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Chain Rule. The chain rule is a formula for the derivative of the composition of two functions in terms of their derivatives. Continuous Function. Critical Point. A critical point is a point in the graph of a function where the derivative is either zero or undefined. Definite Integral. A derivative is the infinitesimal rate of change in a function with respect to one of its parameters. A discontinuity is a point at which a function jumps suddenly in value, blows up, or is undefined.
The opposite of continuity.
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Extreme Value Theorem. The extreme value theorem states that a continuous function on a closed interval has both a maximum and minimum value. First Derivative Test. The first derivative test is a method for determining the maximum and minimum values of a function using its first derivative. Fundamental Theorems of Calculus. The fundamental theorems of calculus are deep results in analysis that express definite integrals of continuous functions in terms of antiderivatives.
Implicit Differentiation. Implicit differentiation is the procedure of differentiating an implicit equation one which has not been explicitly solved for one of the variables with respect to the desired variable, treating other variables as unspecified functions of it. Indefinite Integral. Much later, nonstandard analysis came along, which did put infinitesimals on rigorous foundations, but by then it was not necessary since epsilon-delta definitions were available.
Nonstandard calculus is calculus except based on nonstandard analysis instead of analysis. There are still people who think that nonstandard analysis should replace those definitions, however. I will put an informal introduction to how it works at the end of this post. Would teaching students calculus using nonstandard calculus make it easier to learn? Answers should also take into account future calculus learning, not just learning in the course itself.
I would prefer to focus on "normal" students. I think for bright students, teaching a little bit of both would be beneficial, since I think comparing the approaches teaches the idea that there can be radically different ways to study math and arrive at the same results. However, for most students, I think mainly focusing one or the other would be better.
Here are some of my observations. The most important is probably points 4 and 6, since they are about how standard and nonstandard calculus are different. The other points are about how they are the same. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition.
This has also been published in print by Dover. Kathleen Sullivan.
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The American Mathematical Monthly Vol. My understanding of this question is that it proposes the idea of a "monolingual" freshman calc course in which students mostly learn the language of NSA, and limits are largely or completely neglected. This makes it different from this question by Mikhail Katz, which asks whether it's a good idea for students to be "bilingual.
I have some experience teaching some NSA-based material to first-semester calculus students at a community college. Here is the book I wrote for that purpose. My approach is "bilingual. However, scientists and engineers still use Leibniz notation and manipulate infinitesimals using algebra. They've been doing it for years, and they never stopped doing it just because there was a short gap between the invention of the limit and the invention of NSA. These students will see such manipulations in their physics courses, and they will see and be expected to use them in their careers.
So we should give them some systematic idea of what techniques are appropriate when performing these manipulations. If this was , we would teach them a body of techniques that we knew from experience gave correct results, e. Today we have more secure knowledge that these procedures can be put on a sound logical footing, but that actually has little effect in reality on what body of techniques we use. When you get right down to it, using NSA doesn't really turn out to produce any incredible simplification of freshman calculus.
In the case of the chain rule, you have the issue that the derivative is not the quotient of infinitesimals but the standard part of that quotient. These are complicated to prove and many books don't prove them all , and the complication isn't really reduced very much by using NSA. Keep in mind also that at this level, our students never really get a full-blown introduction to the real number system.
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No commercial textbook I've ever seen systematically introduces and applies anything beyond the first-order properties of the reals. They may state the completeness property, but they never use it to prove things like the intermediate value theorem. This is material that belongs in an upper-division analysis class. Since the hyperreals have the same first-order properties as the reals, there is actually very little that we can meaningfully say about the hyperreals to students at this level.
I do explicitly name the hyperreals and describe how their properties differ from those of the reals this is mainly in section 2. However, I don't think it's a good idea to go into the kind of depth that Keisler does. Speaking as a former student, though an engineering one. It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system as well.
And I don't remember ever needing the rigorous definition of a limit, standard or nonstandard. The most I needed even in complex calculus was an ability to manipulate limits, and an awareness that derivatives and antiderivatives were defined as limits and could be found as limits if necessary. It seems to me that an introductory calculus course isn't going to be an introduction to proving that calculus works, any more than an introductory algebra course is going to start with a rigorous definition of the real numbers—it's going to be an introduction to the concepts of differentiation and integration and how to use them.
Obviously, proving rigorously that calculus works does require rigorous definitions, and at that point something new has to be introduced. But if they're introduced to hyperreal numbers instead, they'll have lingering doubts about whether arguments using them are actually valid. It means another layer of understanding is needed, namely the background theory of the new number system.
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Apart from anything else, this will help them to understand other textbooks in the future, which must likely won't use hyperreals. It's not really that relevant since the bulk of a normal calculus course e. The bulk of the course is about learning derivatives, antiderivatives, methods of integration, classic applied problems, a bit if polar coordinates, bit of series, and small section on ODEs. You are swinging at the wrong opponent with this obsession on the definition of a derivative but not surprising given the theory bent of many math majors.
If you want to make things easier, cut partial fractions, integration by parts, etc. Do nurses and business students really need that? Of course this does mean tracking because science and engineering do. I found your question hard to parse given the notation upside down A. This is ironic given you are planning to make things easier for knuckleheads worse than I. Not showing it here Can't you discuss pedagogy and coverage without such a segue into exploring the math theory like that? Sign up to join this community. The best answers are voted up and rise to the top.
Home Questions Tags Users Unanswered. Would teaching nonstandard calculus in an introduction calculus course make it easier to learn? Ask Question. Asked 7 months ago. Active 7 months ago. Viewed times. Of course, the mathematical foundations of nonstandard analysis would be much too complicated to cover in such a course. The same is true of standard analysis though, so this is not a concern.
Nonstandard calculus is compatible with calculus. By that I mean that anything you can prove in analysis can be proven in nonstandard analysis, and anything you can disproof is analysis can be proven in standard analysis, so there will never be a contradictory result. In terms of provable statements, the only difference is that nonstandard analysis proves some things about infinite and infinitesimal numbers that standard analysis is not concerned with since it does not use such number , and even those can be translated into equivalent statements about real numbers in standard analysis.
The main difference is how they got about proving them.
Definitions in nonstandard calculus involve much fewer quantifiers than those in standard analysis. This however is balanced by the fact that it is much more difficult to do arithmetic with hyperreal numbers than the real numbers. You can not even build a hyperreal number calculator.