Write the first three nonzero terms of the Taylor Series for

Write the first three nonzero terms of the Taylor Series for the following function centered at a = - pi f(x) = x^2cos(5x) Using your solution from part (a), find the first three nonzero terms in Taylor Polynomial form of the following: x^2 cos(5x) dx

Solution

a)given f(x)=x²cos(5x)

f(-)=(-)²cos(-5) =-²

f \'(x)=2xcos(5x) -5x²sin(5x)

f \'(-)=2(-)cos(-5) -5(-)²sin(-5) =2

f \'\'(x)=2cos(5x) -10xsin(5x) -10xsin(5x)-25x²cos(5x)

f \'\'(-)=2cos(-5) +10sin(-5) +10sin(-5)-25(-)²cos(-5)=-2+0+0+25²=25²-2

f(x)=f(-) +(f \'(-))(x-(-))_/1! +(f \'\'(-))(x-(-))²_/2!

f(x)=-² +(2)(x-(-))_/1! +(25²-2)(x-(-))²_/2!

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b)x²cos(5x) dx

=[-² +(2)(x-(-))_/1! +(25²-2)(x-(-))²_/2! ]dx

=[-²x +(2)(x-(-))²_/2! +(25²-2)(x-(-))³_/3! ] +C