# Preface

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The course 18.013A as it appears here is intended as a one and a half term course in calculus for students who have studied calculus in high school. It is intended to be self contained, so that it is possible to follow it without any background in calculus, for the adventurous.
It makes use of some tools that are relatively new, such as applets, which are intended to make the subject easier to learn and more fun.
However it was not our intention to make this course merely an easy calculus course, covering all the same material as the traditional course, but easier to assimilate because of the applets and use of spreadsheets.
Modern labor saving devices in practice do not make life easier and simpler. Instead they save us time that allows us to do more in our lives and to make life more complicated and busier than ever.
In the same sense it is our hope that these new tools allow the student to learn more and learn more thoroughly with the same amount of, and even perhaps more, effort than before.
Thus, the material covered is substantially greater in depth and variety than is normally attempted in a calculus course.
In fact, we have attempted to inject new material into almost every chapter of the text as it appears on this site.
Why have we done so?
In part it is mere human frailty: to keep from going crazy while creating this material.
Also, though, it is in order to maintain interest in the subject among those who have already been exposed to it. Finally, it is to show how easy it now is to absorb and use material that not long ago was utterly inaccessible to students, rarely taught, and when taught rarely mastered.
Obviously some of the extra material contributes mainly to confuse students, some is very badly done though it could be improved, some might suggest to you even better things that might be included.
The authors would be delighted if you find a way to use this material or to learn from it. We would be extremely grateful if you would send us via email (djk@math.mit.edu) any comments you have about it, especially from those who don't like something they see.
It is still at a stage in which it can be changed.

What then is new for a calculus course here?
Chapter 0 which introduces the spreadsheet is entirely new.
The discussion of standard functions in Chapter 1 is new.
The geometric definitions of the trigonometric functions by the illustration is new in Chapter 2, and the section on various metrics (3.8) is new in Chapter 3. (It probably does more harm than good there.)
The main innovations in Chapters 4 and 5 are the applets, though introduction of the concepts of eigenvectors and eigenvalues at this stage is perhaps unusual.
Chapter 6, which involves definition of differentiation in all dimensions is novel; but I think it helps make it possible for students to see why the rules of differentiation are what they are, and why calculus is useful.
Chapter 7 on numerical differentiation is new at least to me.
There is not much new in Chapter 8 except for it applying in all dimensions.

Probably the major innovations in this course apart from the applets, are the sections on numerical analysis, the study of single and multiple variable calculus together, introduction of integration in the complex plane, and the applications to physics, which while commonplace in physics, rarely are discussed in a calculus course.

Our intent here is to cover enough about more advanced areas of mathematics to make students aware of them and to encourage their wanting to learn more about them.

This material could conceivably be studied by a student on his or her own, but this seldom works out. Students tend to get stuck on something, and, having no goad to keep them going, they try to get past it with decreasing energy, and ultimately develop mental blocks against going on. Having an organized course prevents this by forcing them to face obstacles like exams and assignments.

If you attempt this and get stuck, as is almost inevitable, you could try emailing us and we can try to unstick you.