Holes in the graph calculus

Hello, I would like to know an example of a hole in a graph in the real world. I have a lot of students ask me what the hole represents. I can give them the mathematical idea but lack a good example as to where you run into it in nature.
thanks
bill

 

Graph in what sense? Graph of a function or graph in the sense of
vertices connected by edges? And, whichever it is, what do you mean by
"hole"?

 

What's the graph calculus?
Solving calculus problems with graphs?

The ozone hole? That's a real world hole.

Emptiness of course. What else could it possibly represent?

What is the definition of the hole?

----

 

I'm guessing the OP means a "hole" such as in the graph of a function
with a removable singularity like y = sin(x) / x and such.

--Lynn

http://math.asu.edu/~kurtz

 

I fear for his students,

 

You guys are brutal.
To find the lim(x->-3) (x^2 + x -6)/x+3
there is a common factor of x+3 which is a removable hole in the graph at x = -3. The limit is -5. I would like to explain with a real world example of the gap in the graph at (-3,-5).
thanks again
bill

 

You should really write this as (x^2+x-6)/(x+3). The function you
wrote is continuous at -3 (since it is x^2+x-6, divided by x, and 3
added to that result).

How's the issue of "instant velocity" strike you? Seems not only
pretty "real world", but I suspect it may come up again pretty soon in
a calculus class...

It makes perfect sense to talk about average velocities around a
certain time; if p(t) is position at time t, then you can compute the
average velocity at any interval of the form [2,2+h] or {2-h,2] (h>0)
by the simple formula ( p(2+h) - p(2) )/h. This gives you a function
of x that has a hole at h=0, the idea of "what was your velocity at
the instant t=2". i.e., you take a photograph at t=2; how fast was the
car moving at that instant? (Pick your favorite differentiable
function for it, if you like). Or if you want it to look like the one
you have above, then do it as (p(x) - p(2))/(x-2) with the limit as x--

Physics has a lot of functions that compute quantities that stop
making sense at the "extreme" (when the two points in time are the
same point in time; when velocity approaches the speed of light; etc).
Those are excellent candidates.

 

All we asked was for you to clarify what you meant
instead of demanding of us to be mind readers.

If you think clarity is too demanding, then you shouldn't be teaching.

That means (x^2 + x - 6)/x + 3.

There's no common factor.
Do you mean (x^2 + x - 6)/(x + 3) = x - 2 ?

Regarding
.. . (x^2 + x - 6)/(x + 3),
the gap is a point at which it's not defined, ie doesn't exist.

Now explain to your students, if you can,
the technical different between
.. . (x^2 + x - 6)/(x + 3)
and
.. . x - 2
which are almost everywhere the same.

Do you teach your students about the
domain and codomain of a function?

 

At least as long as x is not equal to 3.

And about division by 0?

 

As long as x /= -3.

Clearly we hope, that has already been taught by the brutalized teacher.

Yes, that's needed to discern the implicit domain of
.. . (x^2 + x - 6)/(x + 3).

Happily at x = -3,
.. . x^2 + x - 6 /= 0

so discussion about 0/0 can be avoided.

I've seen middle school students taught 1/0 = oo.

That's disputable because then oo = 1/0 = 1/(-0) = -1/0 = -oo.
It's also disputable because lim(x->0) 1/x doesn't exist as
lim(x->0+) = oo and lim(x->0-) = -oo. Thus twice now, 1/0 = +-oo.

 

If you don't mind Mr. William Eliot, how do you exactly teach a student lim (x-->0) 1/x
or for that matter, the limit of a fraction where the numerator tends to a non-zero constant and the numerator tends to 0?

 

Any function s.t. f(x)=(x+a)(x+b)/(x+a), with a "hole" at x=a, if that's what you mean.

 

Just make something absurd up.

"Have fun with it"

 

"If you don't mind Mr. William Eliot, how do you exactly teach a student lim (x-->0) 1/x
or for that matter, the limit of a fraction where the numerator tends to a non-zero constant and the denumerator tends to 0?"

Since Dr. Eliot has not responded, I will. I suspect he would do exactly what you will find in any text book: show that if M is any negative number, there exist delta< 0 such that if delta< x< 0, then 1/x< M and then show that if N is any positive number, there exist delta> 0 such that 0< x< delta, then 1/x> N.

More generally, if the limit is A/f(x) where A is non-zero and f(x) goes to 0 as x goes to a, show that, given any negative M, there exist x close to a, on one side such that A/f(x) is less than M and that, given any positive N, there exist x close to a, on the other side such that A/f(x) is larger than N.

 

I'm not sure what your question is, but the function that you give
is undefined at x = -a, not at x = a. f(a) = a+b.

 

Yeah. people like me are sloppy.