Taylor Series for Common Functions

Using the representation formula in Taylor's Theorem for a series centered at $a$ (including at $a=0$), $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n,$$ we can derive the power series representation for a number of other common functions.  We call these Taylor series expansions, or Taylor series.

We will compute the Taylor series of several functions, all centered at 0.  Recall from our previous practice the following steps: 
First, compute the first, second, third, etc. derivatives of $f$ until you see a pattern and can write $f^{(n)}(x)$ with some certainty. 
Next, you can compute $f^{(n)}(0)$, and then
finally, find (and simplify if necessary) $\frac{f^{(n)}(0)}{n!}x^n$.  

Or, manipulate a series you know, i.e. if you have the Taylor series for $f(x)$ you can find the series for $f(3x)$ without recomputing.

DO:  Compute the Taylor series for $f(x)=e^x$ and $f(x)=\sin x$.

DO:  Once you have found the Taylor series for $e^x$, find the Taylor series for $e^{2x}$ without recomputing -- just replace all the $x$-values with $2x$.

The video computes these and other Taylor series -- try your own before watching these solutions.