Tuesday May 5, 2009 Lethbridge
5:45 am
I read chapter 2 of "Calculus Know-It-All" yesterday and found it easy going. Now to make a few notes and then to try the exercises at the end of the chapter. This will also give me an opportunity to practice my penmanship.
Calculus Know-It-All
Stan Gibilisco
Chapter 2 is called "Limits and Continuity". One of the reasons I like this book is that it quickly gets to the important ideas.
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- "As the argument (the independent variable or input) of a function approaches a particular value, the dependent variable approaches some other value called the limit. The important word here is approaches." [p. 20]
- A function is called right-hand continuous if three conditions hold:
- the right-hand limit (i.e. the limit as one appoaches the point from the right) is defined at a certain point
- the actual value of the function is defined at that point
- the above two value are the same.
- "When a function is both left-hand continuous and right-hand continuous at a point, we say the function is continuous at that point." [p. 27]
- "A real-number function in one variable is a continuous function if and only if it is continuous at every point in its domain." [p. 29]
- "All linear functions are continuous." [p. 29]
- "All single-variable, real-number quadratic functions are continuous." [p. 30]
- "All single-variable, real-number cubic functions are continuous." [p. 31]
- "All single-variable, real-number polynomial functions are continuous." [p. 31]
- "Plenty of other functions are continuous." [p. 32]
- "A real-number function in one variable is called a discontinuous function if and only if it is not continuous at one or more points in its domain." [p. 32]
I then tackled the 10 exercises at the end of the chapter.
This was a very worthwhile activity as I quickly bumped up against a couple of areas that need a strong review. The basic concepts of limit and continuity presented no difficulty but the general area of pre-calculus mathematics is a problem as I have not looked closely at mathematics in over 40 years. I need a more intensive review of logarithms, exponentials and trigonometric functions as well as the topic of converging and diverging sequences and series. |
An important chapter that covers two of the critical concepts underlying calculus: limit and continuity.
Total time this morning: about 1 hour. |
Tags: mathematics, mathematica
Here is the link to my handwritten work for today:
Books on the Go Today |
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