Thursday, October 20, 2016

Calculus of a Single Variable, Chapter 9, 9.7, Section 9.7, Problem 30

Taylor series is an example of infinite series derived from the expansion of f(x) about a single point. It is represented by infinite sum of f^n(x) centered at x=c . The general formula for Taylor series is:
f(x) = sum_(n=0)^oo (f^n(c))/(n!) (x-c)^n
or
f(x) = f(c) + f'(c) (x-c)+ (f'(c))/(2!) (x-c)^2+ (f'(c))/(3!) (x-c)^3+ (f'(c))/(4!) (x-c)^4+...
To determine the Taylor polynomial of degree n=2 centered at c=pi , we may apply the definition of the Taylor series by listing the f^n(x) up to n=2 .
f(x) = x^2cos(x)
Apply Product rule of differentiation: d/(dx) (u*v) = v*du + u*dv for each derivative.
f'(x) = d/(dx) (x^2cos(x))
Let u = x^2 then du =2x
v = cos(x) then dv = -sin(x)
f'(x) =cos(x) *(2x) + x^2*(-sin(x))
=2xcos(x)-x^2sin(x)
f^2= d/(dx)(2xcos(x)-x^2sin(x) )
=d/(dx)2xcos(x)- d/(dx) x^2sin(x)
For d/(dx)2xcos(x) , we let:
u = 2x then du =2
v = cos(x) then dv = -sin(x)
d/(dx)2xcos(x)= cos(x)*2 + 2x*(-sin(x))
=2cos(x) -2xsin(x)
For d/(dx)x^2sin(x) , we let:
u = x^2 then du =2x
v = sin(x) then dv = cos(x)
d/(dx)2xcos(x)= sin(x)*2x + x^2*cos(x)
=2xsin(x) +x^2cos(x)
Then,
d/(dx)2xcos(x)-d/(dx) x^2sin(x) = [2cos(x) -2xsin(x)] -[2xsin(x) +x^2cos(x)]
= 2cos(x) -2xsin(x) -2xsin(x) -x^2cos(x)
=2cos(x) -4xsin(x) -x^2cos(x)
Thus, f^2(x) =2cos(x) -4xsin(x) -x^2cos(x).
Plug-in x=pi , we get:
f(pi) =pi^2*cos(pi)
=pi^2*(-1)
=-pi^2
f'(pi)=2pi*cos(pi)-pi^2*sin(pi)
=2pi*(-1) -pi^2 *(0)
=-2pi
f^2(pi) =2cos(pi) -4*pi*sin(pi) -pi^2*cos(pi)
=2(-1) -4*pi*0 -pi^2*(-1)
=-2+pi^2 or -(2-pi^2)
Applying the formula for Taylor series centered at c=pi , we get:
sum_(n=0)^2 (f^n(pi))/(n!)(x-pi)^n
=f(pi) + f'(pi) (x-pi)+ (f'(pi))/(2!) (x-pi)^2
=(-pi^2) + (-2pi) (x-pi)+ (-(2-pi^2))/(2!) (x-pi)^2
= -pi^2 -2pi (x-pi)-(2-pi^2)/2 (x-pi)^2
The Taylor polynomial of degree n=2 for the given function f(x)=x^2cos(x) centered at c=pi will be:
P(x) =-pi^2 -2pi (x-pi)-(2-pi^2)/2 (x-pi)^2

No comments:

Post a Comment

Why is the fact that the Americans are helping the Russians important?

In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...