Can Infinitesimals Make Calculus Easier to Understand?

[WizardWill]

I've been playing around with a free PDF Calculus book lately. But, I have no way to check the logic used to get to a particular answer. I've been trying to find the standard part for:

(1/ɛ)((1/sqrt(4+ɛ))-(1/2))

I've tried every way I could think of to algebraically manipulate this in order to avoid dividing by zero (taking the standard part of 1/ɛ). Just by looking at the problem, I would think it would be undefined...but the odd answers tell me otherwise.

Thanks :)

 

[micromass]

Hi WizardWill!

First, put everything in one fraction:

$$\frac{2-\sqrt{4+\varepsilon}}{\varepsilon 2 \sqrt{4+\varepsilon}}$$

Now, multiplicate numerator and denominator by

$$2+\sqrt{4+\varepsilon}$$

 

[HallsofIvy]

I would be inclined to use a power series for $1/\sqrt{4+ x}$:
$$\frac{1}{\sqrt{4+ x}}= \frac{1}{2}- \frac{1}{16}x+ \frac{3}{128}x^2+ \cdot\cdot\cdot$$
so that
$$\frac{1}{\sqrt{4+ \epsilon}}- \frac{1}{2}= -\frac{1}{16}\epsilon+ \frac{3}{128}\epsilon^2+ \cdot\cdot\cdot$$

 

[WizardWill]

Thanks Micromass and HallsofIvy for your replies :)

I began the problem by distributing the 1/ε term. So, what I had looked a bit messy. I'm surprised I didn't notice to use the conjugate of the numerator...so use to looking in the denominator. Problem solved :)

Thanks again.

 

[SteveL27]

As someone who took calc years ago, I was curious to know if the approach based on nonstandard analysis has become so commonplace these days that it can be freely used.

I can see that the OP is attempting to determine the derivative of f(x) = 1/sqrt(x) at x = 4 directly from the definition, and is doing the equivalent of what would classically be written as

$$f'(4) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \rightarrow 0} \frac{\frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}}}{h} = \lim_{h \rightarrow 0} \frac{\frac{1}{\sqrt{4 + h}} - \frac{1}{2}}{h}$$

Of course the nonstandard approach is mathematically rigorous and has been around for 30 or 40 years now. However I wasn't aware that it had achieved so much "market penetration" that it doesn't need to be remarked on.

I've heard that the nonstandard approach has the drawback that it doesn't work as conveniently in the multi-variable case. I don't know if that's true or not. In any event, someone who learns calculus this way will have trouble reading other calculus texts, or going on to Calc II or real analysis. I don't think they've banished limits in higher math yet!

Or perhaps the OP already knows calculus and is just learning the nonstandard approach. I wasn't able to tell from the question.

I wonder if someone can put this into the context of modern teaching for me.

 

[bcrowell]

In any event, someone who learns calculus this way will have trouble reading other calculus texts, or going on to Calc II or real analysis.

IMO it's the other way around. The Leibniz notation was devised to represent infinitesimals. When infinitesimals were banished ca. 1900, it made it harder for calc students to understand Leibniz notation. A student today who learns calc using infinitesimals will have an easier time understanding the literature, which never stopped using the Leibniz notation.

I wonder if someone can put this into the context of modern teaching for me.

There was a book by Keisler published back in the 70's, which did freshman calc using infinitesimals. You can find it online for free now. (It may be the book the OP referred to.) AFAIK it did not become a popular way to teach calculus. (One way you can tell that it probably wasn't popular is that the book went out of print and the copyright reverted to Keisler, making him free to put it online.) Education is very conservative, and the textbook market even more so.

One way to check the result is to go to this online calculator I wrote http://www.lightandmatter.com/calc/inf/ and enter the OP's expression as (1/d)*((1/sqrt(4+d))-(1/2)) , where d stands for an infinitesimal. The leading term is -1/16, which agrees with Halls's result.

 

[SteveL27]

IMO it's the other way around. The Leibniz notation was devised to represent infinitesimals. When infinitesimals were banished ca. 1900, it made it harder for calc students to understand Leibniz notation. A student today who learns calc using infinitesimals will have an easier time understanding the literature, which never stopped using the Leibniz notation.

Yes you're right, dy/dx doesn't make a lick of sense the way it's taught to calculus students. Perhaps the Leibniz notation should be banned. I should mention that I'm a Newtonian

There was a book by Keisler published back in the 70's, which did freshman calc using infinitesimals. You can find it online for free now. (It may be the book the OP referred to.) AFAIK it did not become a popular way to teach calculus. (One way you can tell that it probably wasn't popular is that the book went out of print and the copyright reverted to Keisler, making him free to put it online.) Education is very conservative, and the textbook market even more so.

Yes, that's why I asked the question. I've heard of Robinson's rigorous theory of nonstandard analysis, and I'd also heard about Keisler's book. I wasn't sure if the nonstandard approach had become more widespread since then.

One way to check the result is to go to this online calculator I wrote http://www.lightandmatter.com/calc/inf/ and enter the OP's expression as (1/d)*((1/sqrt(4+d))-(1/2)) , where d stands for an infinitesimal. The leading term is -1/16, which agrees with Halls's result.

You've written a calculus text that incorporates rigorous infinitesimals! I'd have to yield to your judgment about how to present this material to students. I was just wondering what happens when they get to real analysis? Do infinitesimals make it easer or harder to learn limits?

I also hadn't seen the Levi-Civita field before, that's interesting too. How does that relate to Robinson's system?

Thanks for your interesting response.