[Blindmath] Basic mathematical concept question

Sat Mar 24 18:25:37 UTC 2012

I saw your post before going to class this morning and hoped that some math
scholar in the group would have answered your questions by the time I
returned from class.

Since that appears not to have happened, I will try to answer some of your
questions from an engineering viewpoint. (Engineers are similar to
physicists and mathematicians but are often more practical in their
approach to things.)

Some things in life are true simply because they are true and other things
in life are true because someone in authority declared them to be true. It
is important to be able to make a distinction between those two cases.

For example, I was born as a male instead of a female simply because that
is true. My parents had no say in the choice between male and female.

However, my name is Richard because someone who had the authority to do so,
(probably my mother), declared that to be true. She could have declared my
name to be Tom, Harry, Tam, or any of a million other names but she elected
to name me Richard.

Many different ratios can be computed in mathematics and some of them are
so important that they deserve to be named.

There are many different ways to connect three line segments to create a
triangle but regardless of how you create the triangle, the sum of the
interior angles is always 180 degrees. That is true simply because it is
true. No human has any control over that property of triangles. However,
someone in authority decided that the closed shape defined by connecting
three line segments end to end in that matter would be known as a triangle.
(The same goes for hexagon, pentagon, etc.)

At some point in history, someone who had the authority do so so declared
that triangles having one interior angle with a value of 90 degrees will
forever be known as right triangles. (Actually, that name and the other
names that I discuss herein probably evolved and probably weren't the
decision of a single individual.)

The fact that a triangle with one interior angle having a value of 90
degrees can exist is simply true, independent of the likes or desires of
any human. The fact that we call it a right triangle is true because humans
caused it to be true.

Given a right triangle and one of the interior angles, the ratio of the
opposite side to the hypotenuse always behaves in a very predictable way as
a function of the value of the angle. This behavior is so important that
someone in authority declared at some point that in history that the value
of the ratio will forevermore be known as the sine of the angle. They could
have named it the Richard of the angle or the Tam of the angle. Instead,
they elected to name it the sine of the angle. The behavior of the ratio is
true simply because it is true. The fact that we refer to that behavior as
the sine of the angle is true because someone in authority declared it to
be true.

If you walk up a ramp or climb a staircase, you can measure the horizontal
distance x, and the vertical distance y, that you have traveled. Then you
can compute the ratio of those two values. While the length of the ramp or
the staircase can vary, the ratio of y to x won't change so long as the
angle of the ramp or staircase relative to the horizontal doesn't change.
Once again, that ratio is very predictable and is so important that someone
with authority to do so declared that the value of that ratio will
forevermore be known as the slope. That person could have named it the
Richard or the Tam but elected instead to name it the slope. The ratio is
true simply because it is true. The name is true because someone in
authority declared it to be true.

That is the answer to one of your questions. You asked three other
questions. I have no answer to the question regarding an accessible web
site, but I will attempt to answer the questions regarding the derivative
of a function as well as the question regarding continuous and
discontinuous functions.

If a function is continuous, the slope of the function can be measured at
every point on the function. If the function is discontinuous, I believe
there are points on the function at which the slope cannot be measured.
(See more on this later.) For example, I seem to recall that if you plot
the path of a point marked on a wheel as the wheel rolls on a flat surface,
the function described by that path will have discontinuities each time the
point reaches maximum and minimum height (but my memory may be failing me
on that).

You can draw an infinite number of straight line segment that intersect the
plot of a function (a curve) at any point on the curve. That is simply
true. The curve, if it is continuous, has a slope at that point. Only one
of those lines can be parallel to the slope of that curve at that exact
point. That is also simply true.

Someone with the authority to do so declared that such a straight line
segment would be known as the tangent to the curve at that point. The fact
that we call that line segment the tangent to the curve is true because
someone in authority declared it to be true.

Differential calculus is all about finding and dealing with the slope of
curves at points along the curves. That is true because someone in
authority declared that the study of that set of related mathematical
concepts would be known as differential calculus.

Given a point on a continuous curve, you can mark two other points on the
curve that are equally distant horizontally from the horizontal value of
the original point.

Then you can draw a straight line segment through those two points and you
can measure the slope of the line. That slope may or may not be a good
estimate of the actual slope of the curve at the original point. The you
can move the two outer points closer to the original point, draw a new
straight line segment through the two points and recalculate the slope. The
slope of this new line is a refined estimate of the slope of the curve at
the original point.

Theoretically, you can continue this process of moving the two outer point
closer to the original point, drawing a new line segment through the two
outer points, and using the new slope value to refine your estimate of the
slope of the curve at the original point until the distance between the
original point and the two outer points goes to almost zero.

Theoretically, the refined estimate that you get when the distance between
the original point and the two outer points is the smallest possible
non-zero value is the best possible estimate of the slope of the curve at
the original point. Someone who had the authority to do so declared that
the value of the slope at that point would forevermore be known as the
derivative of the function at that point.

The slope of the curve at that point is true simply because it is true. It
is called the derivative because someone with authority to do so declared
it to be true.

Also, since someone also declared that a straight line segment that touches
the curve at that point and is parallel to the slope of the curve at that
point is the tangent line at that point, it follows that the derivative of
the curve at that point is equal to the slope of the tangent line at that
point of its graph.

Now let's return to the topic of continuous and discontinuous. I'm on
really thin ice here so I will rely on information from the web at
http://www.vias.org/calculus/03_continuous_functions_04_14.html This author
states:

"The functions we shall ordinarily encounter in this book will be defined
and have a derivative at all but perhaps a finite number of points of an
interval. The graph of such a function will be a smooth curve where the
derivative exists. At points where the curve has a sharp corner (like 0 in
|x|) or a vertical tangent line (like 0 in x^(1/3)), the function is
continuous but not differentiable. At points where the function is
undefined or there is a jump, or the value approaches infinity or
oscillates wildly, the function is discontinuous."

This description may or may not agree with your calculus book.

As a practical manner, if you can measure the slope of the curve at every
possible point within an interval, it is continuous within that interval.
According to the author at the website cited above, the fact that you can't
measure the slope at a given point is not sufficient to make it
discontinuous.

Hope this helps.

Dick Baldwin

On Fri, Mar 23, 2012 at 9:29 PM, Duong Tuan Nam <tuannamduong at gmail.com>wrote:

> Hello all,
> I don't clearly understand some basic problems.
> How do continuous and discontinuous function
> work.
> Why is the slope of a line is equal y over x?
> Another is that why derivative of a funtion at a point is the
> slope of the tangent line at that point of its graph?
>
> I've searched on Google but most of the Math webpage result can't be read
> with Jaws on IE.
> Can you show me some accessible webpage to read or help me explain these
> problems?
> Thank you very much,
> Nam
>
>
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-- 
Richard G. Baldwin (Dick Baldwin)
Home of Baldwin's on-line Java Tutorials
http://www.DickBaldwin.com

Professor of Computer Information Technology
Austin Community College
(512) 223-4758
mailto:Baldwin at DickBaldwin.com
http://www.austincc.edu/baldwin/