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\begin_body
\begin_layout Standard
Paul Johnson
\end_layout
\begin_layout Standard
Curves and Slopes
\end_layout
\begin_layout Standard
March 1, 2008
\end_layout
\begin_layout Section
Relationships.
\end_layout
\begin_layout Standard
Consider some possible relationships in Figure
\begin_inset LatexCommand ref
reference "curves"
\end_inset
.
\end_layout
\begin_layout Standard
X is an independent variable
\end_layout
\begin_layout Standard
Y is a dependent variable
\end_layout
\begin_layout Standard
The line represents the
\begin_inset Quotes eld
\end_inset
expected
\begin_inset Quotes erd
\end_inset
value of Y as it depends on X.
\end_layout
\begin_layout Standard
When you consider a relationship, you probably ought to consider some basic
questions:
\end_layout
\begin_layout Itemize
is it linear?
\end_layout
\begin_layout Itemize
If it is not linear, how does the relationship depend on x?
\end_layout
\begin_layout Itemize
Are there
\begin_inset Quotes eld
\end_inset
boundaries
\begin_inset Quotes erd
\end_inset
or limits on the values of x and y?
\end_layout
\begin_layout Itemize
Is there a
\begin_inset Quotes eld
\end_inset
global maximum
\begin_inset Quotes erd
\end_inset
?
\end_layout
\begin_layout Itemize
Are there local minima or maxima?
\end_layout
\begin_layout Section
Straight lines.
Can't live with 'em, can't live without 'em.
\end_layout
\begin_layout Subsection
Look at a straight line!
\end_layout
\begin_layout Standard
Consider the straight line graphed in Figure
\begin_inset LatexCommand ref
reference "line"
\end_inset
.
\end_layout
\begin_layout Standard
A point is an ordered pair that is represented by a dot in the plane.
For example, (x1,y1) is a point.
Ordinarily, we would use subscripts on the x's and y's, but my drawing
software does not make that easy, so I'm not putting them in the text either.
(My drawing software does not allow some other things either, but let's
not go on singing that sad song.)
\end_layout
\begin_layout Standard
Comparing two points (x1,y1) and (x2,y2):
\end_layout
\begin_layout Quotation
the difference between them horizontally is Dx=x2-x1.
\end_layout
\begin_layout Quotation
the difference between them vertically is Dy=y2-y1.
\end_layout
\begin_layout Standard
It is pretty easy to see the slope of the line is the ratio of the two changes.
\begin_inset Formula \[
\frac{Dy}{Dx}=\frac{y2-y1}{x2-x1}\]
\end_inset
\end_layout
\begin_layout Standard
When you look at that figure, you see the slope of a straight line does
not depend on how you select either x1 or x2.
\end_layout
\begin_layout Subsection
Slope is a local concept, though.
\end_layout
\begin_layout Standard
Look at the broken line in Figure
\begin_inset LatexCommand ref
reference "brokenline"
\end_inset
.
\end_layout
\begin_layout Standard
What ideas do you have about how to define or interpret slope in this case?
\end_layout
\begin_layout Section
Slope of a
\begin_inset Quotes eld
\end_inset
smooth curve
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Standard
Look at the smooth curve I drew in Figure
\begin_inset LatexCommand ref
reference "DxSmooth"
\end_inset
.
\end_layout
\begin_layout Standard
This is drawn so
\begin_inset Quotes eld
\end_inset
x
\begin_inset Quotes erd
\end_inset
is a particular point and we pick various values x3,x2,x1, that get closer
and closer to x.
\end_layout
\begin_layout Standard
Note how the slope
\begin_inset Formula $\frac{Dy}{Dx}$
\end_inset
adjusts every time the reference point xi changes.
\end_layout
\begin_layout Section
A Derivative is the result of an
\begin_inset Quotes eld
\end_inset
itty bitty
\begin_inset Quotes erd
\end_inset
change in X.
\end_layout
\begin_layout Subsection
Keep making Dx smaller and smaller!
\end_layout
\begin_layout Standard
In Figure
\begin_inset LatexCommand ref
reference "DxSmall"
\end_inset
, it shows what would happen if we kept taking values of xi and making them
get closer and closer to x.
Eventually, the slope would approach a
\begin_inset Quotes eld
\end_inset
limiting value
\begin_inset Quotes erd
\end_inset
.
The limiting value is the slope of a tangent line.
\end_layout
\begin_layout Subsection
A
\begin_inset Quotes eld
\end_inset
tangent line
\begin_inset Quotes erd
\end_inset
is...
\end_layout
\begin_layout Standard
Well, I don't actually know the technical definition.
But I know a tangent line one when I see one! The tangent line
\begin_inset Quotes eld
\end_inset
just touches
\begin_inset Quotes erd
\end_inset
the curve at x.
You create the tangent line by making Dx smaller and smaller.
When Dx gets
\begin_inset Quotes eld
\end_inset
arbitrarily small
\begin_inset Quotes erd
\end_inset
(that means really really small, like as small as you can possibly imagine),
then
\end_layout
\begin_layout Subsection
Derivative.
\end_layout
\begin_layout Standard
There is a special name for the slope of that tangent line.
It is called the derivative of f at x.
\end_layout
\begin_layout Standard
\emph on
I don't expect any math professors will ever read this, so lets just leave
it at that.
\end_layout
\begin_layout Standard
There are various notations people use for a derivative.
\end_layout
\begin_layout Standard
One classic notation is:
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\frac{df(x)}{dx}\]
\end_inset
\end_layout
\begin_layout Standard
or, if it is already known that
\begin_inset Formula $y=f(x)$
\end_inset
then:
\end_layout
\begin_layout Standard
\begin_inset Formula \[
\frac{dy}{dx}\]
\end_inset
\end_layout
\begin_layout Standard
or sometimes people don't want to refer to y, so they say
\begin_inset Quotes eld
\end_inset
f prime of x
\begin_inset Quotes erd
\end_inset
, as in:
\end_layout
\begin_layout Standard
\begin_inset Formula \[
f'(x)\]
\end_inset
\end_layout
\begin_layout Standard
or if you are an
\begin_inset Quotes eld
\end_inset
operator
\begin_inset Quotes erd
\end_inset
minded person, think of D as the derivative operator:
\end_layout
\begin_layout Standard
\begin_inset Formula \[
Df(x)\]
\end_inset
\end_layout
\begin_layout Standard
In case you are a math professor, why don't you take an afternoon and write
a simple thing about continuity and limits and email me a reference :).
\end_layout
\begin_layout Section
Derivative uses:
\end_layout
\begin_layout Subsection
Describe a relationship:
\end_layout
\begin_layout Standard
Look back at Figure
\begin_inset LatexCommand ref
reference "curves"
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.
For each one, say something like
\begin_inset Quotes eld
\end_inset
the effect of x on y is small when x is small, but bigger when x is in the
middle ranges, and smaller when x is large.
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Subsection
Find local maxima and minima
\end_layout
\begin_layout Standard
The single most important use of derivatives is to check to see if f(x)
is a local maximum or minimum.
\end_layout
\begin_layout Standard
See why?
\end_layout
\begin_layout Standard
Hint: What is the derivative at a maximum or minimum point?
\end_layout
\begin_layout Standard
Gotta be 0.
Right?
\end_layout
\begin_layout Standard
So, if you read in any kind of statistics or economics or physics book,
they spend a lot of time/effort translating something into a function that
depends on or more variables, and then they find the place where they find
the
\begin_inset Quotes eld
\end_inset
critical points
\begin_inset Quotes erd
\end_inset
, the points for which the slope is 0.
\end_layout
\begin_layout Subsection
Just a little bit more: second derivatives.
\end_layout
\begin_layout Standard
The second derivative is the change in the slope if you start at x and go
an
\begin_inset Quotes eld
\end_inset
itty bitty
\begin_inset Quotes erd
\end_inset
amount to the right.
\end_layout
\begin_layout Standard
If the slope is
\series bold
getting bigger
\series default
, that means the impact of x is
\begin_inset Quotes eld
\end_inset
accelerating
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
If the slope is
\series bold
getting
\series default
smaller, that means the impact of x is
\begin_inset Quotes eld
\end_inset
decelerating
\begin_inset Quotes erd
\end_inset
or has
\begin_inset Quotes eld
\end_inset
diminishing impact
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
Sometimes notations for the change in the slope are:
\begin_inset Formula \[
f''(x)=D^{2}f(x)=\frac{d^{2}y}{dx^{2}}=\frac{d^{2}f(x)}{dx^{2}}\]
\end_inset
\end_layout
\begin_layout Standard
For some intangible reason, I like the notation: f''(x).
\end_layout
\begin_layout Standard
If the slope is
\begin_inset Quotes eld
\end_inset
getting smaller,
\begin_inset Quotes erd
\end_inset
then
\end_layout
\begin_layout Standard
Think for a minute about each of these claims.
\end_layout
\begin_layout Enumerate
If
\begin_inset Formula $f'(x)=0$
\end_inset
and
\begin_inset Formula $f''(x)<0$
\end_inset
, then x is a local maximum.
\end_layout
\begin_layout Enumerate
If
\begin_inset Formula $f'(x)=0$
\end_inset
and
\begin_inset Formula $f''(x)>0$
\end_inset
, then x is a local minimum.
\end_layout
\begin_layout Section
What to do if you want to learn more
\end_layout
\begin_layout Standard
If you get any elementary book about calculus, you will find a couple of
chapters devoted to the development of the ideas described here.
Two particularly important
\begin_inset Quotes eld
\end_inset
foundation
\begin_inset Quotes erd
\end_inset
ideas are
\emph on
limit
\emph default
and
\emph on
continuous function.
\end_layout
\begin_layout Standard
There are formulas and procedures for calculating the derivative of a function.
Some are stupendously easy!
\end_layout
\begin_layout Section
What to do if you don't want to learn more.
\end_layout
\begin_layout Standard
Don't worry.
All you have to know is that
\end_layout
\begin_layout Enumerate
derivative means
\begin_inset Quotes eld
\end_inset
slope
\begin_inset Quotes erd
\end_inset
of a function.
Slope means
\begin_inset Quotes eld
\end_inset
effect
\begin_inset Quotes erd
\end_inset
and many of our social theories require us to describe the effect of x
on y.
\end_layout
\begin_layout Enumerate
if the derivative is 0, we are at a critical point that might be a maximum
or a minimum.
\end_layout
\begin_layout Standard
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Some Curves
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Slope of a straight line
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Slope is a Local Concept
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Make Dx smaller and smaller
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What if Dx gets really really small?
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\end_body
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