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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Higher Derivatives}
\centerline{November 3, 2015}
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\noindent
{\it Background}
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If $f(x)$ is a function, $f'(x)$ is often called its {\it first derivative}.  (In the 
alternate notation that might be written $\frac{dy}{dx}$ if we're thinking of the 
graph $y = f(x)$.  The 
reason for this is that it is possible to go on and differentiate $f'(x)$ to get 
another new function.  The derivative of $f'(x)$, that is, $(f')'(x)$ is also called
the {\it second derivative} of the original $f$, and written $f''(x)$ or 
$\frac{d^2y}{dx^2}$.  Continuing in the same way, if we can differentiate $f''(x)$, the
result is called the \emph{third derivative} of $f$, and so forth.
The rules for computing these \emph{higher derivatives} are exactly the same as 
the rules for computing $f'(x)$ to start.  Today, we want to practice with these 
and understand why they are interesting.  
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\noindent
{\it Questions}
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\noindent
\begin{enumerate}
\item[(1)]  For each function, use the appropriate short-cut rules to find the first derivative, and then differentiate again to get the second derivative:
\begin{enumerate}
\item[(a)] f(x) = $x^5 + 4x^3 + x$. Also find the third derivative $f'''(x)$, 
the fourth derivative, the fifth derivative, and the sixth derivative for this one. 
(What \emph{always} happens if you differentiate a polynomial function repeatedly enough times?)
\item[(b)] $g(x) = \displaystyle{\frac{x}{x^2 - 1}}$. Your life will be a lot easier 
here if you simplify the first derivative before differentiating again to get $g''(x)$.
\item[(c)]  $h(x) = \displaystyle{(x^2 + x + 1)e^x}$.   
Also find the third derivative $h'''(x)$ for this one.
\end{enumerate}
\item[(2)]  So \emph{why} would we want to be able to differentiate multiple times?  The answer
is that the second derivative $f''$ in particular encodes interesting information about the original
function $f$.
\begin{enumerate}
\item[(a)] Suppose we know $f''(x) > 0$ on some interval $(a,b)$.  Recall that $f'' = (f')'$.  What
can we say about $f'$ on that interval?  Draw pictures illustrating graphs on which $f''(x) > 0$
for all $x$.  What is the name for the property you are seeing (recall today's video)?
\item[(b)]  Now, suppose we know $f''(x) < 0$ on some interval $(a,b)$.  
Recall again that $f'' = (f')'$.  What
can we say about $f'$ on that interval?  Draw pictures illustrating graphs on which $f''(x) < 0$
for all $x$.   What is the name for the property you are seeing?
\end{enumerate}  
\item[(3)]  Do Problem 39 on page 154 of our text.  
\end{enumerate}
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