Math 1571H Honors Calculus Fall Semester 2001

Assignment 10
- Due Thursday 11/15/2001

With revision for the second mid-term exam and the exam itself, I realize that I have specified too much homework to be done in Assignment 9. The problems from section 5.4 will not be due on 11/8/01, and instead appear on this assignment.

Read: Simmons sections 5.4, 5.5, 6.3, 6.4, 6.5.

This looks like a lot of sections, but much of it is general historical stuff and philosophy on which no questions are set. In fact 6.2 is rather interesting to read also, although I have not listed it above. It contains historical background, which is very useful for getting an understanding of what is so significant about the development of differential and integral calculus, why it took so long before anyone thought of it, and what people found difficult before they knew about the calculus. There are no problems set on section 6.2 and no theory is developed there that we will use or you will be tested on, so reading 6.2 is optional (but I recommend it!).

Two things strike me particularly as I read 6.2. One is the assertion on page 190 that the fundamental theorem of calculus is the most important single fact in the whole of mathematics. Can you think of other important facts, and how would you rate them? What about commutativity of multiplication (ab = ba)? Is that more important? Or doesn't that count? Do you think it is wise to make statements such as that in a text book of this kind? The other thing which strikes me is how long it took between when people worked out the basic properties of areas and things like the number ¼ (when was that?) and when the calculus was developed (when was that?). Why do you think it took so long? Is this surprising to you, as it is to me? Is it surprising that a person like Newton could emerge who appears to be head and shoulders above most other people in history?

Coming back to the earlier sections, I forgot to mention in the last assignment to do with section 5.3 that I am not happy with some of the ways Simmons presents things. So on page 171 I would not say that both the formulas (4) are correct. The second formula with the constant c is the one we should use. Also differentials are used at the top of page 174 to introduce integration by substitution, but it seems to me that what Simmons writes there does not have a sound basis. You should really read about 2/3 of the way down page 176 to get a proper justification for integration by substitution. My advice is: be wary. And if something does not seem clear in what you read, try to skip over it.

Section 5.5 is has some quite hard things in it, but the important thing to be taken in from this section is the set of equations which describe motion under constant acceleration on page 183, and how they are derived. Example 3 on page 185 is quite exciting and interesting, but you will not be expected to reproduce the mathematics there from memory.

Exercises:
5.4: 4, 5, 6*, 9, 10*, 11, 12, 13, 20*
5.5: 4, 5, 6*, 7, 8*, 9, 10, 12, 14*
6.3: 2, 2c*, 2f*, 3*, 5
6.5: 1*