Michael K. answered • 04/26/19

PhD professional for Math, Physics, and Computer Tutoring

Using the f(x) = e^{3x} with the center point (x0) at -9, we can start by computing the derivatives of f(x) since it is infinitely differentiable...

f(-9) = f^{0}(-9) = e^{-27}

f^{1}(x) = f'(x) = 3 e^{3x}

f^{1}(-9) = 3e^{-27}

f^{2}(x) = f''(x) = 3 * 3 e^{3x}

f^{2}(-9) = 9e^{-27}

We see the pattern and can now write the nth derivative as...

f^{n}(x) = 3^{n}e^{3x}

f(x) = sum_[lb-->k=0]_[ub-->k=infinity] 3^{k}e^{-27}/k! * (x + 9)^{k}