I saw this meme floating around:

So:

1. Assuming that are real numbers, find all for which each relation is true, OR show why it is impossible.

2. Where appropriate, repeat exercise 1 but for, say, a field or ring.

I saw this meme floating around:

So:

1. Assuming that are real numbers, find all for which each relation is true, OR show why it is impossible.

2. Where appropriate, repeat exercise 1 but for, say, a field or ring.

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Reminder: this series is NOT for the student who is attempting to learn calculus for the first time.

**Derivatives** This is dealing with differentiable functions and no, I will NOT be talking about maps between tangent bundles. Yes, my differential geometry and differential topology courses were on the order of 30 years ago or so. đź™‚

In calculus 1, we typically use the following definitions for the derivative of a function at a point: . This is opposed to the *derivative function* which can be thought of as the one dimensional gradient of .

The first definition is easier to use for some calculations, say, calculating the derivative of at a point. (hint, if you need one: use then it is easier to factor). It can be used for proving a special case of the chain rule as well (the case there we are evaluating at and for at most a finite number of points near .)

When introducing this concept, the binomial expansion theorem is very handy to use for many of the calculations.

Now there is another definition for the derivative that is helpful when proving the chain rule (sans restrictions).

Note that as we have . We can now view as a function of which goes to zero as does.

That is, where and is the best linear approximation for at .

We’ll talk about the chain rule a bit later.

But what about the derivative and examples?

It is common to develop intuition for the derivative as applied to nice, smooth..ok, analytic functions. And this might be a fine thing to do for beginning calculus students. But future math majors might benefit from being exposed to just a bit more so I’ll give some examples.

Now, of course, being differentiable at a point means being continuous there (the limit of the numerator of the difference quotient must go to zero for the derivative to exist). And we all know examples of a function being continuous at a point but not being differentiable there. Examples: are all continuous at zero but none are differentiable there; these give examples of a corner, vertical tangent and a cusp respectively.

But for many of the piecewise defined examples, say, for and for the derivative fails to exist because the respective derivative functions fail to be continuous at ; the same is true of the other stated examples.

And of course, we can show that has continuous derivatives at the origin but not derivatives.

**But what about a function with a discontinuous derivative?** Try for and zero at . It is easy to see that the derivative exists for all but the first derivative fails to be continuous at the origin.

The derivative is at and for which is not continuous at the origin.

**Ok, what about a function that is differentiable at a single point only?** There are different constructions, but if for rational, for irrational is both continuous and, yes, differentiable at (nice application of the Squeeze Theorem on the difference quotient).

Yes, there are everywhere continuous, nowhere differentiable functions.

Though I’ve been busy both learning and creating new mathematics (that is, teaching “new to me” courses and writing papers to submit for publication) I have not written much here. I’ve decided to write up some notes on, yes, calculus. These notes are NOT for the average student who is learning for the first time but rather for the busy TA or new instructor; it is just to get the juices flowing. Someday I might decide to write these notes up more formally and create something like “an instructor’s guide to calculus.”

I’ll pick topics that we often talk about and expand on them, giving suggested examples and proofs.

**First example: Continuity**. Of course, we say * is continuous at * if which means that the limit exists and is equal to the function evaluated at the point. In analysis notation: for all there exists such that whenever .

Of course, I see this as “for every open containing , is an open set. But never mind that for now.

So, what are some decent examples other than the usual “jump discontinuities” and “asymptotes” examples?

**A function that is continuous at exactly one point:** try for rational and for irrational.

**A function that oscillates infinitely often near a point but is continuous**: for and zero at .

**A bounded unction with a non-jump discontinuity but is continuous for all **: for and zero at .

**An unbounded function without an asymptote but is continuous for all ** for and zero at .

**A nowhere continuous function:** for rational, and for irrational.

If you want an advanced example which blows the “a function is continuous if its graph can be drawn without lifting the pencil off of the paper, try the Cantor function. (this function is continuous on , has derivative equal to zero almost everywhere, and yet increases from 0 to 1.

It seems as if the time faculty is expected to spend on administrative tasks is growing exponentially. In our case: we’ve had some administrative upheaval with the new people coming in to “clean things up”, thereby launching new task forces, creating more committees, etc. And this is a time suck; often more senior faculty more or less go through the motions when it comes to course preparation for the elementary courses (say: the calculus sequence, or elementary differential equations).

And so:

1. Does this harm the course quality and if so..

2. Is there any effect on the students?

I should first explain why I am thinking about this; I’ll give some specific examples from my department.

1. Some time ago, a faculty member gave a seminar in which he gave an “elementary” proof of why is non-elementary. Ok, this proof took 40-50 minutes to get through. But at the end, the professor giving the seminar exclaimed: “isn’t this lovely?” at which, another senior member (one who didn’t have a Ph. D. had had been around since the 1960’s) asked “why are you happy that yet again, we haven’t had success?” The fact that a proof that could not be expressed in terms of the usual functions by the standard field operations had been given; the whole point had eluded him. And remember, this person was in our calculus teaching line up.

2. Another time, in a less formal setting, I had mentioned that I had given a brief mention to my class that one could compute and improper integral (over the real line) of an unbounded function that that a function could have a Laplace transform. A junior faculty member who had just taught differential equations tried to inform me that only functions of exponential order could have a Laplace transform; I replied that, while many texts restricted Laplace transforms to such functions, that was not mathematically necessary (though it is a reasonable restriction for an applied first course). (briefly: imagine a function whose graph consisted of a spike of height at integer points over an interval of width and was zero elsewhere.

3. In still another case, I was talking about errors in answer keys and how, when I taught courses that I wasn’t qualified to teach (e. g. actuarial science course), it was tough for me to confidently determine when the answer key was wrong. A senior, still active research faculty member said that he found errors in an answer key..that in some cases..the interval of absolute convergence for some power series was given as a closed interval.

I was a bit taken aback; I gently reminded him that was such a series.

I know what he was confused by; there is a theorem that says that if converges (either conditionally or absolutely) for some then the series converges absolutely for all where The proof isn’t hard; note that convergence of means eventually, for some positive then compare the “tail end” of the series: use and then and compare to a convergent geometric series. Mind you, he was teaching series at the time..and yes, is a senior, research active faculty member with years and years of experience; he mentored me so many years ago.

4. Also…one time, a sharp young faculty member asked around “are there any real functions that are differentiable exactly at one point? (yes: try if is rational, if is irrational.

5. And yes, one time I had forgotten that a function could be differentiable but not be (try: at

What is the point of all of this? Even smart, active mathematicians forget stuff if they haven’t reviewed it in a while…even elementary stuff. We need time to review our courses! But…does this actually affect the students? I am almost sure that at non-elite universities such as ours, the answer is “probably not in any way that can be measured.”

Think about it. Imagine the following statements in a differential equations course:

1. “Laplace transforms exist only for functions of exponential order (false)”.

2. “We will restrict our study of Laplace transforms to functions of exponential order.”

3. “We will restrict our study of Laplace transforms to functions of exponential order but this is not mathematically necessary.”

Would students really recognize the difference between these three statements?

Yes, making these statements, with confidence, requires quite a bit of difference in preparation time. And our deans and administrators might not see any value to allowing for such preparation time as it doesn’t show up in measures of performance.

We’ve arrived at logarithms in our calculus class, and, of course, I explained that only holds for . That is all well and good.

And yes, I explained that expressions like only makes sense when

But then I went ahead and did a problem of the following type: given by using logarithmic differentiation,

And you KNOW exactly what I did. Right?

Note that is differentiable for all and, well, the derivative *should* be continuous for all but..is it? Well, up to inessential singularities, it is. You see: the second factor is not defined for , etc.

Well, let’s multiply it out and obtain:

So, there is that. We might induce inessential singularities.

And there is the following: in the process of finding the derivative to begin with we did:

and that expansion is valid only for

because we need and .

But the derivative formula works anyway. So what is the formula?

It is: if where is differentiable, then and verifying this is an easy exercise in induction.

But the logarithmic differentiation is really just a motivating idea that works for positive functions.

To make this complete: we’ll now tackle where it is essential that .

Rewrite

Then

This formula is a bit of a universal one. Let’s examine two special cases.

Suppose some constant. Then and the formula becomes which is just the usual constant power rule with the chain rule.

Now suppose for some positive constant. Then and the formula becomes which is the usual exponential function differentiation formula combined with the chain rule.

Yes, I know that the proper way to do this is to prove the derivative formula for and then use, say, the implicit function theorem or perhaps the chain rule.

But an early question asked students to use the difference quotient method to find the derivative function (ok, the “gradient”) for And yes, one way to do this is to simplify the difference quotient by factoring from both the numerator and the denominator of the difference quotient. But this is rather ad-hoc, I think.

So what would one do with, say, where are positive integers?

One way: look at the difference quotient: and do the following (before attempting a limit, of course): let at which our difference quotient becomes:

Now it is clear that is a common factor..but HOW it factors is essential.

So let’s look at a little bit of elementary algebra: one can show:

(hint: very much like the geometric sum proof).

Using this:

Now as

we have (for the purposes of substitution) so we end up with:

(the number of terms is easy to count).

Now back substitute to obtain which, of course, is the familiar formula.

Note that this algebraic identity could have been used for the old case to begin with.

Recently, an Oregon university touted graduating someone with Down’s syndrome:

Walking across the stage at graduation was more than just a personal accomplishment for Cody Sullivan as he became Oregon’s first student with Down syndrome to complete four years of college.

Sullivan, 22, received his certificate of achievement at the Concordia University graduation ceremony last month, declaring that while assignments and curriculum were modified for his learning abilities, Sullivan completed all the relevant coursework to make him an official college graduate.

It is every interestingly worded: “certificate of achievement” and “assignments and curriculum were modified for his learning abilities”.

This represents a different point of view than I have.

When a teach a course, getting a certain grade in a course requires that the person getting grade to master certain concepts and skills at a certain level. Those requirements are NOT modified for someone’s learning ability. And getting a degree in a certain subject means (or should mean) that one has established a certain competency in that said subject.

But, well, I wonder if we are moving toward a “meeting a certain competency level isn’t relevant” anymore and just giving “you were here and did stuff” certificates.

There was a time when I thought “aptitude matters” but, well?

I will be talking about teaching limits in a first year calculus class.

The textbook our department is using does the typical:

It APPEARS to be making the claim that the limit of the given function is 4 as approaches 2 because, well, 4 is between and . But, there are an uncountable number of numbers between those two values; one really needs that the function in question “preserves integers” in order to give a good reason to “guess” that the limit is indeed 4.

I think that the important thing here is that the range is being squeezed as the domain gets squeezed, and, in my honest opinion, THAT is the point of limits: the limit exists when one can tighten the range tolerance by sufficiently tightening the domain tolerance.

But, in general, it is impossible to guess the limit without extra information about the function (e. g. maps integers to integers, etc.)