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:
The Taylor polynomial of   $f(x)$   at   $a$   is
$\sum _{n=0}^{\infty }c_{n}(x-a)^{n}$ where $c_{n}={\frac {f^{(n)}(a)}{n!}}.$

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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