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.
provided the limit exists. [What could possibly go wrong? Well, the function might not be defined at a - i.e., f(a) is undefined; or the function might not be continuous, so that the numerator doesn't tend to 0 (whereas the denominator always does).]
The second definition is obtained from the first via a change of variables: if we replace the independent variable x by the independent variable h, linking them by a shift - x=h+a - then we get the second form of the definition.
(the slope of the tangent line to the graph of the position function at P(a,s(a))). We can think of this as our first (and one of our most important) examples of the use of this limit.
average rate of change - - the slope of the secant line connecting two points on the graph, and ). This is often used as an approximation to the instantaneous rate of change.
instantaneous rate of change - .
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.
I didn't find any theorems in this section.
Make a note of any especially useful properties, tricks, hints, or other materials.
Summarize the section in two or three sentences.
Here we catch a glimpse of the importance of limits: they are used to define instantaneous velecity and other important rates of change, which prove important in the sciences, engineering, and the social sciences (e.g. power in physics, rates of reaction in chemistry, and marginal costs in business). Furthermore, we should note that the definition of these slopes involves a limit of a quotient whose denominator has a limit of 0: hence, a trick (e.g. simplifying the numerator, or rationalizing) will be necessary in order to evaluate the limit.
Things observed in the course of working problems.
The coefficients of each term come from Pascal's triangle:
Problems:
Assignment: #3, 4, 6, 8, 13, 16, 22, 24, 25; on the board, #5, 8, and 16.