdcos\theta}} Homework Equations (1 + \epsilon)^{m} = 1+m\epsilon, where...

Homework Statement f(x) = \sqrt{x}, a = 4 Homework Equations f(x) = \sumf^{n}(a)/n! (x-a)^{n} The Attempt at a Solution f(x) = x^{1/2} f^{'}(x) = \frac{1}{2}x^{1/2} f^{2}(x) = -\frac{1}{2}*\frac{1}{2}x^{-3/2} f^{3}(x) = \frac{1}{2}*\frac{1}{2}*\frac{3}{2}x^{-5/2} f^{4}(x) =...

Can someone pls explain hot to compute a taylor expansion for f(x,y) using mathematica

Homework Statement Could someone please explain how the taylor expansion of 1/(r-r') turns into ( 1/r+(r'.r)/r^3 + (3(r.r')^2-r^2r'^2)/2r^5 +...) Homework Equations The Attempt at a Solution

How do you Taylor expand e^{i \vec{k} \cdot \vec{r}} the general formula is \phi(\vec{r}+\vec{a})=\sum_{n=0}^{\infty} \frac{1}{n!} (\vec{a} \cdot \nabla)^n \phi(\vec{a}) but \vec{k} \cdot \vec{r} isn't of the form \vec{r}+\vec{a} is it?

Homework Statement I'm having a hard time following a taylor expansion that contains vectors... http://img9.imageshack.us/img9/9656/blahz.png http://g.imageshack.us/img9/blahz.png/1/ Homework Equations The Attempt at a Solution Here's how I would expand it: -GMR/R^3 -...

Can someone please tell me how to expand \ln(x + \sqrt{1+x^2}) for small x? I'd like to retain terms at least up to order x^5. Thanks!

Homework Statement Develop the Taylor expansion of ln(1+z). Homework Equations Taylor Expansion: f(z) = sum (n=0 to infinity) (z-z0)n{f(n)(z0)}/{n!} Cauchy Integral Formula: f(z) = (1/(2*pi*i)) <<Closed Integral>> {dz' f(z')} / {z'-z} The Attempt at a Solution I have NO idea...

Homework Statement With n>1, show that (a) \frac{1}{n}-ln\frac{n}{n-1}<0 and (b) \frac{1}{n}-ln\frac{n+1}{n}>0 Use these inequalities to show that the Euler-Mascheron constant (eq. 5.28 - page330) is finite. Homework Equations This is in the chapter on infinite series, in the section...

Homework Statement What is the quadratic approximation to the potential function? Homework Equations U(x) = U0((a/x)+(x/a)) U0= 20 a=4 The Attempt at a Solution This is just the last part of a question on my engineering homework, I never learned Taylor expansions before even...

Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4...