61
You may as well ask: Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first? Why teach high-school Algebra without starting with the basics of groups, rings and fields? Why, for that matter, teach children to read first, instead of starting with the fundamentals of grammar and linguistics?
...
51
Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of calculus predated formal definitions of the real numbers (Dedekind cuts, Cauchy sequences) and a formal definition of continuity, individuals must first make ...
40
It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. 376) calls the Fundamental Assumption of School Mathematics.
In terms of the nitty-gritty of classroom instruction, real numbers are handled in K–12 by what ...
32
I suppose my preference would be
4. The differential equations road.
That is, define $x(t) = \cos t$ and $y(t) = \sin t$ to be the solutions to the following initial value problem:
$$
\frac{dx}{dt} = -y,\qquad \frac{dy}{dt} = x,\qquad x(0) = 1,\qquad y(0)=0.
$$
This assumes that you've proven the existence and uniqueness theorem ...
27
What is striking about the Lebesgue integral is how relatively nicely limits play together with this integral, things like the Dominated convergenece theorem are great. I think one can appreciate this result (especially when contrasted with the more clumsy Riemann Integral analogues) right away. One could mention this right at the start, before everything ...
24
As far as I know, there are four basic possibilities for defining the real numbers:
The real numbers can be defined using axioms.
The real numbers can be defined using decimal expansions.
The real numbers can be defined using Dedekind cuts
The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers.
Approach #1 is ...
22
Consider the following observations:
People shorten things with increasing frequency of usage. For example, the most frequent words are short. This is a way of making communication more effective, e.g. see Huffman coding. For example this list has its first 5-letter word at 39th place (first three are for comparison):
$$\begin{array}{c|c|c|c}
\textbf{place} ...
21
Wikipedia always has some great animations, often by User:LucasVB or User:Cmglee.
A few Fourier transform–related ones:
Approximating a square wave with a Fourier transform
(more of these for other functions here: sawtooth, triangle, partial cubic)
Fourier transform time and frequency domains
Continuous Fourier transform of rect and sinc functions
20
Opinion.
There never was a generation of high school students in the US who could jump right into Rudin.
There would be (and still is) a small portion of the top high school graduates who could. And maybe a larger portion of the graduates from a few elite high schools. But that's it.
Baby Rudin would be used (if at all) for advanced undergraduates or ...
20
I would say something like this: "Often in complicated systems one needs to study multiple quantities, each of which varies at rates that depend on the other quantities and on how fast they are varying. For example the rate at which a drug metabolizes may depend not only on how much of the drug is in the body but also on how much blood sugar is in the body,...
19
Isolines and isosurfaces
Isolines and isosurfaces (i.e., lines and areas of equal whatever) correspond to the graphs of implicit functions and are relevant in many sciences, e.g., isopotentials (physics), isobars and isotherms (metereology).
Probably the best-known example of this kind are topographical contour lines (lines of equal altitude, see image ...
18
It was mentioned in other answers that having a more sloppy notation is better to not complicate the communication. This is okay for people who had really understood the concepts of mathematics.
I have the feeling that understanding what a function is one of the those things students don't really understand. Common mistakes (and in my opinion all of them ...
18
Because your core assumption is bad and most students are not aspiring mathematicians
Your opening statement - "Let's assume our students are actual aspiring mathematicians" - is almost certainly a very bad assumption to make about students at the age where calculus is taught. In fact, very few of the students who are taught calculus will go to study ...
18
Although there is a tradition in the U.S. of nominal mention of complex numbers in the high school curriculum, in my observation it is invariably superficial, and complex numbers are not mentioned subsequently. In particular, there is no mention of Euler's identity expressing sine and cosine in terms of complex exponentials, and, therefore, no mention of the ...
17
To distinguish not too strictly between $f$ and $f(x)$ allows to operate more easily with functions built up from other functions.
For example, one might want to say things like:
Let us consider the function $\sin (3 x^2)$ on the intervall $[0,1]$.
This can be considered as sloppy in a formal sense, but I think it is still clear and in some contexts ...
16
In addition to other good answers, I do think that what "calculus" usually refers to is what people did with calculus prior to about 1830, as opposed to the foundations of it. Namely, they solved a lot of physically meaningful/intuitive problems, as well as geometrical problems, problems in mechanics, celestial mechanics, fluid flow, ... and beginning of ...
16
But as far as real numbers were concerned, they were taught very early
without any clear definition was even given.
Were they?
I would say that usually no teacher really speaks of real numbers to students until the first calculus lesson. In almost all pre-calculus exercises, students only meet rational numbers, in the form of fractions or decimal ...
15
First, we have the two "obvious" motivations.
Integrating more functions (e.g. the indicator function for the rationals)
Making the space of integrable functions complete
Second, Stein and Shakarchi give the following motivations in the introduction to their book Real Analysis (volume 3 of the Princeton Lectures in Analysis series).
Fourier Series (...
15
In the United States, complex numbers are a standard part of the typical high school Algebra 2 course. Students (normally in grades 10 or 11, corresponding approximately to ages 15-17) learn to add, multiply, and divide complex numbers; to solve quadratic equations with no real roots; and to find all $n$ roots of an $n$th degree polynomial (usually, ...
14
It is interesting to see how computer algebra systems deal with this kind of thing. In Maple, for example, you can do the following:
Define f := x^3
Enter diff(f,x) to get the derivative
Enter subs(x=3,f) to evaluate.
Alternatively:
Define g := (x) -> x^3
Enter D(g) to get the derivative
Enter g(3) to evaluate
You can convert between the two idioms ...
14
Some suggestions
Michael Spivak's Calculus
This is really an honors level calculus text, but it might be useful to have around. It's pretty expensive, however.
Stephen Abbott's Understanding Analysis
Back in January 2003, in sci.math, I wrote: I think it's the best written introductory analysis book that's appeared in the past couple of decades.
...
13
Though I also like to use the notation "the function $x^3$" when appropriate, sometimes this causes problems. For example when I teach inverse functions, it is difficult to grasp that the $x$ in $x^3$ is not the same $x$ as in $\sqrt[3]{x}$.
I always have problems with first year students who are used to the notation that $x$ is in the domain and $y$ is in ...
12
I've approached this a couple different ways in the past, and I haven't decided what I think is best. But I'd like to point out some arguments in favor of not including any construction of the real numbers (like Dedekind cuts or limits of equivalence classes of Cauchy sequences), and simply stating the axioms which are needed for the rest of the course. (...
12
This aspect is discussed in great detail in §7.2 of A Course in Calculus and Mathematical Analysis by Ghorpade and Limaye. The discussion must be accessible to students undergoing their first rigorous course in analysis.
The discussion is well-motivated: the authors point out in §7.1 that one can integrate all functions of the form $x^r$ for all rational $...
12
Not every property is preserved by limits.
Here is a more basic situation in which the same reasoning is used:
For each natural number $n$, there are only finitely many natural numbers in the interval $[0,n]$. However, letting $n \to \infty$, there are infinitely many natural numbers in the interval $[0,\infty)$. There is clearly no contradiction here, ...
12
in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points.
In beginning Lebesgue measure: the easiest example of an uncountable set of measure zero
In general topology: sets homeomorphic to the Cantor set are useful in proofs
Fractal geometry: many fractals are homeomorphic to the ...
11
At least as a counter-point to other reasonable viewpoints: I am not such an enthusiast about measure theory as "foundation" for integrals, because it seems to me that measure theory over-emphasizes just one smallish part of what we really want. Namely, starting with (ambiguously) nice functions $f_n$, perhaps continuous and compactly supported on a ...
11
By multivariate analysis, I'm assuming that you're talking about a course that covers the following topics in a rigorous way:
Differentiation of functions between vector spaces
The inverse and implicit function theorems, and related topics (e.g. rank of the derivative, immersions and submersions).
Critical points and regular points of multivariable ...
11
This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves teaching a lot of new material*. Real analysis, is mostly (at least in the first quarter/semester), material they have already seen before in calculus - minus ...
11
At German universities, one of the first lectures in mathematics is "Analysis 1" which is a kind of "rigorous calculus" and there one always proceeds more or less like this:
We start with an axiomatic approach to the real numbers. In short: The real numbers are a complete, ordered, Archimedean field. In practice we first introduce the axioms of a field, ...
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