While learning about this theory, I think I relied mostly on inductive reason. Watching Cresswell prove this theory was a bit like watching someone prove the sky is blue or that water is wet. You don't really understand how it is the way it is, you just know it is. The reason this theorem is so fundamental is because it explains why we learned everything from limits to derivatives. It shows us that the limit is related to the derivative is related to the integral, and all just one giant circle of math ad logic. The notation that we see in this theorem is from chapter 2 when we learned about limits and what they mean, it's the integral notation that informs us of the bounds are. Everything connects and relates and can be used to do and undo everything.
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This week we learned about integrals. We started the week by going to the lab and doing a desmos activity involving the average area of various curves and graphs. This is a continuation of using the RAMs. Doing the activity involved a lot of guessing at first but then after we went back to class and actually took the notes it made a lot of sense. This week continued my journey through derivatives, this week was special because we got to deal with inverse derivatives. Which by themselves weren't too bad. They followed pretty much the same formula as the rest of the units. We did some examples, got all the rules for the sinusoidal functions, and then were set to work on the homework. The next day we worked on exponential derivatives which are even easier than inverse derivatives, so yay! Even the quiz that we took on Monday went pretty good for me! So this week is really going pretty well! Can't wait to see what happens this weekend! Still have a review assignment that I have to work on but this entire chapter has gone by pretty quick and easy. So, hopefully the assignment goes about the same way. The test for this entire month long chapter is Monday so we'll see how that goes. I have a good feeling about it though! I'm kind of sad to see this chapter be done. I got comfortable with what we were doing in the class but I'm sure the next unit will probably relate back to derivatives somehow. Especially, because that's all my friends who took this class in past years seem to remember is how long they spent on derivatives. So, here's to the home stretch and may it go well for me! This week was all about more u- substitution and implicit differentiation. Yay! More derivatives! Haha! When will this unit end? It's getting to be a bit much at this point but I'm getting through. I'm starting to really like doing derivatives, they're nice simple math, with definitive rules and regulations. They're amazing. You start to do u-substitution by identifying the u term, then you find u prime, then you make u prime match the rest of the problem, then once you've done all the necessary math, you substitute the original function back in in the place of u, then you've got your brand new derivative. I personally find this method to be unnecessarily difficult in that face of all the other methods we've learned, but whatever floats your boat. We also worked some more on the chain rule this week, which is probably one of the easier derivative rules so far. I didn't super understand it at the beginning but once we did more practice this week, I think I actually got it! The process for it makes way more sense and it's easier to recognize which function is the inside one and when a problem actually requires you to actually use the chain rule. Doing the multiple chain rules in one problem though can get kind of tedious if you ask, but still the end result looks nice. So this week was pretty low key. We took a quiz on Tuesday about the stuff we learned last week and then we learned about the chain rule. Now this was the first time I had ever heard of the chain rule. Which is defined by f'(g(x))*g'(x). Does not look too bad right? It isn't, at least it was not too bad on the worksheet we got right off the bat, but then when I got into the homework things start to get more complicated and everything just starts to look like alphabet soup. So I took a step back from the homework looked over my notes again, then took another stab at it, and it slowly but sure made a little bit more sense by the end of the homework assignment. I looked ahead at next week's lesson plans and I saw that we're working a little more with the chain rule so I'm hoping that as we work with it more and more I'll understand it enough to do well on the next quiz because grade needs some serious help if I'm gonna make it through this trimester... and the next two after that. This week we started derivatives. Well, I guess not started but definitely got into the meat of them. I was a little intimidated because sometimes when class gets into more advanced parts of the topic I can get really lost really fast but surprisingly enough I'm actually doing really well with derivatives and doing stuff with derivatives. Especially all the rules that go with derivatives. The only problem is, is that the algebra can get really tedious especially with the product and quotient rules being used in the same problem. I was really intimidated by the anti-derivatives, at least until Cresswell explained them. They're not too bad either. Another thing I was intimidated by was the three point five lesson because once you throw in sine and cosine, things can get real messy real quick but I took a quick look at the homework and it doesn't look to bad but I guess I'll find out this week and on Monday. But hopefully it won't be too bad or too long because I have to work all weekend and I really don't want to have to stress about this on Monday, which would be really nice. Who knows? This week end I'm either going to get a ton of stuff done or I'll have a lot to do on Monday.
I added a second movable point in the second graph. This point is point f(b), it moves along the parabola in the same way f(a) does. The set up for my own graph wasn't too different than the original two, the only thing I changed was the f(x) equation. I took my creativity to the max. Using limits, we can use a secant line and eventually turn it into a tangent line.
This week was more limits. Not as many limitations though! So we're making progress! It's still taking me a little bit to get through the homework and everything. I also didn't do great on the quiz this week, so here's hoping that I do better on the test! Also I may or may not be struggling to get the homework done on time but that's just beginning of the school year laziness and procrastination. Also adding in a dash of being super busy and just wanting to sleep. I plan on getting all this done this weekend, so come monday I'll e able to focus on the new work and the test! This week we worked on limits. I did limits during the third tri last year and they're pretty much as easy as I remember them being. It was really nice to be able to finish the work so easily especially after going over the review packet and panicking a little bit. Limits were probably one of the easiest things that I did in precalc, working on them in Calc has been good because it means that I haven't completely forgotten everything from the past year. Those three months and the first week of school really brought into perspective just how many limitations I have, especially in math, but doing the limits helped me to also remember that I am very good at getting over those limitations or making the best out of those limitations. Honestly, that's what I really want to do with this class is overcome my issues with math. So, it's the end of the first week of school. I'm finally a senior, and you would think I'd be more prepared for the challenges that come with it, including, but not limited to, AP Calculus and the math that comes with it. I'm not, I'm really, really not, and I'm kind of ashamed of that fact. In all honesty I've never been great at math. Good enough to get through honors courses, with plenty of help from my peers and teachers, but never quite good enough to make math an easy A. It's weird to be in these advanced math classes and o honestly struggle when my peers and friends breeze through it with little more than a question or two. I'm hoping that once we get into learning the actual new material I'll start to feel more comfortable, but for right now looking over the prerequisite packet I'm scared for the coming year. I'll be able to get through it though. I have in the past, so I can do it now. At least that's what I keep telling myself. We'll see what I say when the AP test comes around. That's blog (and week) number one done, here's to a better year than ever before and a successful math class. |
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January 2018
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