find the degree 2 taylor polynomial for f(x)= e^x sin(x) about the point a = 0. bound the error in this approximation when -π/4≦x≦π/4
please steps by steps!
f(x) = e^x sin x
f '(x) = e^x sin x+e^x cos x = e^x ( sin x + cos x)
f ''(x) = e^x ( sin x + cos x) + e^x ( cos x - sin x) = 2 e^x cos x
f ''' (x) = 2 e^x cos x - 2 e^x sin x = 2 e^x ( cos x - sin x )
f(0)=0, f '(0)=1, f ''(0)=2, f '''(0)=2
f(x) ~ f(0) + f '(0) x + [f ''(0)/2!] x^2 + [f '''(0)/3!] x^3= x + x^2 + (1/3) x^3
the degree 2 taylor polynomial for f(x)= e^x sin(x) about the point a = 0
is following
x+x^2 the error (1/3) x^3 when -π/4≦x≦π/4
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f(x) = e^x sin x
f '(x) = e^x sin x+e^x cos x = e^x ( sin x + cos x)
f ''(x) = e^x ( sin x + cos x) + e^x ( cos x - sin x) = 2 e^x cos x
f ''' (x) = 2 e^x cos x - 2 e^x sin x = 2 e^x ( cos x - sin x )
f(0)=0, f '(0)=1, f ''(0)=2, f '''(0)=2
f(x) ~ f(0) + f '(0) x + [f ''(0)/2!] x^2 + [f '''(0)/3!] x^3= x + x^2 + (1/3) x^3
the degree 2 taylor polynomial for f(x)= e^x sin(x) about the point a = 0
is following
x+x^2 the error (1/3) x^3 when -π/4≦x≦π/4