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Section 002 made a successful argument that the lab should be considered "optional"; so everyone gets an additional "Get Out of Homework Free" card. If you did it successfully, you're ahead intellectually, however!
I want to look at a couple more example problems, before moving on to asymptotes. We were finishing up #39....
To some extent I've been pushing these ideas as we've gone along, so this should be familiar-seeming territory.
This is the relationship that we explore and summarize in this section. In addition, we summarize what the second derivative has to tell about the shape of a graph.
Let's look at the graph of #5, p. 220. This is a graph of the derivative (at left):
Questions:
min | max |
$f'(x)$: - to 0 to + |
$f'(x)$: + to 0 to - |
$f''(0)$: + | $f''(0)$: - |
This figure is from Example 5, p. 218:
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.
This is called a "slant asymptote" (not a horizontal asymptote, for obvious reasons! If you get far from the origin, then the difference between the two functions falls away.
We can then replace the more complicated with the simpler.
We use this idea in physics all the time: we assume that gravity is constant at the surface of the Earth. In fact, it varies as distance to the center of the Earth, but we're so far away that we can take this as a constant (its value at about 4000 miles -- our distance to the center of the Earth). We're far enough from the origin (the center of the Earth) that we treat acceleration due to gravity as locally constant.
We can say, however, that $\infty*\infty=\infty$, that $\infty*1=\infty$, that $\infty/1=\infty$, etc. So some of the usual rules apply (and hopefully make sense!).