009C Sample Final 3, Problem 5

Consider the function

$f(x)=e^{-{\frac {1}{3}}x}$

a) Find a formula for the $n$th derivative $f^{(n)}(x)$ of $f$ and then find $f'(3).$

b) Find the Taylor series for $f(x)$ at $x_{0}=3,$ i.e. write $f(x)$ in the form

$f(x)=\sum _{n=0}^{\infty }a_{n}(x-3)^{n}.$

Foundations:

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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