Find the Taylor series for f(x) centered at c.

Question

f(x) = e^3x,    c = −9

Answer 1

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) = f⁰(-9) = e^-27

f¹(x) = f'(x) = 3 e^3x

f¹(-9) = 3e^-27

f²(x) = f''(x) = 3 * 3 e^3x

f²(-9) = 9e^-27

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

fⁿ(x) = 3ⁿe^3x

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