18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 11
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PROFESSOR: OK, we're ready to start the eleventh lecture. We're still in the middle
of sketching. And, indeed, one of the reasons why we did not talk about hyperbolic
functions is this that we're running just a little bit behind. And we'll catch up a tiny
bit today. And I hope all the way on Tuesday of next week. So let me pick up where
we left off, with sketching. So this is a continuation. I want to give you one more
example of how to sketch things. And then we'll go through it systematically. So the
second example that we did as one example last time, is this. The function is x
1 / x
+ 2. And I'm going to save you the time right now. This is very typical of me,
especially if you're in a hurry on an exam, I'll just tell you what the derivative is. So
in this case, it's 1 / (x
+ 2)^2. Now, the reason why I'm bringing this example up,
even though it'll turn out to be a relatively simple one to sketch, is that it's easy to
fall into a black hole with this problem.
So let me just show you. This is not equal to 0. It's never equal to 0. So that means
there are no critical points. At this point, students, many students who have been
trained like monkeys to do exactly what they've been told, suddenly freeze and give
up. Because there's nothing to do. So this is the one thing that I have to train out of
you. You can't just give up at this point. So what would you suggest? Can anybody
get us out of this jam? Yeah.
PROFESSOR: Right. So the suggestion was to find the x values where f (x) is
undefined. In fact, so now that's a fairly sophisticated way of putting the point that I
want to make, which is that what we want to do is go back to our precalculus skills.
And just plot points. So instead, you go back to precalculus and you just plot some
points. It's a perfectly reasonable thing. Now, it turns out that the most important
point to plot is the one that's not there. Namely, the value of x = - 2. Which is just
what was suggested. Namely, we plot the points where the function is not defined.
So how do we do that? Well, you have to think about it for a second and I'll