Calculus is the second-greatest technological invention by mankind (second only to the printing press). When Isaac Newton and Gottfried Leibniz independently invented the calculus, building on the work of earlier mathematicians like Descartes and Barrow, they showed themselves to be responsible for the modern technological age. In particular, the Fundamental Theorem of the Calculus has extremely far-reaching consequences. It is easily the most important theorem in all of mathematics.
I love calculus; it's my favorite mathematics. Aside from simply having the very best name of any branch of math, it's so powerful! There's so much you can do with it. And it is very beautiful.
So I wanted to put forth a few calculus challenges that were a little off the beaten track. Enjoy!
1. Assume a function f is thrice differentiable. If, at a point a we have f''(a) = 0 and it is not the case that f'''(a) = 0, what can you say about f at a? Supposing the second derivative is zero, and the third derivative is also zero, what changes? This is known as the Third Derivative Test.
2. Are endpoints critical points? This is a neat question, asked by one of my students at Tech. The answer is not in any calculus book that I am aware of, though the question is clearly in the realm of calculus.
3. Without looking it up, do you know how to integrate sec(x)?
4. Integrate sqrt(1 - x^2) by using trig substitution, and check your answer using implicit differentiation. (Ok, this one is a bit more mainstream; I just like trig substitution).
5. Examine the following famous ODE, known as the time-independent Schroedinger equation: f''(x)=2m ( V(x) - E ) f(x) / hbar^2, where hbar is Planck's constant, and m is mass. V(x) is simply a function of x called the potential, and E is the energy, a constant. Assume that f is continuous and differentiable everywhere on the real line, and is also nonzero. Show that for bound states (fancy way of saying that f has a finite integral over the entire real line), it must be that E is greater than the minimum value of V(x) for all x.
6. Also in the above problem with the Schroedinger equation, show that if V(x) is an even function, then f can be taken to be either even or odd.
Numbers 5. and 6. are due to Griffiths' Introduction to Quantum Mechanics, 1995, p. 24.