Find the linear approximation to the function f(x)=e^(3x) ∙ sin3x at the base point a = 0.

Question

Find the linear approximation to the function f(x)=e^3x ∙ sin3x at the base point a = 0.

Answer 1

f(x) = (e^3x)sin(3x) f(0) = 0

f'(x) = (3e^3x)sin(3x) + 3(e^3x)cos(3x) f''(0) = 3

so, the linearized function is f(x) = f(0) + f'(0)x = 3x

assuming you are using the Taylor's series truncated prior to the powers of x!