I took calculus courses and have read some books about it. Most of them is similar: There are the definitions and some applications but not enough depth. For example, it is very common to find the problem of the box in calculus books:
Having a sheet of paper of area $a^2$, how can we fold it into a box in a way that maximizes the volume?
This is a really interesting problem and I guess that the books could have a little bit more about it, in a way that almost suggests the study of optimization problems. The problem for me is that calculus by itself doesn't seems stimulating enough. But I have found several titles that make calculus more interesting. Take a look at this result from Chen's: Excursions in Classical Analysis.
I think this is a very interesting and elegant result! One can have all these means as a function based on simple integrals and as a bonus, you also gain a very easy way to deduce inequalities between all them! I have also found
Moll's books are filled with the study of polynomials, Riemann's $\zeta$ Function, Legendre polynomials, Chebyshev polynomials, Hermite polynomials, $\Gamma$ function, Logarithmic Integrals, etc.
Kazarinoff even makes an appeal in his book:
- Steele: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities;
- Gardiner's: Infinite processes, also sold by Dover as Understanding Infinite;
Gardiner's book is IMO an excellent choice to explain why we need a more rigorous analysis. Very soon in the book, he already presents some functions in which a naive usage of ideas in calculus can take one to hazardous consequences.
Shahriari's book is a real gem. Just take a look at its contents. It has a section on dynamical systems!
What I call "interesting" here is the suggestion of using calculus to discover neat problems both in analysis and in others areas of mathematics or a decent and organic explanation of the whys. I felt that calculus was no stimulating because the derivative is basically "a tool for finding lines tangent to functions" and the integral is basically "a tool for measuring the are under a curve", obviously: These are worthy, but where can we go from there? The question is, do you know more books that complement this list?
Behind the scenes: The original title of the question was: "Where to find books with interesting activities using calculus?" but MSE suggested a change for a better title. I changed to this and the warning disappeared: If there is no warning, then this is a better title! (Also, I watched this today.)