Then I gave what Ellie calls an "impromptu" speech about what the Differential Calculus is and what the Integral Calculus is, by giving a general overview of the limit of (one form of) the difference quotient, and then your first (right-endpoint version of) a Riemann Sum (The sigma thingie with the funny-looking "E," which is actually the Greek letter "S," which stands for "sum.), of which we took the limit as the width of the rectangles approached zero.

After that, I left people to their own devices. Only 2 people stayed. I was hoping more would use that time to get work done, but all but 2 made a quick getaway. Then we 3 hung out. There were some spans of several minutes when nobody said anything, and then there was a flurry of questions! Cool!

I ended up working 3 or 4 more or less standard difference quotients of medium-to-high difficulty, including

• Quadratic Function
• Square Root Function
• Cubed Function - Required knowledge of Binomial cubed. Introduced Pascal's Triangle version of the Binomial Theorem
• Cube Root Function - Required a fairly un-obvious pattern recognition and the Difference of Cubes special factoring formula.

• We discussed the "Making Quizzes Optional" plan. People mostly wanted Strategy 1. It's less work to just go with that, for me.

We discussed...
• ... some set notation and logic stuff in a quick, informal, and over-everyone's-heads sort of way.
• ... the finer points of intervals of increase/decrease, and why we include the endpoints.
• ... using the linear regression function for 1.2 #6. It turns out the WebAssign will accept the solution line, as long as the coefficients are within 1% of what they should be, BUT if you don't follow their EXACT instructions on how to round those coefficients and you USE that "OK answer" to calculate new outputs using that "slightly off" answer, then your outputs won't be within the tolerance of the algorithm.

  WebAssign accepted -0.000105357x+14.521428 as your linear model, but if you used that exact model, the answer you got by plugging in x = 85000 was wrong. The CORRECT answer for the model was -0.000100x+14.521429.

  View a PDF of the spreadsheet used for the linear regression question.
  Download a working Excel copy of the spreadsheet.

• ... how to use Desmos to enter and evaluate a function.

• We showed how to graph , which is just a transformed .

  Then we computed some secant slopes fairly efficiently, using Desmos. The idea was to take the x-values closer and closer to x = 7 to see what the instantaneous slope will likely be.

  Finally, we computed the instantaneous slope of f(x) at x = 7.

  Then I wasted time talking about my bad knees. Sorry.

  Fact: If the limit of the difference quotient exists, then plugging in numbers really close to the limiting x-value will get you very close to the instantaneous slope. It's always a good check, but of course, we're going to try to teach you to find its exact value by taking limits.

• Left and right limits. Limit laws. Different styles of limit-taking. Infinite limits.

  Handling absolute values in limits and elsewhere.

  Tangent Problem. Tangent Line. My (preferred) way of writing equations of lines as quickly and efficiently as possible. (Better version of point-slope than the one taught in books and by most teachers.

  Several different ways of expressing the difference quotient. I still want to do an "x - c" version of the limit of (f(x+h)-f(x))/h = (f(c)-f(x))/(c-x).

  We found the equation of the tangent line to the square root of x at x = 16, and we drew the picture for the situation.

  Only scientific calculators permitted (actually REQUIRED) on tests.

  After everybody but one of you left, I was asked a question about 1.3 #11, which in the notes (and hence the videos) is #25.

• I ran over different variations on difference quotients related to the last exercise on the Week 2 Written Assignment.

  I also tried to give people some ideas on how to break trail on new math knowledge. A way to get out of the box you might not realize you put yourself in.

  I keep saying learning is a writing process, and I believe it is, when it comes to learning abstract or complex concepts. I see so many students who try to turn in their first pass at the work, which is slow, laborious, and leads to lower-quality final product.

  You want to be "fast and loose" when you're hitting it the first time. Then, as you figure things out, you can begin writing that "final draft" to be turned in for assessment, and as a touchstone for your own review of the material in the future.

  The better your "final draft," the better of a learning tool it becomes. Also, the better you write it up, the more the knowledge goes into your brain. The act of doing a good writeup reduces the amount of review you need, and it makes that review go much more quickly.

• A quick introduction to Section 1.7, followed by some flailing around. *sigh* My eyes are letting me down.

  You should be able to prove the limit of a linear function, with delta = epsilon/slope.

  Higher-degree polynomials require more analysis. They will be bonus on tests. delta = min{1,epsilon/number}, where number will be determined by the behavior of the other factor on [c-1, c+1] for x-->c situations.

• We had a question from 1.7 about the limit of a cubic. It was all very "Wolframalpha.com."

  I used that 1.7 #19 as a springboard to the formal proof of the question of a cubic-polynomial limit, which WebAssign assumes you're all too stupid to even see. I don't think that way. I want to see. I want to show you the basic technique for ALL, FORMAL polynomial limits, and that I have done.

• We re-visited 1.7 #19. Nobody could get part b. Not sure what WebAssign's trying to accomplish with this one, but finding
  f(x) = L + epsilon and f(x) = L - epsilon, for fixed epsilon was fairly straightforward with Desmos.

• Boy we covered a lot! I talked for over 30 minutes straight.

  This was the last lecture before Quiz 1. I posted a video on how that works from the student perspective: Download and Install Lockdown Browser.

  If you're using an Aims laptop, it's already installed.

  I talked almost the entire time. Sorry about that. We covered:

  • Infinite limits
  • Bye-Bye Shots for Quizzes
  • An application of The Squeeze Theorem
  • Manipulating quadratic functions and equations by completing the square
Also included in the notes are screenshots of my Desmos session.

• We extended Quiz 1 deadline to midnight, tonight.

  We did a poor man's intro to Chapter 2, talking about rates of change and derivatives.

  The talk kind of slopped over between 2.1 and 2.2, which are this week's topics. Section 2.2 sort of streamlines 2.1. Then we'll do a LOT of streamlining in 2.3. The sooner we can get to 2.3, the sooner you will really start to see the power of this calculus stuff, because there are a million shortcuts to computing these limits of difference quotients.

• I briefly discussed some random concepts covered on the Week 4 Written Assignment.

  Phoenix asked about the "cheat" for finding derivatives given in notes and videos. I consider the 2.3 and later stuff will be the way we do derivatives in the sequel. You just had to build your chops on limits, first, so you're not just plugging into formulas with no rhyme or reason.

  I went on about the topologist's sine curve (again), trying to show how you might go about proving a limit does NOT exist, by showing you a function where the definition breaks down.

• I remarked on how slow people are to get started on Chapter 2 stuff. I'm really trying to push more active learning. It's more efficient learning. It's deeper learning. We become more like colleagues, where I'm more of a mentor than a high priest for the Math gods.

  I sneaked (snuck?) Section 2.3 concepts in, today. Differentiation Rules "sneak preview." Can save a lot of time finding derivatives, although we're still working on our algebra-and-limits chops at the foundation of everything.

  Did some sum-and-difference-of-cubes stuff. How to apply it to differentiating cubic functions and cube root functions.

  Introduced Liebniz notation (d/dx) and pointed out that it's a "linear operator," taking us from the power rule to a rapid way to evaluate derivatives of any polynomial.

  Finally, I did some sketches of functions and the tangent line to them at a specified point. I got so caught up in the mechanics of the limits that I never did get around to sketching the cube root function and its tangent line at x = 2. I did, however find the equation of the tangent line.

  I spent some time demonstrating and talking about graphing quickly. It's more of a drafting process, where you rough it in, very rapidly, and then re-do the rough sketch, with key points emphasized and nonessential details ignored ("The highest order of human intelligence is knowing what may safely be ignored!").

• We discussed notation issues in the writing of limits, and that "=" is not the same as "-->".

  New, "50% Off" late written work. You have all semester to turn it in, but it will be at a 50% discount, henceforth.

  Work ahead. Even a little bit. Especially a little bit. Have the new section "roughed-in" in your notes ahead of time. You don't have to understand a theorem to jot it down and leave the rest of the page blank, for when we talk about it in class. Even if you don't understand it - especially if you don't understand it - you're creating a place in your brain for the day's talk to go directly into memory, rather than "I'll figure this out at 10 p.m. when it's due at midnight."

  I fought this for years, where student questions were concentrated the day after the homework was due, and I knew for a fact that the class was a lecture behind where it ought to be, and always trying to drag me 2 days behind. I'm looking for students that are pro-active. It's more efficient to be pro-active. 15 minutes that saves you an hour, later....

  sqrt(x)^2 = x. sqrt(x^2) = | x |.

  I worked a Section 2.3 Product Rule exercise, and showed that the WebAssign is satisfied with the form of the answer (unsimplified) that I want to see on tests and homework. If the derivative, itself, is the final product, and we're not doing anything else with it, there's no reason to simplify, other than for practice. When you're under a time control on the Midterm and Final, you don't want to waste ANY time, and that means not simplifying what you don't need to simplify.

  I teased Section 2.4's derivative of the sine function, which is coming to lecture, Monday.

• We discussed a re-scheduling of some of the WebAssign homework, which looks like it's a bit stretched-out just ahead. I want to compress that a little bit, so that we're finishing 2.3 and 2.4 this week and taking a look at 2.5 by Thursday, hopefully.

  We want to leave ourselves some slack for later, in case we need it, and not go too slowly, in the first half of the semester.

  I did some trig review and showed where the proofs of the derivatives of sine and cosine are in the Homework Videos (and Notes) on harryzaims.com (Click here and navigate to Homework Videos (and Notes).

  Some quick trig review was done. I drew a lot of triangles.

• Brief discussion of how 2.3 and 2.4 work together and links to the proofs of the derivatives of the trig functions.

  Re-scheduling some of the WebAssign as we speak. Crazy how it was set up, before.

  Midterm the week after we get back from Spring Break. We get back 3/24 and the Midterm is the following Monday 3/31 or Tuesday, 4.1.

• Not my best Chain Rule speech, but I did lay out how it works. I think the examples at the end, and especially that last one where you have a product rule with 2 chain rule derivatives inside of it, will provide the most practical insight on the matter. It's an (fg)' situation, where both f and g require the Chain Rule to differentiate.

  If I do the formal proof, I'll put it on video in with the 2.5 Homework notes and videos. No one's ever asked.

  The point was made that you shouldn't try to simplify your derivatives unless you HAVE to. Sure, it builds some muscles but the WebAssign isn't that picky and I certainly would prefer to assess the proper application of the differentiation rules than to grade every simplification move, in detail. There are other ways to test for those skills, and I use all of them. "Differentiate. Do not simplify" is not where I'm looking for simplifications. I'll just deduct points for not following instructions and making more work for both of us.

  Right now, I'm poised to lecture WAY ahead of the class, so it's time to dial it back a little bit at my end, and hopefully see the class dial it up a notch and start beating that schedule. I think the more you attack the material that way, the easier it will get to attack new material.

  After I stopped recording, I went radio-silent and drew a bunch of degenerate triangles corresponding to special values of sine and cosine at quadrant angles.

  We'll keep kickin' the can down the road.

Next week, we'll hit 2.5, again. Maybe do the formal proof of the Chain Rule. It's a bit abstract for this level. I'll probably just do a few Chain Rule examples on Monday, and a quick intro to Section 2.6's Implicit Differentiation.