The book develops this material much more sedately. To me, these ideas all fit together and belong together, with the juicy punchline at the end of those "problematic" antiderivatives we've been denied up to this point.
We did some algebra review. I screwed up the sum/difference of cubes formula. Fixed the notes and the video.
Then I completed the square on a quadratic expression. It's a great way to solve quadratic equations and obtain a super- fast sketch, with vertex and intercepts all shown. Learn this technique and be a lot faster getting math homework done. To motivate the manipulations involved, I brought in an exercise from the future, where you use completing the square to re-write an integral in a form that you can quickly look up in the Table of Integrals in the back of the book.
We finished up with an FTC I question that wasn't trivial, and required some manipulation before you could attack it with the Fundamental Theorem of Calculus, Part 1 (FTC I).
We did some algebra review. I screwed up the sum/difference of cubes formula. Fixed the notes and the video.
Then I completed the square on a quadratic expression. It's a great way to solve quadratic equations and obtain a super- fast sketch, with vertex and intercepts all shown. Learn this technique and be a lot faster getting math homework done. To motivate the manipulations involved, I brought in an exercise from the future, where you use completing the square to re-write an integral in a form that you can quickly look up in the Table of Integrals in the back of the book.
We finished up with an FTC I question that wasn't trivial, and required some manipulation before you could attack it with the Fundamental Theorem of Calculus, Part 1 (FTC I).
I discussed upper and lower estimates and upper and lower bounds on definite integrals.
I wrapped things up by making a long apology for "flipped" courses, where the exposition and a lot of help are online, so that students can work as efficiently as possible, especially when I'm not around. The goal is to help train you to be an independent learner, and get maximum bang for the buck from our time together, live.
I lectured traditionally for 30 years, and winning awards for it, but knowing I was putting my top students and bottom students to sleep, because I was boring to the top students and teaching over the heads of my bottom students, trying to make perfect, humorous, comprehensive, one-size-fits-all lectures to the mythical middle of the class. This was a lot easier to do, face-to-face, because people at the same level could work together. I could group people who were struggling with the same thing together, and help them, while everybody else was able to do their work during class time and save a lot of time, rather than having to pay attention to the clown at the front of the room spoon-feeding them.
It's still a bit of a struggle finding the ideal amount of "lecture" in a remote class. It's a lot harder to break people into groups, and cruise the classroom for where my help is needed and not mess with the people who are doing well and just want to be left alone! I'm sorry if it's not working, perfectly, for some of you. My desire to help often turns into a full-blown lecture, even though everything I might say in lecture is covered in the videos and notes that are available 24/7.
The idea is to train up (or help train up) independent learners, who are able to learn without a teacher lording it over them and force-feeding them, when most students do better working at their own pace. The online resources are meant to be liberating, but students accustomed to daily spoon-feedings don't like it.... until they make their breakthrough and become true, adult learners.
I always felt I'd have had more fun in Calculus I if somebody would just get me to the Fundamental Theorem of Calculus! So that's basically what we did, without proving much of anything, except 1 induction proof to show that the theory being explored actually works for an x^2.
I was just very much off my game, unable to do arithmetic.
We have left, right, midpoint Riemann Sums. We have also have Simpson's Rule for building these sums. Our focus will be on the right-endpoint Riemann Sums, but the WebAssign will make you do a couple other styles, the long way, by hand, at least once.
Revisited a rational function we graphed, previously. Cleaned up an error in my long division of polynomials.
We talked about a trig equation from this fall's Trigonometry Midterm. We discussed finding all solutions between 0 and Pi. We talked about finding all solutions (infinitely many for these periodic functions).
I got the remainder wrong, because I added '9' to '25' instead of subtracting '9' from '25.' It makes no difference, because all we need for the oblique asymptote is the quotient, itself. I fixed this in the notes.
I also added synthetic division method, because 'x - 5' is of the form 'x - c.'
It probably would've done more good to talk about graphing rational functions, using College Algebra/Precalculus skills. That's probably where the holes are in the average student's foundation.
I thought it would be a good idea to show how to build a piecewise function that's half-quadratic and half-linear, that also is continuous and differentiable everywhere. That kind of build would make a good bonus question on the final.
I finished with a very brief intro to Newton's Method, which is an algorithm for finding zeros of functions.
I'm sure we need some graphing clinics. If I told you that you had some algebra deficits, you should be asking me a lot more questions, so we can identify your weaknesses and work on them, together. Even if you were aces in college algebra and trig, you're not going to retain all of it, especially if you haven't taken a math class in a while. I wouldn't panic, but I would try to get help on those prerequisite skills, because they're the most common source of failure in Calculus I.
It's good to see someone handle their failures and move on, without being discouraged. The deeper you go into mathematic, the thicker your skin should get.
All that being said, I never set out to mess up. I just know that it will happen plenty enough times to convey the no-anger and no-discouragement when things don't just go perfectly the first time.
Today, we just took roll and exited. I will do an example or two from Section 3.5, tomorrow. I don't think that Section 3.6 requires much explanation, as we've been graphing with Desmos and wolframalpha.com, almost every day. I will happily answer any questions from 3.6, tomorrow and Monday.
I continued the example begun on Wednesday, and broke it down even farther, explaining how to use DESMOS to analyze the concavity (2nd-derivative test and inflection points).
We answered a few questions from the Spring, 2025 Midterm.
Reviewed epsilon-delta proofs for linear and quadratic functions from last year's midterm. Worked a couple bonus-level questions from last spring's Midterm. I also worked an implicit-differentiation problem.
"Build a rocket on a desert island" was my silly idea. Kallen said a plane or a boat would be more practical, if you were stranded on a desert island. I agree!
I've given modest launch talks on 'most everything in Chapter 2. There's some slack time for the optional Chapter Quiz. If you can take the Chapter Quiz without too much lost time, it's probably smart to take the quizzes and gather all the intel you can for the Bye-Bye Shot at the end. If, at the end of the semester, you have an 'A' without including the Quizzes, I'd can the Bye-Bye Shots, entirely. But if I needed a bump in my grade at the end, the intel from the 1st attempt would be very helpful in the Bye-Bye Shot, so you're more sure of acing the optional category.
I played a video on the proof of the Product Rule. The Quotient Rule is similar. I will show that proof if you want. Just ask me.
So, no video from today.
I didn't cover much, today. It looks like people are forging ahead on Chapter 2, Sections 2.1 and 2.2, this week. The fun stuff begins in 2.3 and 2.4. I gave a sneak preview of this, but 2.3 and 2.4 are due next week. The "Big Proof" in 2.4 is the proof of the derivative formula for sin(x). I made a recording for it that is probably cleaner than what I would do, live.
Section 2.4 Notes and Videos, there are separate sets of notes and separate videos on the theory part. I've never tested over these proofs, but it's really good for a student to see how they go, and to believe that the formulas we're expecting them to learn aren't just some made-up nonsense. It always helped me to remember results when I could understand where they came from.
The other property of linear operators is that when they act on 0 the result is 0.
There was some miscellaneous algebra review/exposition, namely, the 3 ways to solve a quadratic equation.
We wrapped up with the Intermediate Value Theorem, and worked an IVT question from the old Midterm.
Kick-off was a couple of examples where they list limits and you draw the picture. Then I worked a couple of pretty standard limit evaluations, using algebra to factor the quotient involved. The two I did were 0/something and something/0 cases, which are kind of special. I left the 3rd one (from Weekly Written Assignment) to you. Hopefully some thing cancels, because those are the ones were mostly interested in.
I wrapped-up the discussion with another 1.7 "epsilon-delta" proof at Bonus Level. If you can do the "Bonus Level," you're going to have a real leg up on your classmates should you ever be crazy enough to take Advanced Calculus, where you prove, in detail, the legitimacy of the techniques we use in Calculus I.
We talked about a couple of funky WebAssign questions that wanted you to upload a written file. Meh. I should have (and probably did) make those exercises not count. I left them in place, to keep the videos and notes as matched-up to the WebAssign questions as possible.
WRITTEN MIDTERM AND FINAL:
The default is to go to Greeley campus, Horizon Hall, Rm 107, between the hours of 8 am and 6 pm (test start time).
If you are closer to another testing center, then by all means give me a heads-up in e-mail, and we'll hook you up.
If it's a school other than AIMS, there may be a $20 or $30 fee. (Sorry!)
The correspondence between Notes, Videos, and WebAssign questions isn't perfect, but it's pretty close. A good strategy is to always look for the question in the Notes. The exercise # in the notes will be the video #, so that will make it as quick as possible to see and hear what I would write and say, if we were in real-time Q&A.
I hope you like the on-demand help. It's home-made, but so is every traditional, in-person lecture. With the archive of videos, I get to say everything there is to say about everything, and you can slurp up some or all of it, as needed.
When I made the videos, I remembered what had been previously said (or un-said), so that I included the part of the theory pertaining to each new type-problem, so there's a mix of "What's the prerequisite knowledge?" and "What's this problem asking and how do we solve it, knowing what we know?" This is heuristic (problem-based), just-in-time learning
We worked a couple of examples of the Section 1.7 material: The Precise Definition of Limit. This, and the Section 1.8 material on continuity are fundamental underpinnings of Calculus. It's a bit abstract.
Very briefly, continuity means: The limit exists at that point, and it agrees with the value of the function at that point. Continuity usually boils down to DOMAIN questions. Almost any function that we can think of, is continuous on its domain! We really have to be clever to build functions whose limits exist that are not continuous.
The last Question on Week 3 Assignment, that Kory asked about, is one of those "Who thought of this crazy function?" That x*sin(Pi/x) function is undefined at zero, but the limit exists. We "brute-force" continuity by declaring the function in pieces, with limit as x approaches zero substituted for the value of the function at x = 0. This is a classic case of a function with a "REMOVABLE DISCONTINUITY."
I did take the limit of the difference quotient in 3 ways, but there were no questions, and people are making great progress on the WebAssign homework. We'll see, this weekend, how everyone did on the Week 1 Written assignment.
There was some good stuff, there, for dealing with higher-degree polynomials in these epsilon-delta proofs, but the higher-degree polynomial proofs are BONUS only.
The main thing about these proofs is writing them in a cogent manner. I model the proof-writing technique several times. I encourage you to follow my style, rather than try to follow the book method. But suit yourself. I know what a proof is and what a proof isn't. As long as your proof is good, it doesn't have to be just like mine are.
We included some later (out-of-sequence) material on Section 1.7, because a student had a question. It was a somewhat advanced question, but I tried to make it accessible to everyone. I hope it worked!
I would feel bad about talking about Section 1.7 this early, except I believe that the earlier and "oftener" you see these ideas, the better. I'd almost prefer you not do the proofs in Section 1.7 the book/WebAssign way. I especially dislike the "But..." in their proofs. There's no "but" about it! It's a direct implication. More of an "implies" or "then," not a "But." Heck, you're not even supposed to start a sentence with "But..."
Then I added a few more "basic functions" to our list, namely, the special cases of power functions with rational exponents, like 3/5 or 3/8. These have interesting properties at x = 0.