I was tired today and took a long nap. My psychiatric medication is part of what makes this project possible, but it also makes me sleep more than normal. I did manage to read a chapter of Steven R. Lay’s Analysis with an Introduction to Proof, another one of my college math textbooks. (I kept them all.) I’ve decided to try to read it in parallel to my review of introductory calculus, since it covers most of the same concepts, but in a more rigorous way.
Tag: calculus
Stewart’s Calculus
Today was better than yesterday, though I was still not at full capacity. I spent about an hour on algebra review, then thought for a while about the decimal representations of rational numbers. I also read the Wikipedia article on long division, which says that the debate about its place in the curriculum actually dates back to the 1980s. There are some interesting examples there of the ways long division problems are written in different countries, as well.
For anyone interested, here is the review of algebra I am using. It’s made available as a supplement by the publishers of Stewart’s calculus textbooks. There are also reviews of analytic geometry and conic sections, which I plan to work through as well, along with the review of trigonometry provided as an appendix to my textbook. (My textbook has reviews of algebra, analytic geometry, and conics, too, but they are not as complete as those offered on the website.)
I have a lot of affection for Stewart’s calculus texts. They were used in both my high school and my college courses, and represent good times to me. A person I met online once asked me to name a book that had changed my life, and Stewart’s Single Variable Calculus was my choice. Had I not enjoyed calculus so much in high school, I would not have been attracted to economics as a college major, would not have been encouraged to minor in math, would not have taken Discrete Math my first semester, and might never have had my love of math kindled such that I am still carrying a torch for it. His discovery of Ramanujan might have been “the one romantic incident” in Hardy’s life, but my encounter with math was mine, and Stewart’s calculus was one of the sign posts on the way.
Pertinent Points
Today I continued to work through the exercises in the appendix of my calculus book that provides a review of some concepts from algebra.
One series of problems involved finding the intervals on which various expressions containing variables would be less than or greater than zero. One method to do this that was outlined in the textbook was to find the points at which the expression would be equal to zero, use those to divide the number line into intervals, then test a point from each interval to determine whether the expression was positive or negative on that interval. All the time I was doing this I was thinking, “this is all well and good for expressions that correspond to continuous functions, but what about other types?” (The property of continuous functions that makes this work is the Intermediate Value Theorem, I believe.)
Sure enough, near the end of the series of problems, some expressions appeared that would be discontinuous when interpreted as functions. The answer to my question is that you need to treat each point of discontinuity as a point at which to divide the number line, the same as each point where the expression would equal zero. This was never explained or even hinted at in the text, though, which puzzles me. I don’t think of this text as leaving much for the reader to discover. Maybe there is more than I remembered, though. If so, there may be value in doing all of the exercises.
