/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q 29E Question: In Problems 29鈥�34, d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q 29E (page 435)

Question: In Problems 29鈥�34, determine the Taylor series about the point x0for the given functions and values of x0.

29. f(x)= cosx, x0 =蟺

Short Answer

Expert verified

The required expression is 鈭�n-0鈭�(-1)n+1(2n)!.(x-蟺)2n.

Step by step solution

01

Taylor series

For a function f(x) the Taylor series expansion about a point x0 is given by,

f(x-x0)=f(x0)+f'(x0).(x-x0)22!+f'''(x0).(x-x0)33!+...

02

Derivatives of function at x0

We have to calculate the Taylor series expansion for, f(x) = cos x at x0 = 蟺.

Calculating the derivatives of function at x0 .

f(x)= cos x then f(x0) = -1

f'(x)= -sin x then f'(x0) = 0

f'' (x) = -cos x then f''(x0) =1

f'''(x) =sin x then f'''(x0) =0

f''''(x) = cos x then f''''(x0) =-1

03

Substitute the derivatives in Taylor series

Substituting the above derivatives in Taylor series expansion for the function at

x0 = 'then,

cosx-蟺=-1+0.(x-蟺)+1(x-蟺)22!+0(x-蟺)33!-1(x-蟺)44!=-1+(x-蟺)2-2!-(x-蟺)44!+...=鈭�n-0鈭�(-1)n+1(2n)!.(x-蟺)2n

Hence, the required expression is鈭�n-0鈭�(-1)n+1(2n)!.(x-蟺)2n

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Question: In Problems 1鈥�10, determine all the singular points of the given differential equation.

2. x2y"-3y-xy = 0

Question 18: In Problems, find a power series expansion for f(X) , given the expansion for f(x).

role="math" localid="1664283848012" fx=sinx=鈭�k=0鈭�(-1)k(2k+1)!x2k+1

Find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem,x'+(sint)x=0;x(0)=1

Find at least the first four nonzero terms in a power series expansion about x0for a general solution to the given differential equation with the given value for x0,

y''+(3x-1)y'-y=0;x0=-1

Aging spring. As a spring ages, its 鈥渟pring constant鈥� decreases on value. One such model for a mass-spring system with an aging spring is mx"(t)+bx'(t)+ke- 畏tx(t)=0 .

Where m is the mass, b the damping constant, k and 畏 positive constants and x(t) displacement of the spring from equilibrium position. Let m=1 kg, b=2 Nsec/m, k=1 N/m, 畏 =1 sec-1. The system is set in motion by displacing the mass 1m from it equilibrium position and releasing it (x(0)=1, x'(0)=0). Find at least the first four nonzero terms in a power series expansion of about t=0 of displacement.

See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.