f(x)=x^2e^(-x) , n=4 Find the n'th Maclaurin polynomial for the function.

Maclaurin series is a special case of Taylor series that is centered at c=0 . The expansion of the function about 0 follows the formula:
f(x)=sum_(n=0)^oo (f^n(0))/(n!) x^n
 or
f(x)= f(0)+(f'(0))/(1!)x+(f^2(0))/(2!)x^2+(f^3(0))/(3!)x^3+(f^4(0))/(4!)x^4 +...
To determine the Maclaurin polynomial of degree n=4 for the given function f(x)=x^2e^(-x) , we may apply the formula for Maclaurin series.
To list f^n(x) up to n=4 , we may apply the following formula:
Product rule for differentiation: d/(dx) (u*v) = u' *v +u*v'
Derivative property: d/(dx) (f+-g+-h) = d/(dx) f +-d/(dx) g+-d/(dx) h
Power rule for differentiation: d/(dx) x^n =n*x^(n-1)
Derivative formula for exponential function: d/(dx) e^u = e^u * (du)/(dx)
f(x)=x^2e^(-x)
Let u =x^2 then u' = 2x
      v = e^(-x)  then v' = e^x*(-1) =-e^(-x)
d/(dx) (x^2e^(-x)) = 2x*e^(-x) + x^2*(-e^(-x))
                        = 2xe^(-x) -x^2e^(-x)
Let: u =x then u' =1
        v = e^(-x)  then v' =-e^(-x)
Note: c = constant value.
d/(dx) c*xe^(-x) = c*d/(dx) xe^(-x)
                     = c*[1*e^x +x * (-e^(-x))]
                      = c*[e^x-xe^(-x)]
                      = ce^x-cxe^(-x)
d/(dx) c*e^(-x) = c*d/(dx) e^(-x)
                  =c*(-e^(-x))
                  =-ce^(-x)
 
f'(x) =d/(dx) (x^2e^(-x))
           = 2xe^(-x) -x^2e^(-x)
f^2(x) = d/(dx) (2xe^(-x) -x^2e^(-x))
            =d/(dx) 2xe^(-x) -d/(dx) x^2e^(-x)
            = [2e^(-x)-2xe^(-x)] - [2xe^(-x) -x^2e^(-x)]
            =2e^(-x)-2xe^(-x) - 2xe^(-x) +x^2e^(-x)
            =2e^(-x)-4xe^(-x) +x^2e^(-x)
f^3(x) = d/(dx) (2e^(-x)-4xe^(-x) +x^2e^(-x))
            =d/(dx) 2e^(-x) -d/(dx) 4xe^(-x)+ d/(dx) x^2e^(-x)
            =[-2e^(-x)] -[4e^(-x)-4xe^(-x)]+ [2xe^(-x) -x^2e^(-x)]
            =- 2e^(-x) -4e^(-x)+4xe^(-x)+ 2xe^(-x) -x^2e^(-x)
             =- 6e^(-x)+6xe^(-x) -x^2e^(-x)
f^4(x) = d/(dx) ( - 6e^(-x)+6xe^(-x) -x^2e^(-x))
          =d/(dx) (-6e^(-x)) + d/(dx) 6xe^(-x)- d/(dx) x^2e^(-x)
         =[ 6e^(-x)]+[6e^(-x)-6xe^(-x)] -[2xe^(-x) -x^2e^(-x)]
         =6e^(-x)+6e^(-x)-6xe^(-x) -2xe^(-x) +x^2e^(-x)
          =12e^(-x)-8xe^(-x) +x^2e^(-x)
Plug-in x=0 for each f^n(x) , we get:
f(0)=0^2e^(-0)
          =0*1
          =0
f'(0)=2*0*e^(-0) -0^2e^(-0)
           =2*0*1 +0*1
           =0
f^2(0)=2e^(-0)-4*0*e^(-0) +0^2e^(-0)
            =2*1 -4*0*1 +0*1
              =2
f^3(0)=- 6e^(-0)+6*0*e^(-0) -0^2e^(-0)
           =-6*1 +6*0*1 +0*1
           =-6
f^4(0)=12e^(-0)-8*0*e^(-0) +0^2e^(-0)
             =12*1 -8*0*1 +0*1
            =12
Note: e^(-0) = e^0 =1 .
Plug-in the values on the formula for Maclaurin series, we get:
sum_(n=0)^4 (f^n(0))/(n!) x^n
      = f(0)+(f'(0))/(1!)x+(f^2(0))/(2!)x^2+(f^3(0))/(3!)x^3+(f^4(0))/(4!)x^4
      = 0+0/(1!)x+2/(2!)x^2+(-6)/(3!)x^3+12/(4!)x^4
      = 0+0/1x+2/2x^2-6/6x^3+12/24x^4
      = 0+0+x^2-x^3+1/2x^4
      = x^2-x^3+1/2x^4
The Maclaurin polynomial of degree n=4 for the given function f(x)=x^2e^(-x) will be:
P(x)=x^2-x^3+1/2x^4

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