Extreme Average

A recent post on the AP Calculus bulletin board observed that the maximum value of the average value of a function on an interval occurred at the point where the graph of the average value and the function intersect. I am not sure if this concept is important in and of itself, but it does…

2019 CED Unit 8: Applications of Integration

This unit seems to fit more logically after the opening unit on integration (Unit 6). The Course and Exam Description (CED) places Unit 7 Differential Equations before Unit 8 probably because the previous unit ended with techniques of antidifferentiation. My guess is that many teachers will teach Unit 8: Applications of Integration immediately after Unit…

Applications of integrals, part 1: Areas & Average Value

Usually the first application of integration is to find the area bounded by a function and the x-axis, followed by finding the area between two functions. We begin with these problems First some calculator hints Graphing Integrals using a graphing calculator to graph functions defined by integrals Graphing Calculator Use  and Definition Integrals – Exam…

Good Question 8 – or not?

Today’s question is not a good question. It’s a bad question. But sometimes a bad question can become a good one. This one leads first to a discussion of units, then to all sorts of calculus. Here’s the question a teacher sent me this week taken from his textbook: The normal monthly rainfall at the…

Half-full and Half-empty

A thought experiment: Suppose you had a container with a rectangular base whose length runs from x = a to x = b, with a width of one-inch. The container has four vertical rectangular sides. You put a piece of, say, plastic into the container which fits snugly along the bottom and four sides. The…

Most Triangles Are Obtuse!

What is the probability that a triangle picked at random will be acute? An average value problem. The thing here is to define what you mean by picking a triangle “at random.” You could open a Geometry book and take the first triangle you come to, but are the triangles in a Geometry book really…