I mentioned a while back about how I love calculus because it brings back old concepts, and now u-substitution is making it's return. I easily grasped the u-substitution concept when we first learned it, so the u-sub part of indefinite integrals should have been trivial, however it had been so long since I had done u-subs I completely forgot what to do with the du. Adding definite integrals was even more confusing, but I got the hang of it after some practice. It took me a while to realize how the u-interval and x-interval were connected, and to be honest, I forgot how I figured it out, but I did.
I said last week that I liked that things were being brought back from other classes into the things we were working on, and now I see trig is being brought back, and that was my favorite math course. I still don't understand radians though. Putting the derivative in relation to time was genius, who ever came up with that should be acknowledged, I would have never thought of that. So far calculus has been surprisingly easy, with only a few hiccups here and there. This was one of the easier portions of the course in my opinion.
I like how all of these concepts are coming together as we go through the year. Bringing back the ol' substitution rules to figure out optimization was cool, and using concepts we just recently learned like the derivitive zeroes shows that what we're learning actually has uses later on; it's not just learn it, take the test, forget it forever. It gives us motivation and a sense of accomplishment. I hope we do something similar with the anti-derivitives because I quite liked those. Except for the +C part. I also see now how the finding the shortest path assignment we did on the computers goes with this section, although I had completely forgotten about it until I saw it in my folder. I feel like this section was really easy because I did the quiz no problem (I think) even though I only finished half the homework so far (sorry). Maybe it was because of how it all tied into the things we'd already learned.
The worksheet over derivitives and their graphs was easy enough, but we really didn't need the extra day on monday to work on it, I just worked on other homework. As for 4.3, I wasn't completely sure whagt a point of inflection was, and I'm still not certain (but I doubt myself with everything). It's just where the concavity changes right? The part about solving algebraicly where the graph was increasing/decreasing and concave up/down I would have never got if you hadn't explained it, really glad I didn't sleep though that part (I'm getting better about it!). Three day weekends are good, but are kind of moot when there's exams looming over the week afterwards.
Another test week, monday was the test so nothing to talk about there. Tuesday: finding extreme values, simple enough, but finding them algebraicly is still a bit fuzzy for me. I know to get the critical points, but how do you get the absolute extreema? My notes say to evaluate the function at those points, although I was very tired the day I took those notes. I guess I could just plug in those numbers. Wednsday was Mean Value Theorum which is basically the slope formula, which makes sense, because derivitives are the graph of the slope of the original graph; Thursday was more of that. Friday was connecting what we learned, which is a good way to end the week. Graphing f'(x) and f''(x) from the graph of f(x) takes a surprising amount of critical thinking. Basically it showed things like the extrema from the graph are the zeroes of the next graph down. Now I'm thinking of inception and how derivitives are like the different levels of dreams. Anyway, I liked ending the week on a sort of reflection, but perhaps it would be better to do reflections on mondays to keep this fresh in our heads. I also didn't like the
As far as anti-derivitives go, I think I get it, and the tangent line is just the derivitive of the equation right? I really have no problem with sketching f from f' as the book calls it, although when I graph the f' to f, I seem to make silly mistakes. The chapter 3 review was fairly easy, and only a few problems really stuck out as difficult to me. The derivitives of e^u and ln(u) are simple enough, but the derivitive of a^u I will have to write down. I'll also have to write down the derivitives of sin cos and tan, and sin^-1 cos^-1 and tan^-1. I think I know how to do higher order derivitives, but I'm doubting myself again. Looking back on limits, I remembered that I need to know limit of x to infinity of sinx/x = 0 and limit of x to 0 of sinx/x = 1. I have a feeling that will come back on the test.
When I first saw higher order derivitives, I thought "that looks easy!" But now I realize why they're so evil. They expand extremely fast, and I can see how easy it is to lose track of simple things like negatives and distributions. That higher derivitive on the quiz was...interesting. It expanded from xsinx to jfiosadpngapohtruhaolk[asfjoaps. I hope we never have to do that again. Speaking of the quiz, I'm interested to see how that last problem was supposed to be done without calculus. I would have tried for it, but I was already overtime, and I'm sure I would have failed anyway. Chain rule, just like higher derivitives, I thought would be super easy, and just like higher derivitives, the equations get exponentialy more complex. The simple ones were super simple, but the more difficult ones were next to impossible, simply because of the amount of things that had to be kept track of. I guess I'll just have to double check these problems to make sure I don't do something stupid like leaving off a negative. I remembered a youtube channel that Cresswell might like called Numberphile. This is one that sort of has to do with limits. I think I'm getting better at grabbing a whiteboard as soon as I walk into class. One thing I can't remember is the derivitives of cos sin tan cot csc and sec. I can remember that d/dx Sin=Cos and d/dx Cos=-Sin, and I think d/dx Tan=Sec^2 and maybe d/dx Sec=SecTan. I can't remember anymore than that though. I'm not sure why position=S(x), velocity=S'(x), acceleration=S''(x), and jerk=S'''(x). What does jerk even mean? Will we get into this later? I think I understand the Chain rule, but the worksheet that was handed out today (10-18-13) about composite funtion integrals just baffled me. I can follow what the sheet asks, but I feel like I'm doing it wrong; please explain it on monday. Nevermind, I just watched the video we were supposed to watch yesterday, and it answered every question I had.
I was disappointed with my grade on the quiz, but glad to know that the rest of the class did just as poorly. I found anti-derivitives to be quite easy, at least the ones that we've done so far. It took me a bit to remember that y' = dy/dx = d/dxF(x) because of how long ago I put that in my notes. The higher order derivitives are quite simple, they're sort of like F(G(x)), I forget what that's called. Whenever I see things like F'(x)=nx^(n-1), I wonder how someone discovered that that formula worked for every equation. It saves alot of time to use that, as opposed to writing out the four step process, probably would have h
This week was fairly easy, the only problem I had was being unsure of myself when I first started this section. Average rate of change is a fancy way of saying m=(y2-y1)/(x2-x1), and instant rate of change is a long winded formula, but fairly simple once you practice it enough. I'm glad we went over how the slope of the tangent line is the same as instant rate of change, because the book loves to word things in the most confusing way possible. F'(x) was really easy because it was just another name for slope of the tangent line, and graphing it wasn't anything new either. I don't see the point of the "Derivative as a Function Activity" sheet, and was hoping we would go over it more in class. I get that it showed the min and max of the first graph were the same as the zeroes of the second, but I don't quite see how that ties into the rest of this section.
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