**Math 1571H Honors Calculus Fall Semester 2001
**

Assignment 8

4.5: 2, 4*, 5, 8, 17*

4.6: 1, 2a*, 5, 9, 10*, 13

We will be doing questions from section 5.2, but I recommend that if you do read section 5.2 then try not to get hung up on things like the meaning of the term 'differential' on page 164, because it is not precisely explained, and indeed it is problematic to give it a precise meaning. We don't have to work with the notion of differentials as they do in this book. I will show you in class how to do the things they do in section 5.2 without using the differential notation. I take the view that writing things down as differentials is a useful informal way to go about doing some calculations, but in our finished written work we should avoid writing them down. It seems to me that this view is in accordance with history, as described on pages 168 and 169 (and those are two good pages to read).

In section 5.3 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.

Another warning about section 5.3. 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.

5.2: 16, 22*, 24, 27

5.3: 8, 14, 26, 32*, 38, 40, 42*, 44, 48, 50, 52, 54, 56, 58*, 62, 63, 66b*, 69

5.4: 4, 5, 6*, 9, 10*, 11, 12, 13, 20*