The Definition of the Definite Integral. The definition of the definite integrals is: If f is a function continuous on the closed interval [a, b], and is a partition of that interval, and , then The left side of the definition is, of course, any Riemann sum for the function f on the interval [a,…
Another in my occasional series on Good Questions to teach from. This is the Mighty Cable Company question from the 2009 AB Calculus exam, number 3 This question presented students with a different situation than had been seen before. It is a pretty standard “in-out” question, except that what was going in and out was money.…
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…
In keeping with my idea from the last post of sneaking up on ideas, here is a way to sneak up on the other part of the FTC. Consider these three functions F1(t) = 3, F2(t) = 2t and F3(t) = 2t + 3 For each of these three functions do the following: Graph each…
The Fundamental Theorem of Calculus or FTC, as its name suggests, is a very important idea. It is not sufficient to present the formula and show students how to use it. Show them where it comes from. Here is an approach to demonstrate the FTC. I try to sneak up on the result by proposing…
From the last post, it seems pretty obvious that as the number of rectangles in a Riemann sum increases or, what amounts to the same thing, the width of the sub-intervals decreases, the Riemann sum approaches the area of the region between a graph and the x-axis. The figures below show left-Riemann sums for the…
