• I don't think I covered some of the visual exercises in the WebAssign for Section 2.2. I pushed the due date for 2.2 back to 2/16. You're "supposed" to work all of 2.2 by the definition, but there are quite a few 2.2 exercises that you can slay with the Power Rule.

  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.

• We did a few tangent lines and drew their pictures.

  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 answered/worked a 1.7 proof or two. WebAssign is the wrong medium for these epsilon-delta proofs.

  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 went over the basic proof of limits of linear functions, and then took it to the next level, with quadratic and cubic polynomials. That puts us about a week ahead of the 1.7 due date.

  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.

• Written Work deadline dead ahead. Lecturing well into 1.7, already. Knocked off around 11:45. Waiting for questions, now.

  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.

• We continued our discussion of Section 1.7, with a further example: How close to 5/sqrt(pi) does r have to be to produce a disk that has an area within 0.01 square centimeters of 25 square centimeters? Here, the growth rate of the function is not constant like it was with the linear circumference function. It turned out that you could be farther away from 5/sqrt(pi) to the left than to the right, because the pi*r^2 area function is growing faster off to the right and diverges from 25 more rapidly as you go to the right.

  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.

• We picked up the spares for 1.5 and 1.6 with a number of examples and a once-again-through on the limit laws. Then we stuck a toe into the Section 1.7 material, which covers the precise definition of limit. The intro was finding what tolerance on the radius of a disk was needed to keep the circumference withing +/- 0.01 of 25 centimeters.

• I got to mess up the graph of a piecewise function. Awesome! We fixed it.

  I ran through Limit Laws in Section 1.6 relatively quickly.

• We got an intuitive introduction to limits, Section 1.5 in the text. One thing that slipped my mind was drawing the picture for an INFINITE limit. I said some words about it, but drew no pictures and wrote no words.

  Section 1.6 is the nitty-gritty on limits, from a practical perspective. We may or may not get to it, tomorrow.

• We modeled hours of sunlight with a cosine function. We concluded 1.3 with a discussion of composition of functions and their domains. D( f o g ) = { x | x in D(g) and g(x) in D(f) } is the formal definition, but as a practical matter, we just look at f(g(x)) and figure out its domain. The formal way will avoid pitfalls, later on, but common sense generally rules.

  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).

• We opened with a modeling example that I worked incorrectly in my preps last week. I think we modeled the area of the Norman window OK.

  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.

• We took a bite out of Section 1.2, and did a lot of general ideas in algebra and calculus. Completing the square, factoring cubes, tangent lines, tangent line approximation.

  So far, not a very faithful rendition of the text, but all things I want under our belts before we get in too deep.

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The spreadsheet from class

The image of the seismograph from clacc (JPEG)

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