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This week in Calculus, we learned about continuity. This is a topic that has been covered before in pre-calculus so it was mainly a review and a little more in depth information.
There was an interesting problem on in one of our lab assignments dealing with the continuity of salt and pepper functions that I had a hard time with at first. A picture of the problem is below. When I graphed it, I could see that there where infinite places where it was continuous but I didn't initially know how to define those points. After talking with my peers I started to figure it out. To find where the function is continuous, you have to find the x values at which both the rational and irrational sides of the function are approaching the same y value. Since the rational side is always approaching 0, you must find where sin(π/x) equals zero. I like to think of it in terms of the unit circle. Looking at a sin(x) curve, sin(x)= 0 where x =πk. Since the problem calls for π/x not x, we are looking at where π/x =πk. I solved for x and found that x= 1/k where k is any integer. This week we learned about limits. I took the third trimester of precalculus last year, so most of it was review like last week. I'm pretty sure that limits have something to do with derivatives and calculus so it's a good thing that I feel pretty confident finding them. The one thing I was having a hard time with in class was the sandwich "squeeze" theorem. I can understand that it works and even how the example we were given works (pictured below) but I don't know how to apply it to any other problem or equation other than the one we were shown. I can remember it and copy it over, but if I want to be able to translate it over to any other problem, I'm going to need some extra support to conceptualize it.
All of the other limits information we went over I have down from last year, I think that was one of the units I had a good hold on. When finding limits, you are basically looking at where the function is going from both directions at a certain value of x. If the function is continuous at the value of x, all you have to do is plug in the value of x to find the limit. If the graph has a hole at the x value you are looking at, you can manipulate the function algebraically to effectively remove the hole and plug in the x value for the limit. Since this week is review, there hasn't been any new information to learn, but it's definitely been a slow hike back to where I was last year. With a little brain prodding I remember most of the concepts and how to do most of the problems; the struggle has mostly been getting my brain to work at a decent speed. I saw this when we did our first activity, where we found as many numbers between one and forty using four fours and whichever computations we wanted. Even though I was getting to answers, it felt like I was wading through mud in my mind to get there. One the few things I have no memory from last year of how to solve is finding the domain and range of f(x)=√(x^2-1) without a calculator. I'm familiar with f(x)=√(x-a)+b but with the exponent under the root, I'm unsure. I don't know if everything in the packet is going to be relevant to what we learn and are tested on in AP Calculus, but if that is something we need to know in the future I'm going to need some extra help on how to solve that problem. I think that the algebra portion of the review packet was the easiest and came back the quickest, I feel like algebra just clicks for me. I was surprised that I was doing the geometry pretty easily, as that has been one subject of math I have struggled with in the past. Overall, this week has gone fine, but we'll see how confident I feel a trimester in!
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Josephine Swaney's AP Calc Blog Archives
January 2018
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