Looking for a Calculus Textbook I want to start signal processing and I need a book that satisfies my mathematical requirements: I am in the third grade of high school and I don't know any useful thing about limit, differential, ...
Please help me.
 A: As I said, for a rigorous and theoretical approach to calculus, Apostol's Calculus Vols. $1$ and $2$ are very good. Depending on your background, for multivariable calculus, Spivak's Calculus on manifolds is also good. Spivak's Calculus (which does single variable calculus) is also one of my favorites.
I think I first learned calculus from Richard Courant's Introduction to Calculus and Analysis. I think Courant's and Robbin's What is Mathematics? also has good intuitive explanations of differentiation and integration.
For a book more intuitive, and perhaps something that a third grader would have background for, try Silvanus Thompson's and Martin Gardner's Calculus Made Easy.
Of course, a book that worked for me might not work for you. I would suggest that you go to a library and browse through a number of different calculus books (there are a lot of them out there), till you find the one that appeals to you the most. If you really are in the third grade, then I would assume there is no real hurry for you to master calculus, and if there is, then the books above are a good place to start.
A: Differential and Integral Calculus, Vol. I [Paperback]
Piskunov (Author) 
Try this cover to cover and if you finish this you will know more one variable calculus than you will need. 
A: I think one's first exposure to calculus-no matter how gifted or ambitious the student is-should be a physically and geometrically motivated approach that illustrates most of important applications of calculus. Sadly,many people think that means a "pencil-pushing" or "cookbook" approach where things are done sloppily and with no careful explanation of underlying theory. That's simply not true. You can certainly do calculus non-rigorously while still doing it carefully enough to give students the broad picture of the underlying theory.
The best example of this kind of book,to me, is Gilbert Strang's Calculus. Strang's emphasis is clearly on applications and it has more applications then just about any other calculus text-including many kinds of differential equations in physics(mechanics),chemistry( first and second order kinetics),biology (modeling heart rythum) and economics and a basic introduction to probability.But Strang doesn't avoid a proof when it's called for and the book has many pictures to soften the blows of these careful proofs. This would be my first choice for a high school student just starting out with calculus. 
A: Based on what we know about YOU as a student, I recommend not to try one of the "rigor" textbooks. (So "no" on Spivak, Apastol, Hardy, Courant) What we know about you:


*

*Want to use calculus for an application (signal processing)

*Want to satisfy a math requirement

*Upfront about weak pre-calc background (no knowledge of limits, etc.)


My advice to you:
(1) Make sure your pre-calc background is strong (trig and "college algebra").  Also, many pre-calc courses have an easy intro to calculus (think "antiderivative" rather than "integral").  A good text is the Schaum's Outline for First Year College Mathematics by Frank Ayres.  (1958 edition preferred.)
(2) Get a "standard" calculus textbook.  Not one of the "rigor" books that appeal to math grad students and professors.  Not a modern reform textbook.  Granville (or Granville, Smith and Longley) is the first book of this form and was the defacto standard from about 1900 to 1960.  Another good one is Thomas or Thomas Finney (get one from 1982 or earlier, that was the last one Thomas actually helped on, after that it became a brand.)  There are many other comparable (e.g. Swokowski, Stewart), but Granville is actually free on the net.  Thomas/(Finney) is one that I used personally and that many professors of EE and signals prefer for their students training.
Note that the texts (Thomas Finney, Schaum's, Granville), at least the editions I know, HAVE ALL THE ANSWERS in the back.  When you are self studying, you work the problems and check your work.  To use a drill book effectively, you need the answers to check yourself.  
I recommend buying old edition used hardcopies off of Amazon.  This can be quite reasonable.  Avoid reprints (bad quality) or Kindle (even worse quality).
P.s.  If you learn calculus well (and if you work every problem in Granville or T-F, you will learn a TON), you can always go back later and look at the rigor textbooks or do a course in analysis later (really LOW priority for a signals engineer though!)
