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\begin{document}

\title{Math 241, Lecture 2}
\maketitle

\section{ Functions: the basic objects of study in calculus}   

Functions are means for producing one quantity (or set of quantities) from
another.    More formally we have the following.


\begin{definition}
A function assigns to each input, from some set called the {\em domain}
of the function, a unique output, in a set called the {\em range} of the function.   
\end{definition}

Functions can be about measurements of  a quantity, such as 
a cost function which measures for each time the amount
it costs to produce a given  good.    Functions can also
be  algebraic, such as the function which
takes a number, squares it, subtracts $\pi$ from the answer, and then takes
the reciprocal of that quantity.  

Even if you are more interested in functions which measure real-world
quantities, part of the
power of calculus, and of mathematics in general, is connecting those functions
with algebraic functions.   For example, both $F = ma$ (Newton) 
and $E = mc^2$ (Einstein)
changed the world, relating fundamental physical quantities through 
simple functions.     Also, with algebraic functions we can compute
explicitly, which helps develop understanding we can carry with us 
when we do applications.  

 

\section{ Basic numerical functions}   

There are three kinds of basic functions we use in mathematics.   
\begin{itemize}
\item Polynomials, ratios of polynomials, fractional powers.   \\
For example $f(x) = x^2$   or $\frac{x-1}{x+1}$   or $\sqrt{x^5 -3}$.  
\item Exponential and logarithmic functions.   \\
For example $f(x) = 2^x$   or $\log_{10}(x)$, which we will define later
for those of you who haven't seen them.   
\item Trigonometric functions such as $\sin(x)$, which we will not be studying.
\end{itemize}

Understanding basic functions better will be an important outcome of our
study of calculus.  

\begin{question}
What is the better deal: getting one million dollars a day for a month?  
getting $n^5$ dollars on the $n$th day for a month?   getting
one dollar the first day and then on each day getting twice  what one got 
the previous, for a month?   \\

What if a month is changed to a week?   or a year?  
\end{question}

 

\section{ Making new functions from old}

%If you produce $100 + 2n$ widgets on day $n$ and $30 - n$ gadgets,
%what is the combined total of widgets and gadgets you produce on day $n$?  
%Congratulations, you've just added two functions.  

Adding, subtracting and multiplying functions is straightforward to both
execute and understand.  

\begin{example}
``Profit is the difference between total revenue and total cost'' is
translated into taking the difference of functions.  
\end{example}

Taking the quotient of functions can be trickier.   
For algebraic functions, you have to be careful because the domain might change.  

\begin{example}
If $f(x) = \sqrt{x}$ and $g(x) = x^2 - 3x + 2$, what is the domain of the function 
$\frac{f(x)}{g(x)}$?  
\end{example}

For real-world functions, which may contain error, what would seem
to be a small amount of error in the denominator could lead to huge error
in the final answer.  

\begin{example}
To measure speed, we take distance travelled and divide it by time.  
What if we tried to measure the speed of a jet by using a stopwatch
over 100 yards?
\end{example}

 


\section{ Linear functions}

Linear functions are the simplest possible functions,  
so they are used often throughout mathematics.  
One of the main themes of calculus is the understanding
of general functions through related linear functions.  
In short, linear functions are the basis of calculus.  

\begin{definition}
Linear functions are functions of the form $f(x) = mx + b$,
where $m$ and $b$ are constants.  
\end{definition}

The graph of a linear function $y = mx + b$ is a line in the plane.  

\begin{question}
Are all lines in the plane graphs of linear functions?  
\end{question}

There are many ways to describe a given linear function.   
An important skill to develop is the ability to translate between the 
different descriptions.    Key to many of these descriptions
is the notion of slope.  

\begin{definition}
The slope of the line $y = mx + b$ is equal to $m$.   
It measures the change in $y$ if $x$ is increased by one.  
If one is not given the slope explicitly, it can be computed by
$m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$
and $(x_2, y_2)$ are any two points on the line. 
\end{definition}

\begin{example} 
Sketch some lines with slope $1$, $-1$, $2$, $-3$, $\frac{1}{2}$ and $\frac{-2}{3}$.  
\end{example}

\begin{example}
The points $(-1, -1)$ and $(3,7)$ are both on the line $y = 2x + 1$. 
We can verify the formula for $m$ in this case, 
and then take two other points on the line and use them to 
calculate the slope.
\end{example}

The different descriptions we will use are as follows, listed from
simplest to most complicated.  We translate each 
to the standard form, and give 
an example.  

\begin{itemize}
\item Standard, slope-intercept form: $y = mx + b$.  
\item Linear equation form: $ax + by = c$.  
\item Point-slope form:  ``the line which goes through $(x_0, y_0)$
with slope $m$''   namely, $(y - y_0) = m(x - x_0)$.  
\item Two-point form:  ``the line between $(x_1, y_1)$ and
$(x_2, y_2)$,''   namely 
$(y - y_1) = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$.  
\end{itemize}

\begin{example}
Find the equations - in slope-intercept form - of the following lines:  
\begin{itemize}
\item The line $3x + 4y = 7$.  
\item Through the points $(-1, 4)$ and $(2, 3)$.  
\item Through the point $(1, 2)$ with slope $3$.  
\end{itemize}
\end{example}

Two lines are perpendicular if they intersect at ninety degree angles.  
In equations, it means that if the slope of one line is $m$, the slope of the other
is $-m$.  

\begin{question}
 Why does this make sense?  
\end{question}

\begin{example}
Find the equation of the line perpendicular to the line 
$2x - 3y = 3$ at the point $(3,1)$.  
\end{example}



\end{document}







