Riemann Sum & Table Problems (Type 5)

AP  Questions Type 5: Riemann Sum & Table Problems Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, theory and theorems, and position-velocity-acceleration, among others. Numbers and working with numbers is part of the Rule of Four and table problems are one way this is tested. This question often includes…

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…