Difference between revisions of "009C Sample Final 3, Problem 5"

Revision as of 18:19, 4 March 2017

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}.

Solution:

(a)

Step 2:

(b)

Step 2:

Final Answer:
(a)
(b)

@@ Line 3: / Line 3: @@
 ::<math>f(x)=e^{-\frac{1}{3}x}</math>
-<span class="exam">(a) Find a formula for the &nbsp;<math>n</math>th derivative &nbsp;<math>f^{(n)}(x)</math>&nbsp; of &nbsp;<math>f</math>&nbsp; and then find &nbsp;<math>f'(3).</math>
+<span class="exam">(a) Find a formula for the &nbsp;<math>n</math>th derivative &nbsp;<math style="vertical-align: -5px">f^{(n)}(x)</math>&nbsp; of &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; and then find &nbsp;<math style="vertical-align: -5px">f'(3).</math>
-<span class="exam">(b) Find the Taylor series for &nbsp;<math>f(x)</math>&nbsp; at &nbsp;<math>x_0=3,</math>&nbsp; i.e. write &nbsp;<math>f(x)</math>&nbsp; in the form
+<span class="exam">(b) Find the Taylor series for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; at &nbsp;<math style="vertical-align: -5px">x_0=3,</math>&nbsp; i.e. write &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; in the form
 ::<math>f(x)=\sum_{n=0}^\infty a_n(x-3)^n.</math>