Day 28, Mat129: Calculus I

• #15: Please stop writing things like this: $\lim\limits_{x \to \infty}=4$

  $\lim\limits_{x \to \infty}f(x)=4$

  is okay, of course. "lim" takes an argument -- otherwise it's like writing $\sqrt{}$, or $\sin{}$.

  square root of what? sine of what?

  \[ f(x)=\frac{(2x^2 + 1)^2}{(x - 1)^2(x^2 + x)} \]

• #18: and then a serious algebra felony occurred.... \[ \sqrt{9x^6-x} \ne 9x^3-x^\frac{1}{2} \]

  $\sqrt{9x^6-x}$	and with $9x^3-x^\frac{1}{2}$	$\frac{\sqrt{9x^6-x}}{x^3+1}$

  You can tell by the asymptotics:

  • the function on the right is not defined for large negative $x$ values;
  • the one on the left IS defined.

• #46: the first thing to notice is that this function is odd.

  (Did I mention that it possesses symmetry?)

  (Remind me again: What's the first thing to notice?)

  \[ \frac{x}{\sqrt{x^2+1}} \]

  If you've fallen victim to this demon, you've got to exorcise it, ASAP!

• There's an exam on Thursday, covering every section since our last exam: sections 2.3-6, 2.8; 3.1-3.5 (except 3.2). That's nine sections! (2.9 won't be on the exam). So I haven't time to ask a lot about any particular section....

  I'll have to ask questions that cover several sections at a time.

• I've got a bunch of old sections of mat129, and you can look for old exams from each index, like these:
  • http://ceadserv1.nku.edu/longa//classes/2010spring/mat129/
  • http://ceadserv1.nku.edu/longa//classes/2011spring/mat129/
  • http://ceadserv1.nku.edu/longa//classes/2012fall/mat129/
  • http://ceadserv1.nku.edu/longa//classes/2013spring/mat129/
  • http://ceadserv1.nku.edu/longa//classes/2016spring/mat129/

• Let's have that next homework!

• Please put your rectangles away.

Then we began section 3.7, with a classic example: packaging. Making a box with a fixed quantity of material that holds the most it can (maximizing volume).

I'll start with any questions you might have about the material for the exam.

Then we'll continue with that very important section 3.7 -- optimization problems. The best way to explain is to do some examples, but the point is that we're going to be looking for extrema of functions. There are thus two things to do:

1. Come up with the appropriate function, and then
2. find extrema in the ordinary ways.

• Understand
• Plan
• Carry out
• Evaluate

Examples:

• #15, p. 257
• #9
• #27 (what do you imagine the result might be?)
• #20