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\title{Math 241, Lecture 12}
\maketitle

\section{ Limits and their use for derivatives}  

Last time we focussed on intuitive methods for finding limits,  
and introduced a few algebraic methods.     
Let's begin by practicing our algebraic limit techniques.  

\begin{example}
Find the following limits:
\begin{itemize}
\item  $\lim_{h \to 0} \frac{h^3 + h}{3h^2 + 2h}$.  
\item $\lim_{h \to 0} \frac{h^3 + 1}{3h^2 + 2h}$.  
\item $\lim_{x \to 3} 2^x$.  
\item $\lim_{x \to \infty} \frac{3x}{x^2 + 4}$.  
\item $\lim_{x \to \infty} \frac{5x^2 + 1}{x - 3x^2}$.  
\end{itemize}
\end{example} 


For another example, we revisit the famous computation of
the derivative of $f(x) = x^n$.    

\begin{theorem}
For any number $n$, the derivative of $x^n$ is $n x^{n-1}$.  
\end{theorem}

We can now justify this theorem carefully.  The key is to understand a 
general formula for $(x+h)^n$, which starts out $x^n + n \cdot x^{n-1} + \cdots$.  

\section{ Marginal analysis}  

As we have mentioned, the derivative is commonly used in 
theoretical business management and economics.   
One primary application goes under the name of marginal analysis.  
Marginal refers to the idea of trying to understand the costs, benefits,
etc of ``one more.''  

The main idea  mathematically is that since the
derivative is $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$   
the derivative is approximately equal to the difference
quotient when $h = 1$, if $f(x)$ is much larger than one 
(for example, if  $f(x)$ is national debt).   

In other words, sometimes we have the following approximate equality
$$f'(x) \simeq \frac{f(x + 1) - f(x)}{1} = f(x+1) - f(x). $$  
This is equivalent to the expression which we would get from 
approximation by increments, namely
$$ f(x+1) \simeq f(x) + f'(x) \cdot 1.$$  

This leads us to the following definitions.
 

\begin{definition}
\begin{itemize}
\item The marginal cost function $C'(x)$ is the derivative of the cost function.  
It approximates the cost of producing one more item after $x$ items have
already been produced.  
\item The marginal revenue function $R'(x)$ is the derivative of the revenue 
function.  It approximates the revenue generated by producing one more item
after $x$ have already been produced.  
\item The marginal profit function $P'(x)$ is the derivative of the profit
function.  It approximates ...    
% the profit generated by producing one more item
%after $x$ have already been produced. 
\end{itemize}
The relation between these functions is... 
\end{definition}

It may take some getting used to thinking of these quantities as functions of 
the number of goods produced, rather than say functions of time.  
But as we'll see in doing problems, these functions better allow you to make
decisions about production.  

\begin{example}
If it costs approximately $1000 - \frac{500}{x}$ dollars to produce $x$ CD's, what
is the marginal cost of producing the $20$th CD?  
\end{example}

\begin{example}
Suppose the value of the marginal profit function for producing the $1000$th 
jar of peanut butter was $-0.25$.    Should you produce more or fewer jars of 
peanut butter?    In general what does the sign of the marginal profit
function tell you?  
\end{example}

\begin{example}
It costs $\frac{n^2}{4} + 50 n + 100$ cents to produce $n$ rubber duckies.    
Moreover, all $n$ duckies will be sold when the price is  $(200 - \frac{n}{2})$
cents.   
Find the following:
\begin{itemize}
\item The marginal cost for producing the $n$th duckie.  
\item The marginal profit made by producing and selling the $n$th duckie.  
\item The values of marginal cost and profit when $n = 10$.  
\item The actual cost and profit for producing the tenth duckie.  
\end{itemize}
\end{example}

Moral of the story - the marginal cost, revenue and profit functions are
only {\em approximations} to the amounts needed/earned for producing
one more item.  But they are good approximations and (because of
the rules we learn for calculus) are easier to work with than the true values. 


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