This week we learned how to find the areas of intersecting curves and volume of revolved curves. They were both pretty easy. The area of interescting curves was very intutive and I could figured it out myself. For the volume, it wasn't as intutive, and I didn't figure it out myself. It was still very easy to do after it was explained. The end.
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This is the for the blog that was supposed to be done before this weeks. I have no clue what week we are on anymore. We learned how to do u substitution for definite integrals. Pretty much its the same thing as with indefinite integrals except you have to account for the upper and lower bounds. It's easy and all you have to do is plug it into whatever you made the u. Also I thought it was kinda interesting how there's to different ways to do it. We didn't talk about it in class, but on one of the assignments a problem near the end talked about it and showed it. Basically, instead of plugging in the bounds to the u, you just plug the u back in to the formula after you integrated it. This seems more intuitive to me, but I'm guessing it is more work to do usually, or we probably would have learned it.
I think I might have missed a week of a calc blog. I'll probably go back and do that after I post this one, so they might be out of order but oh well.
Besides arguing about what should be on the calc shirt, we learned some more about integrals this week. We also learned out to integrate implicit functions by splitting the derivative. This week was pretty easy overall. Well the prompt asks about inductive vs deductive learning, and I don't know what either means, so I'm going to have to look it up.
Well I'm back. Based on what what I've read, I would say this theorem is more inductive. Basically, it doesn't matter how well it is explained; you still might not get it. You basically just have to keep trying and using calc and eventually it will click and it will make sense why it works. For me, the moment when it all kinda clicked was in the shower. No joke. I don't think I was even thinking about math, but somehow it just clicked, and it made sense why the inverse derivative of an original function was the area of it, and why we use integrals. I would say its fundamental, because half of everything calculus does is based of it (assuming calc is half derivative and have integral stuff). The notation is simple enough and makes sense. One thing I don't really get is how it relates to infinite rectangles. That seems completely unrelated to the integrals to me. We'll its been a while since we've done blog posts so we've learned quite a bit since the last blog. The main thing we've been doing recently is definite integrals. Before we did that, we did the whole concept of using rectangles to find the area under the curve of a function. Using more rectangles is more accurate (obviously). To get a perfect area you'd want to use infinite rectangles, so that's were limits come in. There's this long sum notation thing that equals the integral notation. We did a big long proof which was kind of confusing but kind of made sense. We then learned how to actually solve define integrals by hand which luckily was much easier then the crazy proof. As of right now, I can do all the things we've done so far but I don't completely understand it. I'm basically just following the steps. I don't get how the inverse derivative relates to having infinite amount of rectangles. But I'm pretty sure it will click more as we do it more, which is what usually happens in math if it doesn't make complete sense at first.
We've learned how to do optimization and related rates using calculus. Things are really starting to click for me now. In the past I never really know what dy/dx meant. I would just always use y' instead. Since we started using related rates now I actually understand the whole dy/dx thing.
Were finally almost done with chapter 3. Test is next week for it and I think I'm ready. We learned the exponentiation and natural log rules for derivatives this week too. One thing we learned that I thought was interesting was the derivative of e^x is e^x. I know e is Eulers number, but I have no clue what it's used for besides ln which is just log base e. And even with ln, it had no purpose (that we've learned in high school) other than to get rid of e when simplifying. To solve exponentiation variables questions by had you can just use normal log. Anyways, I was thinking about how e equals 2.71 something. I wonder if e was invented or discovered because its equation that equals itself as a derivative. I think e also is used for some money problems banks use like compounding interest or something. I kind of remember doing it awhile ago if previous math classes but it was so long ago I forgot. So what came first, the chicken or the egg. Was e invented/discovered because its derivative to x equals itself, or is it just chance that it does.
update: I read some of stuff about e and what is and how it works. It makes sense now for interest and stuff. More chain rule!!! What fun!!! Actually it's not bad. This week we did a bunch more practice of that, and learned how to anti derive with the chain rule. Also this is random question I just had now, but I wonder if it's possible to do the limit of h is 0 method to find the derivative of these type of functions. My intuition says yes, but I haven't really thought about it in depth. So far, I actually like calculus. The math behind it I think actually think is cool, like finding the second and third derivatives, and how its acceleration and jerk. Even the math itself isn't bad. The thing I don't like about it so far, is there is so many rules. Like at first I thought now we knew how to find the derivative of any function, and some would just be messier and more math and take longer than others. But it turns out that's not the case. We keep learning more and more rules, which is annoying but there's nothing we can really do about it. But now I'm wondering even at the end of the class will there still be functions we didn't learn how to derive. I hate the concept of like when teachers say about a certain problem, "well on these ones you can't do this formula/method, but don't worry we won't do any like that." It's not the teachers fault, like I'm glad they're not trying to kill us with so much work, but at the same time it annoying because like we can't really find the derivative of everything, just certain equations, catered to our classroom. I don't know if any of that made sense, but yeah lol.
This week we learned the chain rule. I was gone that day so it was a little tricky. I had looked at it the night before to do the book assignment and I learned how to do it, but I didn't really understand it. I didn't get when and why we had to use it. But I eventual figured it all out though. Another little thing that bothers me is I still don't understand the dx/dy thing. It doesn't really matter and I can still do it all, but its just annoying. And we just keep using it more and more for d/dx and du and all these other things I'm probably mis-quoting. Its just do much simpler to do y' or f'(x). Anyways, we also did more derivatives practice and stuff. We did more practice with trig derivatives. I don't know what else to write about. usually I bs stuff and just talk a lot extra about stuff but I really don't want to right now.
This week we did more derivatives. Last week we learned how to do it using the limit 0 of h method, but this week we are learning how to do it faster ways. We also learned how to take the derivative of a derivative. We did it up to 5 levels. At first it sounded completely useless, but of course there was a reason. For position/time graphs, the derivatives can find the velocity, acceleration, and jerk. So far all of this stuff is pretty easy I think. The hardest part is having to memorize a lot of identities and rules. That's one reason I kind of like the limit of h is 0 method. Its just algebra and doing math. You don't have to have all these rules and identities memorized. But I do like that all the rules and things do speed it up a lot. We also did anti derivatives, which supposedly are called integrals Nick said.
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January 2018
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