Taylor Definition and 849 Threads

Consider the IVP: \left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{ze IVP :p} Hypothesis: f(x,y)\subset C^\infty_{x,y}(D)\; \; / \; \;(x_0,y_0)\in D [Note that this condition automatically satisfies the hypotheses of the...

Questions: Is there a quicker way to find the formula for the nth derivative of a function, instead of finding the first several derivatives and trying to find a pattern, and using that pattern to form the equation for the nth derivative? Also, is there a formula for the nth derivative...

I was wondering if someone could help me with Goldstein's equation 6.3 (3rd Edition). It is the chapter of oscillations and all that he has done in the equation is to expand it in the form of a Taylor series. I can't seem to get how all those ni's come to get there.

Homework Statement Find the taylor series of \frac{1+z}{1-z} where z is a complex number and |z| < 1 Homework Equations \sum^{\infty}_{0} z^n = \frac{1}{1-z} if |z| < 1 The Attempt at a Solution \sum^{\infty}_{0} z^n = \frac{1}{1-z} \frac{1+z}{1-z} =...

Homework Statement 1/(4x-5) - z/(3x-2) based @ 0, answers are in those z things.. sigma Homework Equations i think we use sigma of e^x, but idk how... The Attempt at a Solution since tayor sereis of e^x is like 1/x, do i plug 4x-5 in? thanks

Homework Statement sin(x)= sum((-1)^k* (x^(2k+1)/(2k+1)!))k=0 to infinity Homework Equations so if i want to find sin(x)^2, (not sin(x^2), that would be easier though...) The Attempt at a Solution then... do i square the whole thing, like this? sum(((-1)^k*...

Homework Statement how to you find like the answer for f(1.5), or f(1.00001) those kind of question? thanks with like eq. = f(b)(x-b)... am i making sense? thanks

Homework Statement Need to calculate fractional uncertainty f, of M (mass of a star in this case), where f is much less than one. The hint i was given was all i need to know is M \alpha d3, and use a taylor expansion to the first order in f. M = mass of a star, d = distance to star...

not sure I get the Taylor Series... Hello Everyone. I understand that the taylor series approximate a function locally about a point, within the radius of convergence. If we use the Taylor series it means that we do not know the function itself. But to find the taylor series we need the...

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...

Homework Statement find the first four nonzero terms in the power series expansion of tan(x) about a=0 Homework Equations \Sigma_{n=0}^{\infty} \frac{f^n (a)}{n!}(x-a)^n The Attempt at a Solution Well the series has a zero term at each even n (0,2,4 etc) for n=1 I got x, for...

Homework Statement I can either use the alternating series estimation thereom (which i don't really know) or Taylor's Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. sin(x) = x - (x^3)/6 (|error| < 0.01) Do I...

Homework Statement Find the Taylor polynomial T_n(x) for the function arcsin x at a = 0, n = 3 Homework Equations Well, I understand the Taylor poly. for sine, but how do i get arcsine?

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...

Homework Statement using the Taylor Formula, find the series for the function f(x)=e^{2x}Homework Equations \sum \frac{f^{n}(a)}{n!} (x-a)^{n} any help as to where i start would be great. new to series...

Hullo, Somehow, I couldn't get the TeX to come out right. I have been trying to learn scheme theory (algebraic geometry) and completely forgotten how to do this simple calculus type stuff... Homework Statement Let V be a potential of the form [tex]V = \left(\frac{1}{r} +...

Many of you have probably used the book Differential Equations by Lomen & Lovelock. For my class I'm working on Example 2, Page 153. You don't need to see the book, though, to help me out. It's a four-part problem and I'm on the last step not knowing where to take it. In Part B, we...

Homework Statement Let T_(4)(x): be the Taylor polynomial of degree 4 of the function ` f(x) = ln(1+x) ` at `a = 0 `. Suppose you approximate ` f(x) ` by ` T_(4)(x) `, find all positive values of x for which this approximation is within 0.001 of the right answer. (Hint: use the...

Homework Statement The Taylor series for f(x) = ln(sec(x)) at a = 0 is sum_(n=0to infinity) c(sub n) (x)^n. Find the first few coefficients. The Attempt at a Solution I've been trying to figure out where to start by looking it up...I've seen instructions that each coefficient is...

This is for revision purposes (not homework so I am not trying to cheat my way out of it!) and its too late in the week to see my lecturer about this. I don't have much of an attempt at the solution because i haven't got a clue where to start. It looks like just a short one though. Here goes...

Homework Statement im being asked for the first 4 non zero values for the taylor expansion of exp(x) which is simple, but then it asks for the range of x values that are valid for the expansion. i have never come across ths before - any idea?

can anybody help me with this integration? Integral of e to the -2x times x squared dx it expands to 1/4, but I'm not sure how to start.

I need to find the Taylor polynomial of degree 4 expanded about a=4 for the function f(x)=squareroot of (x)=x^(1/2) This is what I've started with but I'm not sure how to proceed and if I even started correctly: f'(x)(-1/2)x^(-1/2)=1/2sqrt(x) f"(x)=(-1/4)x^(-3/2)=-1/4x^3/2...

Homework Statement Calculate the taylor polynom of order 3 at (0,0,0) of the function with well-known series (that means I can't just take the derivatives) f(x,y,z)=\sqrt{e^{-x}+\sin y+z^{2}} Homework Equations The Attempt at a Solution I wrote the functions within the square...

Find the Taylor polynomial of degree 9 of f(x) = e^x about x=0 and hence approximate the value of e. Estimate the error in the approximation. I have written the taylor polynomial and evaluated for x=1 to give an approximation of e. Its just the error that is confusing me. I have: R_n(x) =...

Taylor Polynomial Error--Please help! Use Taylor's theorem to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than .001. e^.3 So is the procedure to take the derivatives and plug in 0 (since c=0) and find an...

Use Taylor's theorem to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than .001. e^.3 I really, really don't know what to do for this one, and I have a quiz tomorrow. I have read through the section in the book, but...

Homework Statement Use Taylor's theorem to obtain an upper bound of the error of the approximation. Then calculate the exact value of the error. cos(.3) is approximately equal to 1 - (.3)^2/2! + (.3)^4/4! Homework Equations The Attempt at a Solution I came up with upper...

Hello, I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :) Find the Taylor polynomial of degree 3 of f (x) = e^x about x = 0 and hence find an approximate value for...

Why does f(x) = abs (0) not have the first Taylor polynomial center at Xo = 0 ? Does it have second Taylor polynomial center at Xo = 0 ?

Hi am trying to solve this Taylor series with 3 variables but my result is not equal to the solution- So i think i might be wrong expanding the taylor series, or the solution is not correct Homework Statement Find an a approximated value for the function f(x,y,z) = 2x + ( 1 + y) * sin z at the...

explain why an estimate of intergal(f(x)) from 0 to .5 using the first 2 nonzero terms of its taylor approximation differs from the actual value of this integral by less than 1/200

Deduce that the Taylor series about 0 of 1/sqrt(1-4x) is the series summation (2n choose n) x^n. From this conclude that summation (2n choose n) x^n converges to 1/sqrt(1-4x) for x in (-1/4,1/4). Then show that summation (2n choose n) (-1/4)^n = 1/sqrt(1-4(-1/4)) = 1/sqrt(2) What I know...

[SOLVED] Taylor approximation Homework Statement I have an exact funktion given as: P(r)=1-e^{\frac{-2r}{a}}(1+\frac{2r}{a}+\frac{2r^2}{a^2}) I need to prove, by making a tayler series expansion, that: P(r)\approx \frac{3r^3}{4a^4} When r \prec \prec a The Attempt at a Solution...

Hi, I want to confirm this: a=8 , b=5 , c=7 Decide the Taylor polynomial of degree 2 in the point (0, a) to the function f (x, y)=sqrt(1+bx+cy). Decide with the aid of Taylor polynomial if the function has a local minimum in (0, a). I used the partial derivates: df/dx =...

Homework Statement f(\lambda) = \frac{8\pi hc\lambda ^{-5}}{e^{hc/\lambda kT}-1} Is Planck's Law where h\ =\ Planck's\ constant\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s; c\ =\ speed\ of\ light\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}; and\ Boltzmann's\ constant\ =\ k\ =\ 1.3806503(24)\...

Homework Statement Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2]. I did the question...

Homework Statement use an appropriate local quadratic approximation to approximate the square root of 36.03 Homework Equations not sure The Attempt at a Solution missed a day of class

Problem Statement Compute the Taylor Series expansion of f(x) = exp(-x^2) around 0 and use it to find an approximate value of the integral (from 0 to 0.1) of exp(-t^2) dt Solution Part1: First to compute the Taylor Series - I am pretty sure about this step so I will not give details...

When I tried to learn the Taylor series , I could not comprehend why a infinite series can represent a function Would anyone be kind enough to teach me the Taylor series? thank you:smile: PS. I am 18 , having the high school Math knowledge including Calculus

Homework Statement Find the Taylor series for f(x) = sin x centered at a = pi / 2 Homework Equations The Attempt at a Solution Taylor series is a new series for me. I believe the first step is to start taking the derivative of the Taylor series. f(x) = sinx f'(x) =...

\biggl(-\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} +\mathcal{O}(x^8)\biggr)-\frac12\biggl(-\frac{x^2}2+\frac{x^4}{24}+\mathcal{O}(x^6)\biggr)^2+\frac13\biggl(-\frac{x^2}2+\mathcal{O}(x^4)\biggr)^3 + \mathcal{O}(x^8)\\ & =-\frac{x^2}2 + \frac{x^4}{24}-\frac{x^6}{720} - \frac{x^4}8 +...

question on solving a very hard limit with taylor i want to solve this limit: (1 - (cos x)^(sin(x^2)) ) / (sin (x^4)) i tried to solve it by tailor the problem is that when i subtitute each trigonometric function with a taylor series i get a series in a power of a series i don't...

Homework Statement Use the Taylor series about x = a to verify the second derivative test for a max or min. Show if f'(a) = 0 then f''(a) > 0 implies a min point at x = a ... Hint for a min point you must show that f(x) > f(a) for all x near enough to a. Homework Equations The Attempt at a...

How to find a polynomial P(x) of the smallest degree such that sin(x-x^2)=P(x)+o(x) as x->0? Do I have to calculate the first six derivatives of f(x)=sin(x-x^2) to get Taylor polynomial approximation?

[SOLVED] power series and taylor Homework Statement Let f be a function defined by f(x)=\frac{1+c x^2}{1+x^2}, and let x be an element of R for c\neq1, find the taylor series around the point a, and find the radius of convergence of the taylor series Homework Equations for power series...

I am trying to find the maclaurin series for f(x) = (1 + x)^(-3) --> what is the best way of doing this--to make a table and look for a trend in f^(n)?

Find the Taylor polynomial of degree 10 about x=0 for f(x)=sin2x (show all work) This is what i have: M10= f(0)+f\hat{}1(0)x+f\hat{}2(0)x\hat{}2/2!+f\hat{}3(0)x\hat{}3/3!+...+f\hat{}10(0)x\hat{}10/10! f(x)=sin2x f(0)=sin2(0)=0 f\hat{}1(x)=2cos2x f\hat{}1(0)=2cos2(0)=2...

We are supposed to use taylor's inequality to estimate the accuracy of the approximation of the taylor polynomial within the interval given. so, f(x) = cos x , a = pi/3, n=4 and the interval is 0<= x <= 2pi/3 the fifth derivative is -sin x to get the M in taylor's inequality, wouldn't...

Homework Statement Find the Maclaurin series of the function f(x) = 5(x^2)sin(5x)Homework Equations \sum(Cn*x^n) The Attempt at a Solution I'm supposed to enter in c3-c7 I already know that c4 and c6 are 0 because the derivative is something*sin(0)=0 but for the odd numbered c's I am having...