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I'm teaching for a small group of future undergraduate science students. I'm from Brazil, so here, we don't see any calculus in school, Just definition of function and some properties of most known functions of a real variable, everything very simplified.

I) I want to reintroduce this functions in a way that it will be more useful for a future calculus course at university. (Pre Calculus or first chapter of calculus books).

II) I want to teach the concepts of limits, and derivatives with some formal approach.(Because they will study sciences). I want to give some interesting applications of functions, limits and derivatives(Because they are not going to a pure math course).

What I need It's a list (1,2 or 3) of books that fits in:

1)Mathematically strong calculus books, with some good pre calculus on it: (Write your list 1, 2 or 3 books).

2)Book of calculus that has some good applications in physics/economics: (Write another list 1,2 or 3 books ).

OBS: I don't know english language calculus books, so I don't even can compare this books.

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  • $\begingroup$ Stewart is the standard text in Canadian universities. It has a variety of applications as well as precalculus review. $\endgroup$ – TheDayBeforeDawn Nov 22 '16 at 22:24
  • $\begingroup$ Do you need a book in English or Portuguese? $\endgroup$ – user137731 Nov 22 '16 at 22:25
  • $\begingroup$ in English. Some books, I heard are very good, and don't have translation. Doesn't matter, I will make the notes of the course, so I need good reference. I prefer in English. $\endgroup$ – Gustavo Viana Nov 22 '16 at 22:28
  • $\begingroup$ Courant's book has some good physics applications exercises. $\endgroup$ – user137731 Nov 22 '16 at 22:30
  • $\begingroup$ Good, thanks. .. $\endgroup$ – Gustavo Viana Nov 22 '16 at 22:31
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I would recommend Paul's Online Math Notes. These notes cover a wide variety of preliminaries and I believe they will prove to be useful to your students even in their undergraduate education.

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I recommend "Calculus: Early Transcendentals" (Stewart).

I'm currently studying by it, and while I haven't gotten very far, I do like the teaching style: abstract when appropriate, yet filled with concrete examples.

Here's the ToC:

  1. Functions and models
  2. Limits and derivatives
  3. Differentiation rules
  4. Applications of differentiation
  5. Integrals
  6. Applications of integration
  7. Techniques of integration
  8. Further applications of integration
  9. Differential equations
  10. Parametric equations and polar coords
  11. Infinite sequences and series
  12. Vectors and the geometry of space
  13. Vector functions
  14. Partial derivatives
  15. Multiple integrals
  16. Vector calculus
  17. Second-order differential equations

In case you want a more detailed ToC, "peek" into the book in the link above.

Regarding the introduction (chapters 1 and 2): Stewart does a great job not expecting much prior knowledge from the student, aside from algebra. Limits are well-covered in chapter 2.

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