Make a list of all definitions in the section (a few words each is fine). Summarize any lengthy definitions introduced in this section in your own words.
Didn't notice any!
Make a list of all theorems (lemmas, corollaries) in the section (a few words each is fine). Summarize each one introduced in this section in your own words.
For the following, we assume that f and g are differentiable functions.
A constant function is flat, horizontal; its slope is 0:
The slope of the line y=x is 1:
where n is an integer. This rule is very useful, since polynomials include so many terms of this form.
The product rule may seem a little counter-intuitive; it certainly isn't as simple as a product of derivatives (however much we'd hope so):
I just recently learned a rhyme to remember this formula:
``Lo dee hi minus hi dee lo, over the denominator square we go!''
It's sometimes represented this way:
where is a real number. An important special case is the following:
where n is an integer.
Make a note of any especially useful properties, tricks, hints, or other materials.
``The theorems of this section show that any polynomial is differentiable on and any rational function is differentiable on its domain.''
Examples:
Summarize the section in two or three sentences.
This section is replete with formulas for some of the most important functions we will be working with (these formulas must be committed to memory!). It began with a few special cases, then began extending to general cases: several of these follow our intuition, e.g. the sum rule (the derivative of a sum is the sum of the derivatives); others are not so intuitive (the product and quotient rules are not obvious). There were many examples which demonstrated how one uses these formulas.
Proofs are given for these special formulas, and, for the most part, are not too complicated. One proceeds directly from the definition of the derivative function as the limit
and one hopes that things just fall out!
Problems:
Problems pp. 156-158, #4, 13, 17, 24, 32, 44, 60, 63, 80; Board: 3, 18, 16, 37