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Chapter 8: Q-30E (page 435)

Question: In Problems 29鈥�34, determine the Taylor series about the point X0for the given functions and values of X0.

30.f(x)=x-4x0=1

Short Answer

Expert verified

The required expression is .鈭�n-0鈭�(-1)n(x-1)

Step by step solution

01

Taylor series

For a function the Taylor series expansion about a point is given by,f(x-x0)=f(x0)+f'(x0).(x-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) = x-1 at x0=1.

Calculating the derivatives of function at x0.

f(x)=x-1thenf(x0)=1

f'(x)=x-2thenf'(x0)=-1

f''(x)=2x-3thenf''(x0)=2

f'''(x)=6x-4thenf'''(x0)=-6

f''''(x)=24x-5thenf''''(x0)=24

03

Substitute the derivatives in Taylor series

Substituting the above derivatives in Taylor series expansion for the function at x0=1, then,

(x-1)-1=1-1(x-1)+2.(x-1)22!-6(x-1)33!+24.(x-1)44!+....=1-(x-1)+(x-1)2-(x-1)3+(x-1)4+....=鈭�n-0鈭�(-1)n.(x-1)n

Hence, the required expression is=鈭�n-0鈭�(-1)n.(x-1)n

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Most popular questions from this chapter

In Problems 15-17, solve the given initial value problem t2x"-12x=0. x(1)=3 and x'(1)=5.

Question: In Problems 29鈥�34, determine the Taylor series about the point X0for the given functions and values of X0.

33. f (X)= x3+3x-4, x0= 1

In Problems 15-17,solve the given initial value problem x2y"+5xy'+4y=0.

y(1) =3 and y'(1) = 7

In Problems \(5 - 14\) solve the given linear system.

\({\bf{X'}} = \left( {\begin{array}{*{20}{c}}{{\rm{ 0 2 1}}}\\{1{\rm{ }}1{\rm{ }} - 2}\\{2{\rm{ }}2{\rm{ }} - 1}\end{array}} \right){\bf{X}}\)

In Problems 13-19,find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

y''-e2xy'+(cosx)y=0y(0)=-1,y'(0)=1
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