Okay so here’s the thing about calculus and why I never want to do it. It looks super scary when you have no idea what is going on. That was the case for me with 7.2 and finding the area in between curves. Obviously I’m still playing the catch up game from my marking period from hell and I have never been so traumatized by missing math in my entire life. I literally have been putting off everything in my life because I didn’t want to work on learning it because it was stressful (and now I’m treating my reflective learning blog like a diary for christs sake). Looking back that was ( a normal reaction from me for large amounts of stress ) stupid, because I just had to ask for help.
As demonstrated by this photo I took from a random youtube video, the concept of finding the area between curves is really not that bad. To find the area between a curve (in respect to x) on a set interval it’s just the integral of the top curve minus that of the bottom. This changes as it right minus left when the function is in respect to y. Honestly that part that looked really bad on paper but was actually really chill was finding area using sub regions. It’s just two calculations used together to find a total area of two parts.
Alright, let’s just start off with something that is blatantly obvious by the posting time of this blog as well as the knowledge held by Mr.Cresswell and myself, I have had one heck of a marking period. I think I’ve been in class for literally half of it due to illness and school commitments and now I am paying the price. If you’ve read a lot of this blog before then it’s also pretty obvious that I struggle with learning the material in calculus enough even when I’m there. But I’m trying my best to catch up and I think I’m finally getting somewhere. However one thing that will forever doom me is the ever present fact that math is cumulative, and it’s becoming ever apparent that chapter 7 is going to become somewhat reliant on antiderivatives from chapter 6, and if you are an avid reader of this blog you will know that chapter 6 and I, well to say the least, did not agree with one another. One thing I do find rather agreeable and familiar about chapter 7 is looking at how there is a relationship between derivatives and the functions of position, velocity, acceleration, and jerk. If position is F(x), then Velocity is F’(x), acceleration is F’’(x) and so on.
At last something I understand! I mean that's probably only because it's similar to stuff I learned about in pre calculus, but none the less I see this a progress in my calculus confidence. This week we studied exponential growth and decay. I liked this because it was a topic It’s like a baby step from what I had learned last year. Previously the content surrounding exponential growth and decay utilized a the formula y=a(1+r)^x or y=a(1-r)^x and that was pretty much it. This time around derivatives were incorporated in the form of Separable Differential Equations and The Law of Exponential Change. Separable Differential Equations allow multiple variable equations to be isolated on either side and anti differentiated for each variable. To describe growth the differential equation that is used to describe growth is dy/dx=ky where k is either the growth constant or the decay constant depending on whether it is positive or negative. By breaking it down and separating the variables you can anti differentiate both sides into ln |y| =kt+c after exponating both sides you get |y|=e^kt+c. After the property of exponents is applied it becomes the law of exponential change, y=ysubscript0 e^kt.
This week in calculus we continued our study of integrals by revisiting U substitution. We first covered U substitution briefly during chapter 3 when studying antiderivatives. By using U Substitution we were able to turn complex indefinite integrals into something much easier to evaluate. In definite integrals U Substitution is used to make a more complex integral much easier to evaluate. I am going to be completly honest, I have literally no idea what is going on in calculus right now. I can do the problems but I cannot tell you what they mean or why. I am following a system of steps and I don't now what they mean and I have only a base understanding of what is happening. I wish I didn't feel so lost but I can't seem to figure out what is and isn't clicking in these topics. I understand the process of substitution, it’s the same thing we studied at the beginning of the year. I think the main cause of my distress is that this topic is very closely intertwined with what we learned in chapter 5, which after failing the test last week, is probably where my downfall lies. I need to revisit the topics from chapter 5 and recheck my understanding and maybe everything else in this coming chapter will finally click.
While Working on the Fundamental Theorem of calculus I used both inductive and deductive reasoning. At first while working out the problems I would try piecing together the information I thought I already knew and moved from there, this applied to learning how to apply the theorem and finding antiderivatives. This only worked about 25% of the time seeing as every three in four times I checked the solutions manual I was definitely very wrong. After that I started getting really knit picky and pulling out notes from chapter 3 as well as going over examples in the book over and over again just to make sure I finally understood what I was trying to tackle. The Fundamental Theorem of calculus is so fundamental because it shows the link between derivatives and integrals. Through it we can evaluate integrals exactly instead of trying to compute the limit of riemann sums. Through it was can also calculate area. The notation used connects to when we looked at integrals by themselves in chapter three along with u substitution in order to get a better grasp on antiderivatives.
This week to continue our study of finding the area under a curve we looked at Definite integrals and we revisited antiderivatives. As a class we have already kind of touched on antiderivatives in the past to supplement our derivative studies in chapter 3 with U substitution. This week we used antiderivatives in relation to definite integrals. The anti derivative is like the indefinite integral of the function. In order to find an integral you can use antiderivatives, by using the formula
Where F(x) in the antiderivative of f(x). During our study of definite integrals we also looked at finding the average of a function on a closed interval. We can do this by looking at the definition of the average value which multiplies the whole definite integral by one over b, the right bound of the interval, minus a, the left bound of the interval. When applied this definition is the basis of The Mean Value Theorem for Definite Integrals.
Shout out to Pauls Online Math Notes for like the sixteenth week in a row for helping me to put this into words after not looking at it all weekend and forgetting everything from the lecture.
This week in Calculus we studied related rates. I found this section interesting because I feel like it’s one of the first things we’ve looked at as a class that shows applications of derivatives past more abstract math stuff. Optimization fits into that category for me too. However, Even if I am finding this slightly more interesting than previous subject matter, I am still absolutely shocked and upset that the tedious and painful problems from last chapter have now been put into words and turned into some kind of sick puzzle game. Related rates is an application of implicit differentiation and when I thought math couldn’t get worse, oh boy, was I wrong. Once I got past the puzzle aspect of the solving process related rates problems really are not that bad. I found the list of steps Mr.Cresswell gave us extremely helpful and probably the best way anyone could have started off that lesson. First It’s important to draw a diagram of the situation the problem is explaining a label it. Then find a mathematical model that relates the things that are changing. Once that is done take the derivative of the model. Write out all of the information you have on the value of the variables and plug them in.
I forgot to take pictures of anything relavant so please enjoy this picture of Thomas from our whiteboarding activities:
Oh my god I think we finally finished the derivative unit. How long has it been? 3 weeks? 4 weeks? 6 WEEKS?!?! I feel accomplished. In this time I’ve gone through, what derivatives are, how do figure out if functions are differentiable, basic rules for differentiation including exponents, product and quotient rules, derivatives in trigonometric functions, derivatives in composite functions, derivatives in exponential functions, implicit differentiation, and how to do all of it backwards. Turns out there are all sorts of little math rules and nuances that make it cool but also really, really annoying.
This week we wrapped everything up with a lesson on Implicit differentiation and derivatives of exponential functions. Implicit differentiation kind of ties everything together because it shows how to take the derivative of a function not written explicitly as y equals x. A common application of this is in circular functions where x squared plus y squared equals radius squared. Even though I was a sleepless, miserable wreck this week I feel like I still got a handle on the material really well after I actually finished my homework assignments and had time to process the information. Using the solutions manual to make sure I was doing the problems correctly helped a lot too.
This week we continued our exploration of derivatives, again. After learning about the chain rule last week, we learned the concept of U Substitution, which I guess is the inverse of the chain rule. One undoes what the other does. The Chain rule is the formula to find the derivative of a composition function. The U rule simplifies what the Chain rule does and makes it easier to identify its parts to antiderivative a derivative.
This week I think I have finally pinpointed why I am struggling with math more so this year that I have in previous years. I am a very read to learn kind of person. Due to my generalized anxiety I have a hard time staying focussed on a lesson and usually retain read information better. In years past my math classes have often had pre prepared notes where the mathematical concepts were put into words and taught that way. This year class has been taught mostly through examples written on the board and maybe a sentence rule here or there, and then most of the information was spoken lecture style as opposed to being written out and projected. There is nothing wrong with this, I just need to rethink my note taking system to better reflect the information being spoken and note just writing what is on the board.
This week we continued expanding our knowledge of derivatives. Most of the new content learned this week centered around the chain rule of derivatives. The chain rule is a formula for computing the derivative of the composition of two or more functions. This adds to our understanding of derivatives and expands our knowledge of what we can do with them. Last week we learned how to take the derivative of multiple part functions in the form of the quotient rule and the product rule. By building up an arsenal of these skills in the form of breaking larger functions into smaller pieces we are making it easier to solve problems involving derivatives. I found that this week I struggled more with this topic than in weeks previous. I believe that this is because I often have a hard time identifying when I may need to use more than just one technique to find the derivative of a function, whether it might be needing to use both the quotient and chain rule together of the product and quotient rule together.