I'm a math team coach and homeschooling parent looking for some extra depth for the math learners in my care. Most of the young people on my math team get their main math instruction from the U. of MN Talented Youth Mathematics Program, using the Stewart Early Transcendentals textbook with an excellent supplementary materials. Even though they've got some good instruction already, I'm looking for suggestions of calculus problems that really make a learner THINK about calculus concepts and go beyond the usual school homework exercise. That's the best way to help those kids thrive in their main program and go on to higher mathematics study. What I especially need, as someone largely clueless about calculus myself, is to find exercises that have numerical or symbolic answers, or better still full solutions, so that I can check them even if I can't (as is likely) work through them myself from first principles. My students work beyond my own level of ability, and they are eager to learn more.
Here's a list of calculus books I own, from which it would be especially easy to find problems that you suggest are good, challenging problems. The list is ordered by publication date.
Burington, Richard Stevens & Torrance, Charles Chapman, Higher Mathematics: With Applications to Science and Engineering (New York: McGraw-Hill, 1939).
Randolph, John F., Analytic Geometry and Calculus (New York: Macmillan, 1949).
Finney, Ross L., Thomas, George B., Demana, Franklin & Waits, Bert K., Calculus: Graphical, Numerical, Algebraic: Single Variable Version (Reading, MA: Addison-Wesley, 1995).
Ostebee, Arnold & Zorn, Paul, Calculus from Graphical, Numerical, and Symbolic Points of View Volume 1 (Fort Worth, TX: Saunders College Publishing, preliminary ed. 1995) (and separate selected answers volume for this title).
Priestley, William McGowen, Calculus: A Liberal Art (New York: Springer-Verlag, 2nd ed. 1998).
Anton, Howard, Calculus: A New Horizon (New York: John Wiley, 6th ed. 1999).
Hughes-Hallett, Deborah, et al., Calculus (New York: Wiley, 4th ed. 2005).
I have had some other really good books out of libraries, such as Apostol, Strang, and Courant. And if you know of great problems with solutions from online sources, I'm sure that would be of interest to many of us. Please let me know about your favorite problems for the student who wants to go beyond the usual syllabus.