This week in AP Calculus we learned ways to find more complex areas under and between curves. There are more areas to be found rather than a simple area between the x-axis and a curve. First of all, you can also find the area between a curve and the y-axis. To do this, you have to isolate the x and treat the y as you would generally treat the x in the integral. You can also split the area into different parts, which we had already did a little of when we found total area, but this time we can split it up however we want whenever we need to. We also learned about finding areas between two different curves, which requires a good understanding of the area you're trying to find, what the integral of both curves mean, and using those integrals with the right bounds correctly to find the area between them. This article explains how to do it.
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This week in AP Calculus, we started to learn about how to apply the integrals and derivatives we had been learning about into word problems. This week has made the idea that the area under the graph is the integral of the function mean more. For example, when looking at a velocity graph, the area under it is not just a function that happens to be the integral/antiderivative, but it's the total amount of distance traveled. This has made it easier to visualize what the integral really means. This lesson helps to explain how position, velocity, and acceleration are connected by derivatives and integrals.
This week in Calculus, we learned about slope fields. Slope fields are a way to represent slopes of a function at every single point on a graph. This makes it easy to visualize the antiderivative of functions with implicit differentiation, as you can plot dy/dx as a slope at each (x,y). Since in the antiderivative each (x,y) has a slope of dy/dx, you can see the general trends of the function family, not just the function, since you have to remember +C. For a more in-depth description on how to create a slope field, click here. I'm not exactly sure what we have coming up next that slope fields will be especially useful for, but it is an easy way to visualize graphs, albeit a little time-consuming. We also started on antiderivatives of separable functions, which will lead into exponential functions, which the slope fields may be useful for, but I don't really know yet. To find the antiderivative of separable functions, you have to put all y's and dy on one side, and all x's and dx on the other side, and find the antiderivative of both.
This week in AP Calculus we learned how to apply derivatives to story problems. This allowed me to see what we were doing in class in a different way, but I don't think that I'd use derivatives in everyday life like the problems would make you think. From maximizing and minimizing stuff, we get to see what the best dimensions are, but in all honesty, you wouldn't need to have the best dimensions in real life. All you would need them for would be if you were an engineer or something and every little bit of money is actually a lot of money. Nevertheless, once you get the hang of it the problems aren't very hard. This page can help to understand the concept, but in essence you have to plug the variables into multiple equations that describe what you want. One has to have two variables and the other has to have those two variables and another variable that's what you want to maximize/minimize. You could also do it with only one equation and two variables, and you could expand that to however many variables you want, provided you have enough equations. To find the maximum or minimum, you would just have to find the derivative and make it equal to zero and finding what the variable is equal to, as well as finding where the variable doesn't exist. Of course it helps to put this into a sign chart to make sure that you really are maximizing or minimizing, depending on what you need.
This week in AP Calculus, we learned about the chain rule in derivatives, used for when there is a composite function. This is found by finding the derivative of the outside and multiplying it by the derivative of the inside. This is kind of a little bit similar to how we treat exponents around exponents, multiplying them together instead of treating them as two different terms, like how we treat the derivatives. We also learned how to do this backwards using substitution. This was easy for me to understand, since my precalculus teacher really drilled subbing in u into our heads, telling us to have it always on our mind in any problem, as it would be used a lot in calculus. It didn't seem like this class had the same kind of ease with u-substitution as I had, so maybe I have a leg up in that respect. Lastly, we learned how to find and use the derivative of y. The fact that it becomes dy/dx makes sense because every time we found the derivative of a function that began with y= it became dy/dx=. It's also easier to see why you need the chain rule when y is squared because y is a function, not a variable.
This week in AP Calculus we learned about the chain rule of finding derivatives. The chain rule is used when there is a composition of functions, which there almost always is. Learning how to do the chain rule has opened up a chasm of possibilities for future problem solving, allowing more and more complex problems to be solved. This is the point where math starts to get fun, because we're taking what we had in the prerequisite packet and merging it with multiple concepts that we've learned so far, creating a sort of puzzle. This makes the more difficult problems, such as those that use a triple chain rule or mix the chain rule with other rules such as the product rule and the quotient rule, the most interesting problems we've done. This article, which takes four whole sections to tell one detail, starts to talk about how we are thinking about math wrong, and that makes it unfun. When we think about it as trying to get the one right answer, it's a chore, but when it's a multi-step process of puzzle solving, it can be engaging to do and fun, as I'm finding what we're doing now.
This week in AP Calculus we made the annoying process of finding a derivative much easier with rules on how to simplify the experience. Considering this, there is absolutely zero reason to do what we have done so far. What's even the point of understanding how the math works when we can just take shortcuts? In fact, what was even the point of all the math classes we took up till now, when this is what it all amounted to? Why not just start here? Obviously, this hypothesis is overdrawn, and it really is important to know why things work instead of just doing them. Mr. Cresswell even said it this week: "In other math classes you may be able to skate by without knowing the details, but that's not going to work in calculus." Knowing why things work makes math so much easier, which is honestly why I feel I'm doing well in calculus. My precalculus teacher was very heavy on understanding and making us apply our knowledge to hard things, and this has made calculus super easy. This page gives an easy chart of different rules for finding derivatives, including a bunch we haven't learned yet. Maybe if I started looking through it, I could get ahead in class, but I don't think that's necessary.
This week in AP Calculus we learned about how to find a derivative. It was a long equation that we used to find the derivative (lim h🡒0(f(x+h)-f(x))/h) and to be honest I don't remember how we got to that exact equation, so I'm going to figure it out again. I remember that it had to do with how we were zooming in on an infinitely small sample to find the instantaneous slope. Since I know that slope = rise/run, I can figure that h is the run, or how far to the right the sample goes. If I think about it that way, it makes sense that the limit would have h🡒0, since we want the run to be as small as possible to find the instantaneous slope. Also, I know that if x is the x value of a point, then f(x) is the y value of that point. So the rise would be the second f(x) minus the first f(x). How do I find what those two points are? Well, I know that I'm trying to find the slope at x, so the first y value must be f(x). Then I want a point that's infinitely close to that point to find the instantaneous slope: x+h! That's how I get lim h🡒0(f(x+h)-f(x))/h. This article also does this proof.
At the beginning of the hour, we did an activity that helped me understand the first graph. It was simply a point that had a changing slope, which we could put in point-slope form and vary the slope. The point was simply the slope put into the function for the parabola, f(x)=.5x^2. One thing I struggled with was getting the second graph to work correctly, and actually, I created the third graph through experimentation before getting the second graph. For the second graph, I used the two point form using a, b, f(a) and f(b) for the x and y values. Secants can be valuable for finding tangents because you can approximate them until they have the same slope and position as a point on the graph.
This week in AP Calculus, we learned about limits that involve infinity, which seems like it would be oxymoronic, since infinity is by definition limitless, but it strangely is very much how math works. It seems weird that a number could get closer and closer to a horizontal line without ever touching it. I think that it's interesting that in order for this to work the slope would have to be changing for eternity, becoming infinitely small fractions. What would the number 1/infinity even look like? Is it possible to know? In that case, what does infinity itself even look like? It's easy to draw a little arrow at the end of a line and say it goes on forever, but to actually think of that line as going on forever is hard to think about. It reminds me of this video I saw a while back, where even though the infinite hotel is filled there is always a way to add more people. This is getting off topic, though. I thought it was interesting that a limit going to infinity meant a horizontal asymptote, which I used in algebra and precalculus, but now I understand more about it than I ever did when I was using it then.
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