> - Why are limits defined the way they are (with epsilons and deltas)?
> - The book will probably touch lightly upon the Mean Value Theorem -- why is this important? What's the point?
> - Why is the chain rule true? It reads dy/dx = (dy/du) (du/dx). Yay! This is just cancelling fractions, right? Any "respectable" calculus book will insist that it's not, but most students will cheerfully ignore this, still get correct answers to the homework problems, and sleep fine at night.
Which "respectable" book(s) would you recommend for those who want to dive into this details? Is Tom Apostol's Mathematical Analysis? good for learning these kind of details? (They say this book is "respectable", but I would like to hear your thoughts about it. Thanks).
I don't know anything about Apostol's Mathematical Analysis. My guess would be that it demands a fairly sophisticated background of the reader, and does an excellent job of covering calculus from an extremely rigorous point of view.
I have heard that Apostol's Calculus is an excellent choice, probably somewhat more accessible to beginners, but still offering a rigorous, highbrow perspective. I've also heard the same of Spivak. I'd probably opt for one or both of these.
> - The book will probably touch lightly upon the Mean Value Theorem -- why is this important? What's the point?
> - Why is the chain rule true? It reads dy/dx = (dy/du) (du/dx). Yay! This is just cancelling fractions, right? Any "respectable" calculus book will insist that it's not, but most students will cheerfully ignore this, still get correct answers to the homework problems, and sleep fine at night.
Which "respectable" book(s) would you recommend for those who want to dive into this details? Is Tom Apostol's Mathematical Analysis? good for learning these kind of details? (They say this book is "respectable", but I would like to hear your thoughts about it. Thanks).