Saturday, February 15, 2014

Calculus of a Single Variable, Chapter 9, 9.10, Section 9.10, Problem 41

Maclaurin series is a special case of Taylor series which is centered at a=0. We follow the formula:
f(x) =sum_(n=0)^oo (f^n(0))/(n!)x^n
or
f(x) = f(0) + (f'(0))/(1!)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)x^3 +(f^4(0))/(4!)x^4 +(f^5(0))/(5!)x^5 +...
To list of f^n(x) up to n=10 , we may apply the Product rule for differentiation: d/(dx) (u*v) = u'*v +u*v '.
f(x) = xsin(x)
f'(x) = xcos(x)+sin(x)
f''(x) = 2cos(x)-xsin(x)
f'''(x) =- xcos(x)-3sin(x)
f^4(x) = xsin(x)-4cos(x)
f^5(x) = xcos(x)+5sin(x)
f^6(x) = 6cos(x)-xsin(x)
f^7(x) = -xcos(x)-7sin(x)
f^8(x) = xsin(x)-8cos(x)
f^9(x) = xcos(x)+9sin(x)
f^(10)(x)= 10*cos(x)-x*sin(x)
Note: d/(dx)x=1 , d/(dx) cos(x) =-sin(x) , and d/(dx) sin(x)=cos(x) .
Plug-in x =0, we get:
f(0) = 0*sin(0)
=0*0
=0
f'(0) = 0*cos(0)+sin(0)
=0*1+0
=0
f''(0) = 2cos(0)-0*sin(0)
=2*1-0*0
=2
f'''(0) =- 0*cos(0)-3sin(0)
=-0*1-3*0
=0
f^4(0) = 0*sin(0)-4cos(0)
=0*0 -4*1
=-4
f^5(0) = 0*cos(0)+5sin(0)
=0*1+5*0
=0
f^6(0) = 6cos(0)-0*sin(0)
=6*1-0*0
=6
f^7(0) = -0*0cos(0)-7sin(0)
=-0*1-7*0
=0
f^8(0) =0*sin(0)-8cos(0)
=0*0-8*1
=-8
f^9(0) = 0*cos(0)+9sin(0)
=0*1+9*0
=0
f^(10)(0)= 10*cos(0)-0*sin(0)
=10*1-0*0
=10
Note: cos(0)=1 and sin(0) =0 .
Plug-in the values in the formula, we get:
f(x) = 0 + 0/(1!)x+2/(2!)x^2+0/(3!)x^3+(-4)/(4!)x^4+0/(5!)x^5
+6/(6!)x^6+ 0/(7!)x^7+(-8)/(8!)x^8+0/(9!)x^9 +10/(10!)x^10+...

=0 + 0/(1)x+2/(1*2)x^2+0/(1*2*3)x^3-4/(1*2*3*4)x^4
+ 0/(1*2*3*4*5)x^5 + 6/(1*2*3*4*5*6)x^6+0/(1*2*3*4*5*6*7)x^7
-8/(1*2*3*4*5*6*7*8)x^8 + 0/(1*2*3*4*5*6*7*8*9)x^9 + 10/(1*2*3*4*5*6*7*8*9*10)x^(10)+...

=0 + 0+2/2x^2+0/6x^3-4/24x^4 + 0/120x^5 + 6/720x^6
+0/5040x^7 -8/40320x^8 + 0/362880x^9 +10/3628800x^(10)+...

=0 + 0+x^2+0-1/6x^4 + 0 + 1/120x^6
+0-1/5040x^8 + 0+1/362880x^(10)+...

=x^2 -1/6x^4 + 1/120x^6 -1/5040x^8 +1/362880x^(10)+...
Therefore, the Maclaurin series for the function f(x) =xsin(x) can be expressed as:
xsin(x)=x^2 -1/6x^4 + 1/120x^6 -1/5040x^8 +1/362880x^(10)+...

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...