This chapter, and most chapters, I relied on both inductive and deductive learning. I learn the 'big idea' parts of the section by what the teacher does on the board, and the explanation he gives. Afterwards, I look in the book, or notice what a peer is doing to solve specific questions. The process is learned inductively, the big idea is learned deductively. In the end, they all put together what I need to know (at least i hope).
The fundamental theorem of calculus is called such because it is so basic. It simply says that a function F is the same as the derivative of the integral of F(x). There is also the second part which describes how to compute the area under the curve on an interval. Finding these is what makes calculus calculus, which is why the theorem is fundamental to the subject. It has been a while since we had to write blogs for this class I think. Tests, break, etc. Now we have finally moved on to the next chapter where we begin by learning about indefinite integrals. The first thing we did was much the same as the first thing we did with derivatives, which was to help conceptualize what is happening. For derivatives, we saw how a secant line approaches a slope of zero becoming the tangent. For integrals, we calculate the area under a curve with an (theoretically) infinite amount of boxes that we know the area of and adding them up. The method is called the Rectangular approx method, shortened to RAM, so expect a lot of Random access memory jokes in everyone's blogs. Starting with showing us this type of conceptual meaning is important, and probably more helpful than i realize, but I think that we took an excessive amount of time to practice it.
The other part of this chapter is finding the actual area under each curve using integrals. Riemann sums were used to simulate a theoretically infinite number of boxes under the curve, then finding the area of those over an interval. It has been a while since we had to write blogs for this class I think. Tests, break, etc. Now we have finally moved on to the next chapter where we begin by learning about indefinite integrals. The first thing we did was much the same as the first thing we did with derivatives, which was to help conceptualize what is happening. For derivatives, we saw how a secant line approaches a slope of zero becoming the tangent. For integrals, we calculate the area under a curve with an (theoretically) infinite amount of boxes that we know the area of and adding them up. Starting with showing us this type of conceptual meaning is important, and probably more helpful than i realize, but I think that we took an excessive amount of time to practice it.
The different types of RAM is interesting but it's hard to figure out which to use sometimes. I guess it doesn't matter often, but it just is kind of strange to me. Finished chapter 3, including inverse trig functions, and logarithmic derivatives. We also took the test over chapter 3. I always do terribly on tests when I think I know the subject matter, and this week was no exception. I'm really good at getting in my own head, and making stupid mistakes on tests. Maybe I'm just really lazy, maybe I'm the exact opposite, and too focused on certain things to see my mistakes in other areas. The trig functions, and logs aren't much different from the other derivatives we were solving for, just different formulas for solving. The main trend throughout chapter 3 is that the set up is super simple, and then the simplification with algebra is way too long. I'm really not good at seeing ways to simplify things, so it has been difficult. Also, +C. I adopted Eric's strategy, but the main flaw is that I don't look at my pink sheet
Initially, I understood implicit differentiation as soon as I saw it. It just made sense that differentiating y was chain rule outputting the derivative of x. However, as Mr. Cresswell was doing the first example of implicit differentiation of the second derivative on the board, I had no idea what to do. It took me more time than it probably should have to figure out that he was taking the derivative of the first, and plugging it into the dy/dx spots in when taking the second derivative. Once I saw this however, everything seemed so simple. U substitution was another topic this week. Basically if you substitute something for U, and you have the derivative of U, you can write it in terms of du. Count the amount of times "u" was used in that sentence. I just find it interesting how someone can come up with all of these different strategies to solve with substitutions. Taking derivatives of composite functions requires use of the chain rule, in which we learned this week. If each part of the composite function is differentiable, you can take the derivative by using the chain rule in which you take the derivative of y relative to the outside function, then multiply it by the derivative of the inside function. This is especially useful with finding the derivatives of trig functions, because we have not learned any other way to do so yet as all trig functions can be looked at as a composite function. Tan(x^2+x) where the outside function is tan(x), and the inside is x^2+x. Using the chain rule, we get sec^2(x^2+x) * (2x+1).
This assignment was meant to show that when the two points relative to the secant place in a function are the same, it creates a tangent. The first graph, and more so the second graph, show the secant as the relative points approach one another, and how it gets closer and closer to the tangent line. The activity where we found the slopes of multiple points along a function to show how when the difference between two of them is zero, it is the tangent line, we did in class before this assignment was plenty for me to understand the concept. After doing that, the desmos assignment felt unnecessary. I felt this assignment actually didn't help me understand how secants and tangents relate all that much, as it is a fairly simple concept. It is most defiantly necessary to see a visual representation of this concept, however the activity was mostly me trying to figure out how to use desmos more than thinking about the relation between secant and tangent.
Continuity as was described in class feels contradictory. I talked to a few people from class and they all said they felt that it was odd as well. A function can only be non continuous within it's own domain. Therefore even if there is an obvious discontinuity, as a function it is still continuous. So even if there is an infinite discontinuity, it can still be considered a continuous function. What's the point of having discontinuities if it's only within the domain? Is there a specific reason we call a 1/x function continuous? It has a discontinuity, why not just call it a discontinuous function? There are many types of discontinuity.
The other major thing we did this week was salt and pepper graphs. How is this any different than a standard piece wise function? The equation was only continuous where the rational and irrational graphs intersect, these are the only place where the limit is the value of the equation. This week began getting back into the routines of a typical math class, last week being a review week. By this I mean note taking, quick lectures explaining the concepts, and book assignments. Over the many years I have been doing this routine, I have definitely noticed an area where I need improvement. Note taking. I am very unorganized, and taking notes in a manner that I can easily look back at something is something I have yet to get the hang of. Hopefully, with my clean notebook, I can find a system this year, or maybe the "card stock" given to everyone will be more helpful than I expect.
Other than finishing the review packet and the quiz, this week in calculus was mostly working with the limits of functions. Most of limits is review from pre-calculus, but there was one theorem that was new: The Squeeze Theorem. The squeeze refers to finding two functions that the function you want to find is between. If you can prove that the limits of the two functions that flank the function you want are the same, that means that the function must have the same limit. |
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