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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{The Derivative Function, part 2}
\centerline{October 23, 2017}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
On Tuesday, we were working with the \emph{derivative} of a function 
$f$ at $x$:
$$f'(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h},$$
provided the limit exists. With today's video we know the following facts about derivatives
\begin{enumerate}
\item[(1)]  If $f(x) = x^n$ for any number $n$, then $f'(x) = n x^{n-1}$.
\item[(2)]  If $f'(x)$ exists, then $(kf)'(x)$ exists and $(kf)'(x) = k f'(x)$.
\item[(3)]  If $f'(x)$ and $g'(x)$ both exist, then so does $(f + g)'(x)$, and 
$(f + g)'(x) = f'(x) + g'(x)$.  
\item[(4)]  If $f(x) = e^x$, then $f'(x) = e^x$ (that's right, it's the same
function(!))
\end{enumerate}
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\noindent
{\it Questions}
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\noindent
\begin{enumerate}
\item[(1)]  Compute $f'(x)$ for $f(x) = x^7 + 2 e^x$.
\item[(2)]  Compute $f'(x)$ for $f(x) = x^4 + 3 e^x + 7 + 2x^{-1}$.
\item[(3)]  Make up your own example function of the same type as the one in (2) and
compute $f'(x)$ using points (1), (2) and (3) in the Background.
\item[(4)]  What is $f'(x)$ if $f(x) = e^\pi$?  (Be careful!) 
\end{enumerate}
\noindent
The video for today gave the idea for showing point (4) in the Background
because for any exponential function $f(x) = b^x$, 
$$f'(x) = \lim_{h\to 0} \frac{b^{x+h} - b^x}{h} = b^x \lim_{h\to 0} \frac{b^h - 1}{h}.$$
We said that $b = e$ was the base for the exponential that made 
$$\lim_{h\to 0} \frac{b^h - 1}{h} = 1.$$
\begin{enumerate}
\item[(5)] Take $b = 2.7$ and estimate 
$\lim_{h\to 0} \frac{(2.7)^h - 1}{h}$ numerically.  Then repeat with 
and $b = 2.8$ and estimate the limit $\lim_{h\to 0} \frac{(2.8)^h - 1}{h}$. 
What do you conclude about the number $e$?
\item[(6)]  Now repeat part (5) with $b = 2.75$.  Is $e$ in the interval $[2.7,2.5]$
or $[2.75,8]$?  Note that you could repeat this process over and over to \emph{zero in}
on the decimal expansion of the number $e$.
\end{enumerate}
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