My father was as self-taught man. He taught himself Calculus from "Calculus made Easy". Because I found the book on his bookshelves, I also taught myself calculus from the same book, and got 'A's for my exams. I cannot find it quickly in my library (some pupil may have purloined it!). However I downloaded a copy of it to my harddrive today, courtesy of Gutenberg.

http://www.gutenberg.org/files/33283/33283-pdf.pdf

Highly recommended!!!

See also http://en.wikipedia.org/wiki/Calculus_Made_Easy

Start with the hands on (mechanics), and lead on to the Maths. Few students are theoreticians. Delta before differentiation. Small square counting before Integration. Discontinous function plotting before continuity. Etc.

Teaching Calculus should always be done with the highest quality possible from a theoretical and practical standpoint, depending on the type of student to whom it is addressed.

For students of Mathematics and Physics, Calculus should be given with the maximum theoretical rigor, deepening the concepts, principles, properties and theorems associated with the concepts of limit, derivative, differentiation, and integration, just to mention the most important issues within Calculus and consolidating theoretical knowledge with practical examples in which students can clearly see the application of these concepts in the various branches of knowledge. These students must be able to understand all the theoretical elements associated with the concepts, principles, properties and theorems associated with Calculus and to explain them to other professionals.

For engineering students, the teaching of Calculus should be based on the concepts, principles, properties and theorems associated with this subject, but without going into the details of the theoretical aspects associated with these elements and emphasizing the practical application thereof in various branches of engineering by performing specific examples where you would use them.

For other students from other university courses, including economists and the professorial careers, Calculus should be taught using the same concepts, but on a more practical manner and without going too deeply into the theoretical aspects associated with the same elements. Students of medicine should also understand the general elements of Calculus but in a very practical way, so that they can understand the general principles of operation of sophisticated equipment that are now available in hospitals for diagnosis and treatment of certain type of diseases.

Of course, textbooks for teaching Calculus must be different and take into account the type of student to whom it is addressed.

Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry.

The most important way to foster conceptual understanding is through the problems. But some students start by trying their homework problems and read the text only if they get stuck on an exercise. So a better plan is to read and understand a section of the text before attempting the exercises.

There is a really amazing book on the subject, with actual projects for Calculus students, you may find it interesting: Student Research Projects in Calculus

by Edward D. Gaughan. I read it looking for new and different ideas for projects in Statistics. http://www.amazon.com/Student-Research-Projects-Calculus-Spectrum/dp/0883855038

My father was as self-taught man. He taught himself Calculus from "Calculus made Easy". Because I found the book on his bookshelves, I also taught myself calculus from the same book, and got 'A's for my exams. I cannot find it quickly in my library (some pupil may have purloined it!). However I downloaded a copy of it to my harddrive today, courtesy of Gutenberg.

http://www.gutenberg.org/files/33283/33283-pdf.pdf

Highly recommended!!!

See also http://en.wikipedia.org/wiki/Calculus_Made_Easy

The limit definition currently used is entirely unrelated to the conceptions of algebra, trig, early analytic geometry/coordinate algebra, etc., that students have. Such subjects concern mainly the rote application of rules to compute answers given some problem set. A single, intro course in formal logic would do more to prepare students for analysis than most of pre-calculus education, as analysis (calculus being the first step) is founded upon the application of expressing, analyzing, deriving from, and manipulating formal (symbolic) expressions; set theory; and mappings (functions).

Of these, only the last is really taught prior to calculus, but taught as just another element of "math"; no more or less important than division or exponents and largely unrelated to trigonometry or polynomials (or any number of other functions) unless these are expressed with some notation like "f(x)=" or "y=". Pre-calculus courses do not, in general, contain anything resembling the kind of first-order logic that is the basis for limits, and 40+ years of research in linguistics, psychology, and cognitive science have shown over and over again that conditionals do not come easily to most people, let alone reasoning about embedded conditionals and extra quantifiers defining some abstract proximity of two equally abstract inequalities.

This is why, even if one cannot use the entire textbook due to a student body without the necessary skills and/or experience, every first course in calculus should begin with the first chapter from Spivak's Calculus (4th Ed.). Most calculus texts start with something like functions or a review, while Spivak begins with addition. Yet it is not a review. It is the first exposure that is not centered around How to apply some rule WHEN some seemingly arbitrary condition holds true, but WHY these "rules" exist and what they MEAN.

More "hands on", intuitive approaches are nice, but unfortunately one reason limits alone are so hard for many students to grasp is because math courses before calculus have eschewed rigor for rules and traded derivations for definitions. The continuation of this means students can pass 2 or 3 semesters of calculus and not understand most of it because the bulk of the "applications" consisted of e.g., how a farmer can set up a fence with minimal length, how a carpenter might ensure a wooden box has a certain approximate volume that no one on earth could care about, how to know when some upward trending projectile starts falling, or any other useless "real world" applications that nobody would ever us.

The adoption of the current limit definition was not the intuitive notion motivating the great minds behind analysis (Newton, Leibniz, Cauchy, Euler, etc.). It was simply the first sufficiently rigorous definition, and even after Weierstrauss textbooks continued to us of infinitesimals. This intuitive approach came at a price- rigor.

However, rigorous formulations of infinitesimals (non-standard analysis) has existed for the past ~50 years.

Back to Classics: Teaching Limits Through Infinitesimals

(http://arxiv.org/pdf/1108.4657.pdf)

The Marriage of Intuition and Rigor: Teaching Conceptual Calculus with Infinitesimals

(https://scholarworks.alaska.edu/bitstream/handle/11122/1285/Pittenger.pdf?sequence=3)

Elementary Calculus: An Infinitesimal Approach

(http://www-mdp.eng.cam.ac.uk/web/library/enginfo/textbooks_dvd_only/keisler/keislercalc1.pdf)

Foundations of Nonstandard Non-linear Analysis

(http://ria.ua.pt/bitstream/10773/2937/1/2008001418.pdf)

The limit definition, though, has nothing on current methods of teaching integration:

"we teach the Riemann integral...when the necessity of integrating unbounded functions arise, we teach the improper Riemann integral. When the student is more advanced we sheepishly let them know that the integration theory that they have learned is just a moldy...concept [replaced] a full century ago.

We do not apologize for the fact that we have misled them...we likely will not even mention the fact that the improper Riemann integral and the Lebesgue integral are quite distinct...we just drop the subject entirely."

from The Calculus Integral, a free textbook among others like it (also free) found here:

http://www.classicalrealanalysis.info/com/FREE-PDF-DOWNLOADS.php

It is essential to STOP teaching students to be calculators. The current curriculum is based on the 2+ centuries in which physics essentially WAS science and calculus was its instrument. Currently, the ubiquity of statistical software packages that rely heavily on concepts covered in passing in calculus courses (but in detail for those who take linear algebra courses) mean the bulk of calculus is entirely wasted. We teach the trapezoid rule in first semester calculus (and pre-college calculus) as if virtually any students are going to go on to use numerical approximations, but we don't teach logic, statistics, or the better half of the foundations for multivariate statistics (linear algebra). For the students that continue on, they will have spent hours manipulating algebraic expressions to evaluate limits that don't in general increase their understanding of limits and are useless because even their calculators can compute these.

We can retain rigor in calculus and vastly simplify mathematics by ridding our reliance on computationally difficult methods (e.g., hours spent learning rote mechanisms to rationalize equations so that if a student takes calculus they can evaluate a limit just as mindlessly). The formulae will be forgotten as will the rules, but the logic, inference, deduction, and derivation of solutions, proofs, and rules remain. Most of what is learned need not be as it will never be used and does not teach the concepts, and their are alternatives to the standard limit definition as well as the mostly pointless integral method.

Calculus made easy! I bought that. Alas, the divide was to great between how calculus is taught and how the book presents it (which, I found out later, was an infinitesimal approach, albeit before infinitesimals were rigorously defined). It provides a better conceptual understanding, but in a world in which one relies on statistical software packages, graphing calculators, and modelling environments, concepts do not avail as much as they once did. Humans are not capable of most of the computations in the sciences, and not knowing the formalism is an obstacle to success. Yet that book does have modern (or more modern) analogous texts (such as Keisler's or perhaps the best teach-yourself calculus concepts text I've ever encountered: The Calculus Direct: An Intuitively Obvious Approach to a Basic Knowledge of the Calculus for the Casual Observer)

At the weblabs project, We used programming and collaborative inquiry as a pedagogical framework for teaching children about sequences, convergence, limits and infinity.

Our learners where secondary school students, but I think some of the ideas may be relevant for older students as well:

https://www.researchgate.net/publication/226841054_Young_Students_Exploring_Cardinality_by_Constructing_Infinite_Processes

https://www.researchgate.net/publication/32231649_Constructing_structure_in_number_sequences

https://www.researchgate.net/publication/32228150_Programming_as_mathematical_narrative

Article Young Students Exploring Cardinality by Constructing Infinit...

Article Constructing structure in number sequences

Article Programming as mathematical narrative

It is better to take help of diagrams while teaching elementary calculus. For example, while explaining Rolle's Theorem, we can take help of a diagram. For y = f(x), for x in [a, b], if f(a) = f(b), then if the curve is increasing at x = a, then it must come down again by the time x = b, if it is decreasing at x = b, then it must go up by the time x = b, or it may remain a constant throughout the interval. In other words, dy/dx changes from either positive to negative or from positive to negative within the interval, and accordingly it must be zero somewhere between a and b. If the function is a constant, the differentiated value is zero anyway. Therefore dy/dx will have to be zero somewhere within the interval at least once. This can be explained geometrically first. That way, the students would be able to understand the theorem easily.

I support the position of Andrew Messing! I would like to add that my experience tells me that using infinitesimals, not even rigorous, to conceptually introduce basic notions of calculus, then one my proceed to introduce limits and other classical notions. This methods I found it has the best results.

with the support of CAS I have taught introductory calculus beginning with area and numerical integration, introducing the idea of limits in an effort to get a more accurate result for the area under a curve. See my article:

Herbert, S. (2013). Challenging traditional sequence of teaching introductory calculus. Computers in the Schools. 30 (1-2), 172-190.

Hi Costas,  H. J. Keisler's "Calculus with Infinitesimals" is downloadable for free. 

Thank you Paul, I had Keisler's books many years ago!

Calculus with Infinitesimals is a very interesting book.

Thanks!

The topic of limits is one of the most important topics in calculus. Unfortunately most people who take calculus often confuse the fundamental concept of a limit

The concept of limit is  as x gets closer to “a” what is f(x) getting closer to, NOT what f(x) is at a.

This is mainly because most of the texts on Calculus start as under

lim ( 2x2+4) = 4

X-> 0

we substitute the value of x =0 and calculate the value of the function.

I think it is essential that we should be able to give a good example to illustrate the concept of limit namely

When we say that t X-> 0 , it means that the value of X tends to be 0 but it will never be 0.

To have a conceptual view of this let us take 95 ml of water and add to it 5 grams of sugar( or salt). Let us assume for our convenience 5 gram of sugar(salt) = 5 ml. Then now we have a sugar(salt ) solution of 100 ml where the sugar(salt ) is 5/100 = 0.05

Now let us add 900 ml of water to this solution. With this we get I liter of Sugar(salt) solution. The amount of sugar(salt) now is 0.005.

Let us now add 9 liters, 90 liters,900 liters etc of water, we will see that the sugar(salt) present in the solution goes further down to 0.0005 to 0.00005, 0.000005 etc.,

Now let us imagine that all the water available in the whole universe is added to this solution and when we determine the amount of sugar or salt present we can see that it is 0.0000...................................5.

You can see that the value approached to zero but it will never be zero.

I think the above example will help students to understand the concept of Limit in a better way.

( From my forthcoming book on  Operations Research( Quantitative Techniques).

 Would be grateful for you feedback on this  to my mail [email protected]

You might find something useful in the book from a project I was part of that looked at developing ways of teaching calculus with technology. The book is free to download at the link below.

http://eprints.soton.ac.uk/372030/