Taylor Definition and 849 Threads

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...

1. Solve y'=3t^2y^2 on [0, 3] , y0 = −1, using Euler method and Taylor method of order 3. Compare your solutions to the exact solution. y(t)=(-1/((t^3)+1)) I DONT KNOW WHAT IS WRONG WITH MY PROGRAM! PLEASE HELP =D Homework Equations http://en.wikipedia.org/wiki/Euler_method...

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...

Homework Statement The first three terms of a Taylor Series centered about 1 for ln(x) is given by: \frac{x^{3}}{3} - \frac{3x^{2}}{2} + 3x - \frac{11}{6} and that \int{ln(x)dx} = xlnx - x + c Show that an approximation of ln(x) is given by: \frac{x^3}{12} - \frac{x^2}{2} +...

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've spent all day on this problem and am wasting precious time needed for other work - please give any input you can! The question: given two wages, w1 and w2 where w2 > w1... a. the difference between the wages as a proportion of the lower: a = (w2 - w1) / w2 b. the difference between the...

Homework Statement Not much has gotten me in this class, and I almost want to say this has to be a typo, but I want someone else to check it out first. Homework question is that we need to show that cos(cos θ)*cosh(sin θ) = Ʃ(-1)ncos(nθ)/(2n)! for n>=0 There is a similar one involving...

Part 1 of Pharaoh's Taylor series and modified Euler question from Yahoo Answers The Taylor series expansion about \(t=0\) is of the form: \(y(t)=y(0)+y'(0)t+\frac{y''(0)t^2}{2}+.. \)We are given \(y(0)\) and \(y'(0)\) in the initial condition, and so from the equation we have: \(y''(0) =...

Homework Statement Expand f(x) = x/(x+1) in a taylor series about a=10. Homework Equations f(x) = Ʃ (f^n(a)*(x-a)^n / n! The Attempt at a Solution I'm having a hard time arriving at the correct answer..I think I'm definitely getting lost somewhere along the way. Here's what I've...

Hi Guys, Looking at some notes i have on conformal mapping and I have the following where z is complex and z* denotes its conjugate, R is a real number z* = -iR + R^2/(z-iR) and my lecturer says that using the taylor series we get, z* = -iR + iR(1+ z/iR + ...) I've been...

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.

Anyone bored enough to want to help me out with some calculus? I got to deliver this in 6 hours and can't work these out. Help would be SO much appreciated, I've been at it all night and can't make it out. 1. y^2 - e^sin(x) + xy = sin (x)* cos (y) +3 assume y= y(x) and find y ' (0) 2...

If I want to find the taylor series at x = 0 for sin(x^2)+cos(x)... sin(x^2) = x^2 - x^6/3! + x^10/5! - x^14/7! ... cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! ... So why does sin(x^2) + cos(x) = 1 + x^2/2! + x^4/4! + 121x^6/6! ...? Thanks!

1. Prove that the MacLaurin series for cosx converges to cosx for all x. Homework Equations Ʃ(n=0 to infinity) ((-1)^n)(x^2n)/((2n)!) is the MacLaurin series for cosx |Rn(x)|\leqM*(|x|^(n+1))/((n+1)!) if |f^(n+1)(x)|\leqM lim(n->infinity)Rn=0 then a function is equal to its Taylor series...

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...

This might be a weird quest and in the wrong section but does anyone know of a list with errors and corrections for the book "Advanced Calculus" by Taylor & Mann 3rd ed? Thanks

Homework Statement W(x,s)=(1/s)*(sinh(x*s^0.5))/(sinh(s^0.5)) Find the inverse laplace transform of W(x,s), i.e. find w(x,t). Answer: w(x,t)= x + Ʃ ((-1)^n)/n) * e^(-t*(n*pi)^2) * sin(n*pi*x) summing between from n=1 to ∞ Homework Equations An asymptotic series..?! The...

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...

1. Problem: if f(1,3)=7, use Taylor expansion to describe f(1.2,3.1) and f(.9,2.8) if the partials of f are give by df/dx=.2 d^2f/dx^2=.6 df/dy=.4 d^2f/dy^2=.9 (you do not need to go beyond the second derivative for this problem) 2. I know from class how to do this if one variable changes...

Homework Statement The course is Computational Physics, but in a sense this is a pretty straight computer science or even mathematical challenge. The first part of the assignment - the relatively easy part - was to write a Fortran program to take two variables - the number to which e...

Homework Statement Here is the question: I don't quite know what I did wrong. My method is below. Homework Equations The Attempt at a Solution f(x)=√x f'(x)=\frac{1}{2(x)^{1/2}} f''(x)=\frac{-1}{(2)(2)(x^{3/2}} a=4 f(a)=2 f'(a)=1/4...

Use the error estimate for Taylor polynomials to find an n such that | e - (1 + (1/1!) + (1/2!) + (1/3!) + ... + (1/n!) | < 0.000005 all i have right now is the individual components... f(x) = ex Tn (x) = 1/ (n-1)! k/(n-1)! |x-a|n+1 = 0.000005 a = 0 x = 1 I don't know where to go from here

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.

Homework Statement I was given the following problem, but I am having a hard time interpreting what some parts mean. We're given the equation sinθ+b(1+cos^2(θ)+cos(θ))=0 Assume that this equation defines θ as a function, θ(b), of b near (0,0). Computer the Taylor polynomial of...

Homework Statement Derive the following formula using Taylor series and then establish the error terms for each. Homework Equations f ' (x) ≈ (1/2*h) [4*f(x + h) - 3*f(x) - f(x+2h)] The Attempt at a Solution I honestly have no idea how to go about deriving this. The professor did...

$1+v_{t+1} = (1+v_t)\exp\left(-rv_{t-1}\right)\approx (1+v_t)(1-rv_{t-1})$ The book is linearizing the model where we generally use a Taylor Series. How was the expression expanded in the Taylor Series to get the approximate answer? Thanks.

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...

Bit stuck on this. I tried writing 1/(1-z^2) as taylor series then Cos z as taylor series, then substituting one into the other but it looked a bit dodgy. Can one simple substitute like this?

I have to find the first three non zero terms of this series by hand. I know the answer and it is -(z^3/3) - z^7/2520 - z^11/19958400 Which will take ages to get to by brute force. Is there a quicker way?

I am trying to find the Taylor series for $$\displaystyle \dfrac{\left(\dfrac{1}{z-i}\right)}{z+i} $$ where z is a complex number.There is a reason it is set up as a fraction over the denominator so let's not move it down.

Hi there, I was hammering out the coefficients for the Taylor Series expansion of f(x) = \frac{1}{\sqrt{1-x^2}}, which proved to be quite unsatisfying, so decide to have a look around online for alt. approaches. What I found (in addition to the method that uses the binomial theorem) was...

Homework Statement The magnitude of the gravitational force exerted by the Earth on an object of mass m at the Earth's surface is Fg = G*M*m/ R^2 where M and R are the mass and radius of the Earth. Let's say the object is instead a height y << R above the surface of the Earth. Using a...

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...

Hello, I have two functions say f1(β) and f2(β) as follows: f1(β)=1/(aδ^2) + 1/(bδ) + O(1) ... (1) and f2(β)= c+dδ+O(δ^2) ... (2) where δ = β-η and a,b,c,d and η are constants. Eq. (1) and (2) are the Taylor series expansions of f1(β) and f2(β) about η...

Under what circumstances is it correct to say of the function u(x) \in L^2(-\infty,\infty) that u(x-t) = u(x) - \frac{du}{dx}t + \frac 12 \frac{d^2u}{dx^2}t^2 - \cdots = \sum_{n=0}^\infty \frac{u^{(n)}(x)}{n!}(-t)^n.

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

Hello all, My question is in regards to the Taylor series expansion of f(x)=e^x=1+x+x^2/(2!)+x^3/(3!)... I calculated the value of e^(-2) using the first 4 terms, 6 terms, and then the first 8 terms. I then calculated the relative error to compare it to the true value, depcited by my...

Homework Statement Hi! I have a couple of problems on Taylor Series Approximation. For the following equations, write out the second-order Taylor‐series approximation. Let x*=1 and, for x=2, calculate the true value of the function and the approximate value given by the Taylor series...

http://uploadpic.org/storage/2011/dGTvcFGgvl4VYoVB40HpLSMxH.jpeg Can somebody guide me through this? I know how to apply Newton Raphson Method, but the x^* symbol and "argmin" function are kinda new to me. I am re referring to part (c). Thanks.

Homework Statement Taylor's theorem can be stated f(a+x)=f(a)+xf'(a)+(1/2!)(x^2)f''(a)+...+(1/n!)(x^n)Rn where Rn=fn(a+y), 0≤y≤x Use this form of Taylor's theorem to find an expansion of sin(a+x) in powers of x, and show that in this case, mod(\frac{x^n Rn}{n!})\rightarrow0 as...

Can somebody guide me through this? I know how to apply Newton Raphson Method, but the x^* symbol and "argmin" function are kinda new to me. I am re referring to part (c). Thanks.

Homework Statement Find the taylor series of Log(z) around z=-1+i.Homework Equations The Attempt at a Solution So I have for the first few terms as \frac{1}{2}*log(2)+\frac{3\pi i}{4}+\frac{z+1-i}{-1+i}-\frac{2(z+1-i)^{2}}{(-1+i)^{2}}+\frac{3(z+1-i)^{3}}{(-1+i)^{3}}- But the correct...

http://bildr.no/view/1030479 The link above, it is my own and it is a bit disorderly, I think should explain taylor polynomials. In one assignent one had an assignment to derive taylor polynomials for cost^2 If one use the derivation rules with chain one get 2t for first derivative and...

I am asked to solve the taylor expansion of sin x around the point -pi/4 to the fourth term. I got sin(-pi/4)+cos(-pi/4)(x+pi/4)-.5sin(-pi/4)(x+pi/4)^2-1/6(cos(-pi/4)(x+pi/4)^3 but I am getting it wrong and can't see my mistake.

Homework Statement Hello all, I have been working on a 3rd order taylor series, but the formula I have does not seem to get me the right answer. The formula I was given is for a taylor polynomial about point (a,b) is: P_3=f(a,b) +\left( f_{1}(a,b)x+f_{2}(a,b)y\right)...

Homework Statement Find the Taylor series of e^(x^2) about x=0 Homework Equations Taylor Series = f(a) +f'(a)(x-a) + (f''(a)(x-a)^2)/2 ... The Attempt at a Solution So, the first term is pretty obvious. It's e^0^2, which is zero. The second term is what got me...

Homework Statement Find the Taylor series expansion of f(x) = (x-1)/(1+(x-1)^2) about x=1 and use this to compute f(9)(1) and f(10)(1) Homework Equations The sum from n=0 to infinity of f(k)(c)/(k!) (x-c)k The Attempt at a Solution I'm not sure how to approach this...

Homework Statement A translation operator T(a) coverts ψ(x) to ψ(x+a), T(a)ψ(x) = ψ(x+a) In terms of the (quantum mechanical) linear momentum operator p_x = -id/dx, show that T(a) = exp(iap_x), that is, p_x is the generator of translations. Hint. Expand ψ(x+a) as a Taylor series...