Taylor Definition and 878 Threads

So I'm computing a second order Taylor series expansion on a function that has multiple variables. So far I have this I(x,y,t)=dI/dx(change in x)+dI/dy(change in y)+dI/dt(change in t)+2nd order terms Would it still be a better approximation than just he first order if I included some...

I'm trying to learn about finite difference methods to solve differential equations. I'm using Advanced Engineering Mathematics 9th Ed., and in explaining Euler's method he claims the following Taylor series: y(x+h) = y(x) + hy'(x) + \dfrac{h^2}{2}y''(x) + \cdots He then truncates that...

Now although this is a problem for my EE course, it is more of a calculus question so I figured I would receive the best answers by posting it in this section. I have just started on the problem but could use some input on my thoughts. So here we go (there are two parts): (problem screenshot is...

When a Taylor Series is generated from a functions n derivatives at a single point, then is that series for any value of x equal to the original function for any value x ? For example graph the original function (x) from x= 0 to x = 10. Now plug into the Taylor Expansion for x , values...

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...

Homework Statement series expansion at c=2 of ln(x^2+x-6) Homework Equations The Attempt at a Solution After substituting y= x -2 we get ln(y^2+5y) = ln(y) + ln(y+5) but I am not kinda sure how to use the taylor series of ln(1+x)...

I have a question about Taylor expanding functions. For both cases I can't get my head around why things are the way they are. I just don't see how one would perform Taylor expansions like that. The first: The starting point of a symmetry operations is the following expansion: f(r+a) = f(r)...

I read wikipedia article also but I can't find the proof of taylor series and from where it came from??

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...

Usually to do the remainder we take Rn(x) = (f differentiated n+1 times at a ).(x-c)n+1/(n+1)!, but when my function is sin(x) do i take (f differentiated 2n+2 times at a ).(x-c)2n+2/(2n+2)!? Thanks

Homework Statement How many terms of the taylor series of the cosine function about c = 0 are needed to calculate cosine 2 to an accuracy of 1 / 10000 The Attempt at a Solution I have said that |Rn(2)| = |cosn+1(a) 2n+1/(n+1)!|<2n+1/(n+1!) Now i can't do it ...

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.

I consider an array of lattice points and construct a vector at each lattice points. How to convert this discrete system into a continuum one by using the Taylor series expansion by considering the lattice distance say \lambda? thanks in well advance?

in general I'm trying to figure out a way to work with taylor series more efficiently. this means i want to be able to write down the taylor series of a complicated function just by knowing the taylor series(es?) of the component functions. I've figured out how to do products and quotients...

How do I use Taylor Series to show f(P) is a local maximum at a stationary point P if the Hessian matrix is negative definite. I understand that some of the coefficients of the terms of the taylor series expansion are the coordinates of the Hessian matrix but for the f_xy term there is no...

Homework Statement Hi there. I have this exercise which I'm trying to solve now. It says: Using that \displaystyle\sum_{n=0}^{\infty}x^n=(1-x)^{-1} find one Taylor development for the function f(x)=\ln(1-x) So, I've made some derivatives...

Solved Thanks guys

Homework Statement Let v(λ) = ∫0∞ λe-λy f(y)dy a) Prove that limλ→ ∞ v(λ) = f(0). b) Determine and prove limλ→ ∞ λ(v(λ)- f(0)). α β γ δ ε ζ η θ ι κ λ μ ν ξ ο ° π ρ ς σ τ υ φ χ ψ ω Ω ~ ≈ ≠ ≡ ± ≤ ≥ Δ ∇ Σ ∂ ∫ ∏ → ∞ Homework Equations Assume there exists some constant M where maxx∈ℝ...

Homework Statement Find the Taylor Series of 1/x centered at c = 1. Homework Equations \sum_{n=0}^{\infty} f^n (c) \frac{(x-c)^n}{n!} The Attempt at a Solution I made a list of the derivatives: f(x) = 1/x f'(x) = -1/x2 f''(x) = 2/x3 f'''(x) = -6/x4 f(1) = 1 f'(1) =...

Homework Statement a. Find the first four nonzero terms in the Taylor series expansion about x = 0 for f(x) = (1+x)^.5 b. Use the results found in part (a) to find the first four nonzero terms in the Taylor series expansion about x= 0 for g(x) = (1 + x^3)^.5 c. Find the first four...

Homework Statement The Taylor series of function f(x)=ln(x) at a=7 is given by: f(x)=\sum^{\infty}_{n=0}c_{n}(x-7)^{n} Determine the interval of convergence The Attempt at a Solution I have worked out that the series would be of the form...

Do you just replace the x's with (x-3)'s? Since e^(-x^2) is defined as the taylor series though, it seems like the answer should be the same as the series about x=0. Thanks! P.S. does anyone know how to resize images? :$

Homework Statement I have to approximate sin(1/2) with the taylor inequality Homework Equations taylors inequality |Rn(x)| ≤ M/(n+1)! | x-a|n+1 The Attempt at a Solution Im not really sure what the significance of this is, but ill do the derivatives f(x) = sin(x) f'(x) = cos(x) f''(x) =...

Homework Statement The function f(x)=ln(10-x) is represented as a power series: \sum^{\infty}_{n=0}a_{n}x^{n} Find the first few coefficients in the power series. Hint: First find the power series for the derivative of . The Attempt at a Solution Okay, start seems fairly...

Homework Statement Determine the limit and then prove your claim. limx\rightarrow\infty (1+\frac{1}{x^2} }) xHomework Equations I know that the formal definition that I need to use to prove the limit is: {limx\rightarrow\infty (1+\frac{1}{x^2})x=1}={\forall \epsilon>0, \exists N > 0, \ni x>N...

I have a couple of general questions, combined with this one specific question Homework Statement Find the Taylor or MacLauren series centered about the given value for the following function, determine the radius of convergence Homework Equations \mathrm{Ln}\ z, 2 The Attempt at a Solution...

Homework Statement Let f be a function that has derivatives of all orders for all real numbers. Assume f(1) = 3, f'(1) = -2, f''(1) = 2, and f'''(1) = 4 a. Write the second-degree Taylor polynomial for f about x = 1 and use it to approximate f(0.7) b. Write the third-degree Taylor...

Homework Statement find the taylor series of ln(1+x) centered at zero Homework Equations from 0 to infinity ∑ cn(x-a)n cn = f(n)(a)/n! The Attempt at a Solution f(x) = ln(1+x) f'(x) = 1/(1+x) f''(x) = -1/(1+x)2 f'''(x) = 2/(1+x)3 f''''(x) = -6/(1+x)4 f(0) = 0...

Homework Statement Find the Taylor series for f(x) centered at the given value of a. (Assume that f has a power series expansion. Do not show that Rn(x)--> 0.) f(x) = x^3, a = -1Homework Equations f(x) = f(a)+f'(a)(x-a)+(f''(a)/2!)(x-a)^2+(f'''(a)/3!)(x-a)^3+...+(f(nth...

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...

Homework Statement Expand f(z) = (3z+1)/(15+2z-z^{2}) at z=1 and find the circle of convergence. Homework Equations The Attempt at a Solution I think this is pretty straight forward, but I want to make sure I'm doing everything correctly. I used a power series...

Homework Statement [PLAIN]http://img822.imageshack.us/img822/427/scangj.jpg Homework Equations The Attempt at a Solution Hi, could anyone help me with part b of this question, part a I have completed, however I seem to be drawing a blank on the second part

First off, I apologize if this is in the spot. I thought about this off and on since this morning on where to put it. There are probably four sections this could go. Feel free to move it if needed. I am awful at recognizing and solving taylor series and approximation type problems. Since...

Homework Statement I have this equation: T=(1+\frac{U_{0}^{2}}{4E(U_{0}-E)}sinh^{2}(2 \alpha L))^{-1} Where α is given by: \alpha = \sqrt{ \frac{2m(U_{0}-E)}{\hbar^{2}}} I have to show that in the limit αL>>1 my equation is approximately given by...

Homework Statement Suppose that: sum [a_n (n-1)^n] is the Talyor series representation of tanh(z) at the point z = 1. What is the largest subset of the complex plane such that this series converges? Note: 'sum' represents the sum from n=0 to infinity Homework Equations tanh(z) =...

One of the greatest movie legends we've ever had. http://news.yahoo.com/s/ap/us_obit_taylor

Homework Statement Expand the function f(E) as a Taylor series.Homework Equations f(E)=E/(KT)+(Ec/E)1/2 The Attempt at a Solution E=Eo So it says that F(E)~Ao+A1(E-Eo)+A2(E-Eo)2... I need to find out what Ao A1 and A2 are, but not sure how to do that. It says as a hint that A1=0 becasue f(E)...

Background: I'm trying to transform the gaussian distribution from flat space to curved space. I start with the flat, 1D gaussian distribution in the form \[{\textstyle{1 \over {{{(\pi {\Delta ^2})}^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em...

I'm a 2nd year Physics undergrad and am ashamed to admit I don't fully understand the taylor expansion. I have seen the Maclaurin expansion derived which I understand but the Taylor expansion is a little weird to me. I don't really understand what it means to expand a function "about a...

Homework Statement approximate sin13 by using the Taylor series using the TI84. Add for n=150 Homework Equations the infinite sums for sinx is ((-1)^n)(x^(2n+1)/(2n+1)!) The Attempt at a Solution I'm new to programming so i don't have any idea on where to start. I was...

how would you use Taylor Series to answer this: Find a value of h such that for |x|<h implies sin(x)=x-x^3/6 +x^5/120 + R where |R|< 10^-4?

Homework Statement Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder. The Attempt at a Solution I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2]...

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...

Homework Statement find the 2nd, 3rd, and 6th degree taylor approximation of: f(x) = 10(x/2 -0.25)5 + (x-0.5)3 + 9(x-0.75)2-8(x-0.25)-1 for h = 0.1 to h = 1, with \Deltah = 0.05 and where xo=0; and x = h Homework Equations N.A The Attempt at a Solution I just need to...

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?

Homework Statement Find the Taylor polynomial of degree 3 of \frac{1}{2+x-2y} near (2,1). Homework Equations The Attempt at a Solution I have already solved this problem by evaluating the R^2 Taylor series; I'm mostly curious about another aspect of the problem. By substituting u = x-2y, it...

find the first three non zero terms in the Taylor Series about z=0 of exp(z sin z) i have little idea how to even start on the question because it is exp to the power of z sin z and it just looks too complicated. i hav tried looking thru txtbooks for something similar but no similar question...

I'm aiming to calculate ln(x) numerically. I'm using the following procedure for this: 1) If x is greater or equal to 1, use Newton's method. 2) If x is smaller between 0 and 1, use Taylor series expansion. Newton's method works good, but I have problems with Taylor series expansion method...