Last time | Next time |
I graded #33 from 2.9, and 46 and 64 from 3.1. We'll look at those.
Lot's of bailing out, so I'll add a general GOHF and we'll go over some of these problems (especially the story problems at the end).
Some of you don't like story problems, but that's why we're doing all these practice problems without stories -- so that, when you hit a story, you'll be able to step up and express it as a calculus problem! The only way to get ready for those is to practice, practice, practice.....
This is the relationship that we explore in this section.
Let's look at the graph of #5, p. 220. This is a graph of the derivative:
Questions:
Concavity test:
You might notice that the function itself looks cubic, and hence think to yourself that the derivative probably looks quadratic....
Another example with a horizontal asymptote is knowledge as a function of time -- #51, p. 222. We might guess that accumulated knowledge in studying for an exam looks something like this:
We might imagine that this physical process becomes less productive from hour to hour as the evening wears on (the law of diminishing returns).
Other Examples:
More generally, If $r>0$ is a rational number, then
then if the degree of q exceeds that of p, there is a horizontal asymptote, and the value of the asymptote is given by examining the approximating function given by the ratio of leading terms alone.