Continuity

Karl Weierstrass (1815 – 1897) was the mathematician who (finally) formalized the definition of continuity. Included in that definition was the epsilon-delta definition of limit. This definition has been pulled out, so to speak, and now is usually presented on its own. So, which came first – continuity or limit? The ideas and situations that…

Intermediate Weather

I found a curious fact in a textbook today that relates to the Intermediate Value Theorem [1]. It claimed that if you draw a circle of any size on a map, there will be two diametrically opposite points on the circle at which the temperature will be the same! This might make an interesting example…

Proof

When math books present a theorem they almost always immediately present its proof. I tend to skip the proofs. I assume they are correct. I want to get on with the ideas in the text. Later I may come back and read through them. Is this a good thing to advise students to do? I…

Theorems

Theorems are statements that summarize the results that are true in mathematics. Theorems are statements that have been proved true; but the emphasis in AP Calculus is not on proof. Rather, it is on what the theorems mean and how to use them. Theorems have two parts: the “if …” clause called the hypothesis and…