This week we studied finding volumes of solids created by rotating a formula on a certain interval around an axis or line, usually the x or y axis. It's a pretty speedy process! We are using the disk method and you basically take pi multiplied by the integral of the formula squared on the said interval. You just have to plug the integral into NINT and multiply the result by pi!! And that's not just pulled out of the air, it comes from the volume formula V= pi*hr^2. I can do the problems pretty quickly and easily which is a fun time! Hooray for understanding math!
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This week in calculus we learned how to find the area between two curves. Like you said, this is fairly intuitive, so it made sense even before the lecture when we tried the exploration. You just subtract the integral of the bottom function (the area under the curve closer to the x axis) from the integral of the upper function (the area under the curve farther from the x axis). I think it actually doesn't matter which function you subtract, because the area between the curves would always be positive so if you end up putting the wrong one first and you get a negative number, just flip it to positive and you are good to go! Our other lesson
In calculus this week we learned a whole bunch of things! First we learned about definite integrals and anti-derivatives and all their rules, including the sum and difference rule, the additivity rule, and the domination rule. The rules are all pretty straightforward and follow common sense, so I feel pretty good about them. The only thing that I have to remember is one part of the sum and difference rule. When you take the integral of the sum or difference of a function and a constant, that constant acts as its own linear function and you have to use the function rule. This was one of the questions I got wrong on the AP practice quiz that we did in class. This week we also learned about finding the average value of a function on a closed interval. We started this by doing a lab. I got the concept of what is above the average equaling the distance of values below the average, but I didn't get to the formula for finding the average value of a function. It makes sense, I just couldn't get it on my own. The final thing we learned this week (and seemingly an incredibly important and crucial concept in calculus!! Our math lives have been leading up to this point!! I am not really sure why exactly it is so important but that's the vibe I was getting.) was how to find the definite integral with anti-derivatives. This one was kind of confusing and hard to follow during the notes but once I was doing it in the homework it just clicked and started making sense. Yay! At the beginning of this week, I had some struggles. We did a worksheet on more chain rule problems, and even though I didn't have any problems with the chain rule, for some reason I was having a lot of problems. First of all I kept taking the chain rule over and over. When I was supposed to just multiply by the derivative of the inside, I starting multiplying the answer by the second derivative. It was a mess and I ended up doing significantly more math than was necessary and making my life hard.
Our new lesson this week was implicit differentiation. Implicit differentiation is when the dependent variable is not isolated and the equation is not a function. You have to derive both sides of the equation, and the derivative of the dependent variable, 'y' in most cases, is a function, so the derivative is dy/dx. It's a great time! And that's only 80% sarcasm! Then, we learned about taking higher order derivatives of implicit equations. So we take the second derivative of the equations. It's pretty straight forward. You just take the derivative and then when you end up with dy/dx, you just substitute in what you just found dy/dx to be!!!!!!! Here's a link to a video explaining implicit differentiation: https://www.khanacademy.org/math/ap-calculus-ab/ab-derivatives-advanced/ab-implicit-diff/v/implicit-differentiation-1 This week we learned more rules for solving derivatives. The week's lesson was pretty much just focused on learning the chain rule. It is a pretty simple concept, but when you add multiple chain rules inside each other and mix in other rules for derivatives, it can get a little confusing and complicated. You just take the derivative of the outer function with keeping the inner function intact, and then you multiply that to the derivative of the inner function. This week we learned more about derivatives!!!!!!! WOOT WOOT!!!! We learned a bunch of rules for solving more complex derivatives including for quotient and multiplicative derivatives, trigonometric derivatives, anti-derivatives, and higher order derivatives. I feel pretty good about this unit so far. I have already learned the quotient rule and the product rule by heart from the homework. The one thing that I have been having difficulty with isn't even conceptual- I just have a very hard time remembering the +C that you have to include on the end of antiderivatives. In a derivative there is no evidence to tell you if there is a constant in the original function. This is because constants do not affect the slope, they affect the position of the function in respect to the y-axis, and since the derivative is the slope of the tangent at any point, it only takes into account the parts of the function that affect slope. For this reason the original function of a derivative could have any constant in the function. It is unclear which, so when doing antiderivatives you have to include +C. I need to find a way to remember the +C.
Below I have included a method of rearranging a problem so that you can avoid using the quotient rule over and over again.
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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