We got MOST of the way through calculus-based graphing techniques. There are some formalities I skipped, and just went straight to graphing an example, from start to finish.
I will finish the graph of the trigonometric polynomial
f(x) = 2sin2(x) - sin(x) -1.
We "know" the shape it has. We know where its max/min points are and inflection points are. Our rough sketch doesn't find the actual HEIGHTS (y-values) corresponding to these key points, so our graph needs to be refined.
On homework, you can always get a pretty graph from technology, and THEN just fill in the key points on the perfect graph. On a test, you might not get much farther than we got in class, today, where the final details, like the actual y-values haven't yet been computed. Those details are time-consuming, and may not be worth the points earned for the time expended. They're worth points, but not very many. They're details you might leave for your 2nd pass at the test.
On a test, when you reach a point on a problem that you know will be time-consuming, just leave the rest of the page blank, and start a fresh sheet of paper for the next problem! Do this for every problem you want to come back to, if you have time at the end.
Tomorrow, we will discuss some of the minutiae of the theory, much of which you already understand, intuitively.
Theory we want to cover, tomorrow:
I gave a loose overview of Chapter 3: Extreme Value Theorem, Rolle's Theorem, and Mean Value Theorem. I proved Rolle's Theorem and showed you the idea and the visual behind the Mean Value Theorem.
Have a great Spring Break!
We worked an Intermediate Value Theorem problem, after I finally cooked it up, right. Notice that before I tweaked the question, we didn't get different signs from the two y-values, even though we knew there were 4 zeros in the first version of the question we worked, unsuccessfully. The key to building the question was picking intervals where one input evaluates to negative and the other evaluates to positive.
Midterm is Wednesday, March 11th, from 10 am to 6 pm. Test takes 2 hours. Bring pen/pencil, scientific calculator, and photo ID.
Then we began reviewing for the Midterm test. Amazing how much you've learned, so far!
As soon as everyone left, I realized my mistake on the "On what interval is
| f(x) - L(x) | < 0.1?"
question. I solved f(x) - L(x) = 0.1
and
f(x) - L(x) = - 0.1
What messed me up was that the first equation gave BOTH endpoints and the 2nd equation had no solution! That's why the Desmos worked, but the Wolframalpha didn't.
I pushed back Section 2.9 one more day. I will have your Week 04's and maybe your Week 05's back by tomorrow.
We moved on to Section 2.9 - Linear Approximations and Differentials.
We pushed back Week 06 assignment to tomorrow, to give people more time.
These problems flow naturally from the Section 2.7 material. I think it's less intimidating if you see a few, relatively easy models from geometry (cylinders, squares, cubes, etc.). The next level is finding things like "How fast is the spotlight moving along the treeline if the angle of the beam is changing at a given rate?"
We finished with more discussion of rational functions. We had the slack in the schedule to give it decent coverage.
We now have seen Proper Rational functions and the Two Types of Improper Rational Functions: Those with horizontal asymptotes (Ridiculously easy) and those with slant asymptotes (requires polynomial division). You haven't seen any with quadratic or higher-degree asymptotes. Slant asymptote: Degree of top = Degree of the bottom plus 1.
Quadratic Asymptote: Degree of top = Degree of bottom plus 2. Cubic Asymptote: top = bottom + 3, etc...
We kicked off Section 2.7 - Derivatives in Science with a general discussion of a cost function for a producer as a function of the number of items produced. I didn't work a specific example, nor do I see a marginal-cost question on WebAssign. I just always make the point that economics is actually easier than economists make it, and basic math ideas they're using can be hidden behind their ridiculous ways of representing things, mathematically.
If you take Econ, don't be afraid to look things up in your math books and math notes to make sense of the Econ.
I apologize for struggling with the mechanics of the force applied to a mass in motion. I should have organized it, better, because of the complexity of the derivatives and the complexity of the simplification steps. I think there's some value in watching a person trouble-shoot what they're doing, on the fly. If I were doing that problem for credit, I would re-write the entire thing.
Students can get a false impression of what this kind of work looks like, in real life, because the teacher makes as perfect a presentation as possible. They can get paralyzed when they're not sure how to proceed. There's a lot of talking to yourself. A lot of "What does this mean?" They think they're failures, because they can't just crank out good work on the first attempt. And all their teachers pride themselves on their flawless presentations.
Math isn't carpentry, where you measure twice and cut once. Math is writing, where you go ahead, make a cut, and if it was a bad cut, you make another cut. See how that turns out. Critique it. Check it against known solutions, when available. Grow a thick skin. Always pursue the truth, which means always questioning what you're doing. Rather than being sad that you made a mistake, be glad that your found your mistake! VICTORY! That's how I always looked at it, as a student. If you always win the game, it's not much of a game. It's more rewarding if you lose, at first, and then win!
This applies to test-taking, as well. If you can just start writing and putting down what you do know, you'll be amazed at how much credit you get, even though you never got the right answer. Writing a lexicon is 3 out of 10 on a 10-pointer. That's something you can do, and often it is what helps you frame all the work you do on the problem. Just start writing, and often times, the solution just sort of reveals itself, but you wouldn't have seen it if you didn't have something written down to look at.
I'm pretty sure my ax = by idea was bogus. Instead, I think the idea is to find y' for both equations, find where the two graphs intersect, and then substitute that point or those points into both expressions for y' and confirm that they are negative reciprocals (mperp = -1/m intersect.
I concluded my relatively short talk with a pretty straightforward example from 2.8. Lexicon matters.
I was off my game, today. People were patient and kind.
I plan to continue pushing the lecture envelope as much as I dare, so everyone will have time to take a breather, the week before the Midterm, and prepare ourselves thoroughly for it. Minimal preparation would be just getting all the homework done.
One thing I didn't show you in this regard is copy-pasting its output into a window and subtracting your version from it (by typing in your answer), and seeing if you get zero!
Week 5 postponed. Nobody wanted to do that. :o)
Then I did a somewhat lengthy review of rational functions. We'll see more on these, later, but the better your chops on the College Algebra stuff, the better your understanding when the topic arises again in Chapter 3.
Worked a bunch of examples from Section 2.5 Chain Rule.
Worked some examples involving trig functions and the rules for differentiation we've been given, so far. Extended the product rule to products with 3 or 4 factors. It's basically just what you'd hope it would be.
Worked examples that incorporated trig derivatives, power rule, quotient rule, and product rule. Things are really opening up. The Chain Rule finishes it all off. The rest is doing stuff with this derivative thingie.
We finished in the last 15 or 20 minutes with a pretty weak proof of the Chain Rule. I think the essential parts of the proof were presented, but just not very well. There's a lot lurking in the machinery of continuity and differentiability that make it all work, so that we can say with confidence that the ε1 and ε2 are continuous functions of Δx.
We gave Section 2.3 a better treatment, today, focusing on Product Rule and Quotient Rule. The "in-demand" Power Rule was already covered pretty well, and you were permitted to use it and the other two rules for your 2.2, if you wanted.
As I said for the 2/12 notes, you didn't have quite enough knowledge to cheat all of Section 2.2, without the Chain Rule. I gave you a very quick "How-To" for Chain Rule that would probably be sufficient, if you went over the 2/12 notes and followed along in the video, but we're far from PROVING the Chain Rule.
We'll do the Chain Rule Tuesday or Wednesday, and then maybe have an in-class work session or two.
You have to be really good with the Chain Rule before you can really do Implicit Differentiation, which is one of its big applications beyond just taking derivatives of big, messy functions.
Since Bradley asked, the meeting before, we did some Section 2.3 How-to's for power rule, product rule, quotient rule, and chain rule, in one force-feed, because WebAssign doesn't care if you cheat it with 2.3 stuff. The problem is that there are some exercises in 2.2 that require techniques beyond the mere Power Rule. I tried to cram the Chain Rule in there, to give an enterprising student a way to apply advanced techniques, rather than grinding out difference quotients by hand.
I felt like I had to, because to "cheat" Section 2.2 with the power rule (and other rules) from Section 2.3, you're also going to need the Chain Rule to explain the derivative of the square root of the quantity "6 - x," or "sqrt(6 - x)," in computer English.
I think there's probably 2 weeks' worth of material in today's lecture. It might be worth watching and/or reading more than once. I made some important asides that were not made in the notes. I wanted to give you the full cheat code for Section 2.2, once I got my first "Do we have to keep doing it this way?" from the class. That's kind of the class's tell that they're hungry for the power rule (especially when they mention it by name!) and there's an opportunity, there, to teach a lot in a short time, because there are exercises in Section 2.2 that aren't amenable to the mere Power Rule without the Chain Rule to explain how you handle the "6 - x" inside the power. (That's why the derivative is negative.)
I think I reached in the last 5 minutes, talking about the geometric proof of
sin(h)/h --> 1 as h --> 0.
in the proof of (d/dx)(sin(x) = cos(x)
that we will see early next week. Sorry I kinda flubbed it, but I think I've almost proven the whole thing, in bits and pieces thrown at you when I was refreshing myself on the construction, outside of class.
I took you into a land of extreme pathology with the "topologist's sine curve" and its cousin. x sin(1/x) and x^2 sin(1/x). These are classic examples that test the edges of the theory. We'll do most of our work on domains where everything is nice and smooth, except where there's a division by zero or a square root of a negative. (or a 4th root or 6th root or 8th root or ...)
This very special piecewise function has a way of cropping up in the bonus section of tests.
We covered 1.8: Continuity. It's our first pass. Hopefully, people will have questions, tomorrow on this. I do want to jump into Chapter 2, but this is a natural pause point, to catch up on homework, too
We knocked off early and worked the rest of the hour. I can't find the acoustic Jimi Hendrix I've been looking for. I've got a pretty good CD that's just Jimi, and I'd like people to hear that, because he was a blues man long before he blew up Rock.
WebAssign Section 1.7 #s 19 and 20 are, I would say, beyond the scope. Something that would take most of a lecture to explain and for people to absorb. They're the kinds of problems you'd see in Advanced Calculus, if at all.
I'm pretty sure I made the intractable parts count for zero points, so they won't hurt you, unless you waste too much time on them without result.
I think we have all we need to prove limits of linear functions. Next up will be designing proofs for quadratic and higher-degree functions, which will be bonus material on tests.
Then, I'm not sure if it won't be a good time for some kind of clinic on graphing or algebra skills. Calculus I is when students really start understanding all their algebra and becoming fluid and FAST on those skills.
I ran through Limit Laws in Section 1.6 relatively quickly.
Section 1.6 is the nitty-gritty on limits, from a practical perspective. We may or may not get to it, tomorrow.
Section 1.4 tries to convince you that the tangent slope is the limit of the slope of the secant line. A lot of tedium in these exercises. I want you to get the thrust of the lesson, but I don't want too much busy work. But these aren't too bad, if you can build the slope function m = (f(x) - f(x1))/(x - x1) and you can build the tangent line once you find the limit as x -> x1 by plugging in x's closer and closer to x1 pretty quickly. Then build y = m(x - x1) + f(x1).
The rest of the time was spent on graphing by transformations on a known basic function. The example we worked was a square root that we stretched verticall, shrank horizontally, shifted right, and then shifted down.
I mentioned reflections, but didn't do any.
Next time, we'll model hours of daylight as a function of the day's date.
So far, not a very faithful rendition of the text, but all things I want under our belts before we get in too deep.
