This week we started working on chapter six. So far chapter six seems fairly easy to me. A lot if the stuff is almost review. We started with 6.2 which was basically just u substitution again. This came pretty easy to me. There was one tricky part though. We you are finding a definite integral and use u substitution you have to change the bounds. You simply do this be plugging in the original values into what you have for your u equation. The reason you have to do this is because the original bounds are for x but when you use u substitution you are changing the function inside the integral so therefore your bounds must be different. This is a step that I forget every once in awhile so it is an important step to memorize. After 6.2 we worked back to 6.1. In 6.1 we learned about slope fields. With a slope field you graph the slope at every point of a derivative function. The resulting image looks similar to what the antiderivative would look like. These slope fields are helpful for problems that we can't easily antiderive. Some slope fields had patterns that you could pick up on. Others had both an x and a y. That leads into 6.4. In 6.4 we had to antiderive when there was an x and a y in the function. This wasn't as challenging as I origanlly thought it was going to be. Basically all you have to do is get all the y's to one side and then multiply both sides by dx so you can then antiderive both sides. When you antiderive both sides you still aren't done yet. Don't forget that you have to add a plus c to the x side. Most of the problems we had also gave a point. For those you have have to plug the coordinates back in and solve for c. I believe that is all we learned this week.

 

This week we didn't have any school on Monday so we only had four days of actual work. If I remember correctly on Tuesday we just learned some more stuff about the mean value theorem and prepared for the quiz. The quiz we took was an AP style quiz. A lot of people failed it. I got a 9.5 which is still failing but not as bad as some others. The quiz didn't seem too incredibly hard at first but when I got my score back I realized that it was more difficult than I had origanlly thought. The multiple choice questions were the part that really got me. The first two were pretty tricky but after we went over them they made sense. The last part I did mostly correct but I didn't really show my work so I know now that I need to work on that. On Thursday and Friday we started 5.4 and the fundamental theorem of calculus. This fundamental theorem talks about how every continuous function is the derivative of some other function, every continuous function has an anti derivative, and integration and differintiation are inverses of each other. We did a couple proofs to prove these things and make them easier to understand. I sort of understand them but I haven't fully grasped what they mean. All these concepts are pretty new to me so I'm trying to take them all in and comprehend them. That AP style quiz made me realize that the AP is going to be harder than most other tests that I've taken. I usually don't study very hard and still have pretty high scores. Like on the ACT I scored a 27 and didn't study at all but this AP test might be something I study more for. It will be good preparation for college I think.

 

This week was broken up by a snow day so it went by pretty fast. Even though we had the snow day we still managed to get in some quality learning. We started working on chapter 5 this week. 5.1 was about finite sums and using lram, mram, and rram to estimate areas under curves. Basically we just made rectangles and added up all of their sums. This was a pretty easy concept. Next in 5.2 we learned about definite integrals. In 5.2 all we really did was work on changing sigma notation into a definite integral. We also learned that when finding net area all the area above the x axis is positive and all the area below the x axis is negative. In 5.3 we learned about rules for integrals and the mean value theorem. The first rule was that the integral of f(x) on [a,b] is equal to the negative integral of f(x) on [b,a]. The second rule was the integral of f(x) on [a,a] equals zero. The third rule was that if you can factor out a number then you can put that number out front and multiply it back in at the end. There are three other rules but I don't really want to type them all out. They all make sense to me though. We learned about the mean value theorem on Friday and I still haven't fully grasped that but I also haven't done the homework for it yet so that is understandable. The mean value theorem says that there is a point a c on the interval [a,b] that will make the area. The function is 1/b-a times the integral of f(x) on the interval [a,b]. Again I don't fully understand this but I'm assuming that I will get it all figured out soon.

 

This week we worked with related rates and optimization and we also took a quiz and a test. Optimization was kind of hard to understand at first. You have to find two equations in the problem and then solve for one variable and it just got confusing at times. It didn't help that all the problems were story problems either. Some problems were worded confusingly and that also hindered me at times. I did really bad on the quiz because I was gone the day we took it and I waited too long before I retook it and I completely forgot how to optimize. I did get to do a mastery though and that really helped. This was my first mastery that I've done and it actually helped me out a lot. I liked that I had to explain myself so that I really knew what I was doing and I wasn't just button punching on my calculator. Also getting the points back that I missed will be really helpful for my grade. I found related rates to be a lot easier than optimization. For related rates I wrote down a four or five step process that made it really easy to solve the problems. These problems were also story problems so they did get confusing at times. The hardest part about these problems for me was finding and the information they told me in the problem and interpreting what it meant. Once I got that figured out it was mostly pretty easy. The test wasn't very difficult it just took me a really long time. I feel like I got most of the problems right. It's getting hard with wrestling to do my homework every night and to remember to do my blogs but I'm trying. It's also hard to find time for everyone in my group to meet and work on the can project. The project itself isn't very difficult it's just finding the time to actually do all the work is a challenge.

Tri 2

12/8/2013

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This week was pretty interesting. We only learned section 4.4 because it was a pretty hard section but we did learn a lot of stuff. The problems we went over were mainly like real life problems that we might actually have to solve some day. I like these kind of problems because they make you think. I plan on being an engineer so the problems that talked about maximizing volume and stuff I will probably have to use at some point in my life. When I did the homework I understood probably just about half of the problems that were assigned. Obviously this is a bad thing but when we worked together in class it made it a lot easier for me. It helped to have input from people on spots where I had gotten stuck in problems. I like how we work together on problems because sometimes the best way to learn is to teach others. When I can teach things to my classmates it really helps me understand the concept. This week we almost had to find equations hidden in the problem and figure out how they related. Then we had to make the equations only have one variable. This made it possible to find the max or min of that variable. This wasn't too hard of a process I just had to make sure I had the equations right in the first place. I missed the quiz because of a business management field trip. It was a pretty fun field trip and I learned a lot. But now I'm going to have to retake the quiz with Kiegan. I'm feeling pretty confident about this quiz though. I really started to understand the material after we went over things in class. I kind of liked section 4.4. It was pretty fun.

 

This week we learned about the derivatives of trig functions, the derivatives of inverse trig functions, and the derivatives of exponents and logarithms. The derivative of sin(x)=cos(x). The derivative of cos(x)=-sin(x). The derivative of tan(x)=sec(x)^2. The derivative of cot(x)=-csc(x)^2. The derivative of sec(x)=sec(x)•tan(x). The derivative of of csc(x)=csc(x)•cot(x). These are all pretty to apply to the problems because you can still use the chain rule and u substitution and things with them. As for the inverse trig functions I have them all written down in my notes and everything but I don't want to type them out. They are pretty long fractions and it would probably just be a waste of my time to write them all down into my blog. But they are also pretty easy to use. You can basically just plug them in. Sometimes the inverse trig functions can get a little tricky with all the square root stuff and one of them has absolute value and if you don't plug the numbers in cortectly you could make a mistake. As far as the derivatives of exponents the derivative of a^u=a^u•ln(u)•du/dx. This is a pretty easy concept for me. The derivative of e^u=e^u•du/dx. The derivative of ln(u)=1/u•du/dx. All three of these were pretty easy to understand and apply. This whole chapter wasn't very hard it was just knowing how to use what, and when. I got an 18 on the quiz but it was mainly because I made a few small errors and then overthought one problem and made it a lot harder than it really was. I'm glad the test got moved to Monday because I was absent Friday on a field trip. I think the weekend will be good to help me study and give me a little time to finish the review sheet.

Week 8

10/25/2013

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This week we learned about the chain rule, u substitution, implicit differentiation and higher order implicit differentiation. The chain says that when you take the derivative of a composite you first take the derivative of the outside and then multiply that by the derivative of the inside. Mathematically that is (fog)'(x)=f'(g(x))•g'(x). It looks more complicated than it really is. U substitution is for finding antiderivatives of composites. Basically you plug in u for whatever is in the parenthesis then find the derivative and find the derivative of u. I understand how it works it's just hard to explain in words. Once I got the hang of these u substitution problems they were kind of fun to do in the homework. Implicit differentiation is when you have y in the problem such as x+2y=5x-y. For this there is like four steps you have to do. First you have to take the derivative of both sides while treating y as a function. Then collect all the terms with y', which is equal to dy/dx, onto one side of the function. After that you have to factor out the y' and then do the algebra so the function is set equal to y'. For the higher order implicit differentiation problems all you have to do is do implicit differentiation as many times as it asks you to. These concepts didn't seem very hard to me but if you miss one step or make one little mistake then it screws up the entire problem. The quiz was somewhat challenging. Number nine was really a struggle. Overall I think I have a pretty good understanding of the concepts though. Hopefully the test won't be too challenging. There was a problem on the quiz with a circle that was kind of confusing but when the equation was written on the board it made it a lot easier to solve.

Week 7

10/18/2013

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This week we learned about higher order derivatives, the derivatives of trig functions, and the chain rule. The higher order derivatives we talked about had to do with position, velocity, acceleration, and jerk. Basically you start with position and then take the derivative as many times as you need, like once for velocity, twice for acceleration, and three times for jerk. Next we learned about the derivatives of trig functions. These are pretty easy but it's just hard to memorize all of them so I wrote them on my cheat sheet. I will most likely have them memorized soon if we keep working with them, which I'm assuming we will.We took a quiz on 3.3 and 3.5 and I think that I did pretty good. None of the questions seemed really hard to me so hopefully I got a good score. I think that I did. After the quiz we started to learn about the chain rule. Basically the chain rule is

(f o g)'(x)=f'(g(x))•g'(x). In simpler terms you just have to take the derivative of the outside and just ignore the inside, then multiply that by the derivative of the inside. This concept was fairly easy and I understood the homework pretty well. Then on Friday we learned about how to use the chain rule and u substitution to find the antiderivatives of composite functions. This was confusing. I watched the video and then tried to do the worksheet but I struggled. I understand what u is but and I get how to take the derivative of u but then substituting u in is where I get lost. I don't really understand how that works. I am going to try and do the worksheet and see if it sarts to make some more sense. I will probably try and watch the video over again too. That should hopefully help me out a bit.

Week 6

10/11/2013

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This week we continued to learn about derivatives and how they work. We learned about graphing the derivative and what the graph means. When the graph of the derivative is below the x-axis then that means that the original graph is decreasing. When the derivative is above the x-axis the original function is increasing. The zeros of the derivative graph are the maxima and minima of the original graph. This I a concept that I grasped pretty well. After looking at multiple examples it really made sense to me. We also learned about other ways to denote a derivative. The most common is dy/dx which reads "the derivative of y with respect to x." Also there is d/dx which reads "the derivative of f at x." Next we learned about where a derivative fails to exist. Those places are corners, cusps, vertical tangent, and discontinuities. This was a good thing to know because on the quiz there was a question that dealt with several of these things. Next we learned a shortcut to finding the derivative which would have been helpful when we first started learning about derivatives. The short cut is this... f(x)=x^n then f'(x)=xn^(n-1). That makes it really simple to quickly find the derivative of any function. Also we learned about the nderive feature on the calculator. This makes it easy to graph a derivative or find the derivative at a point. Then we took a quiz which I got a 28.5/30 so I was pretty happy. Today we learned about the product and quotient rules for derivatives. So far those don't really make much sense but I haven't had any practice with them yet so I'm sure they will make sense soon. The product rule reads... d/dx(u•v)=dv/dx(u)+du/dx(v). The quotient rule reads... d/dx (u/v)=( du/dx(v)-dv/dx(u))/v^2. That one is pretty complicated and doesn't make much sense yet. Finally we learned about higher order derivatives which is basically just taking the derivative of a derivative. It's pretty easy. Overall this was a pretty productive week I would say.

 

This week we dove into chapter 3 and derivatives. First we learned what a derivative was. It's the slope of any given on a function. We learned about taking the slope of tangent lines. We learned that to find the derivative at a point you have to use the limit as h approaches 0 of f(a+h)-f(a)/h, where a is any given x value. To find the derivative of a function you have to use the limit as h approaches 0 of f(x+h)-f(x)/h. We did a lot or practice with this and we did several worksheets to go with it too. There was a four step process to go along with this. First find f(x+h). Then write f(x+h)-f(x) and simplify. Next find f(x+h)-f(x)/h and simplify. Finally find the limit as h approaches 0 of f(x+h)-f(x)/h. That will give you the derivative of the function. This week I understood pretty much everything. At first things were a little sketchy but once I figured it all out it got a lot easier. One thing I struggled with a little bit was understanding the graphs of the derivatives. It was hard to understand what they meant but after awhile I started to figure it out. Also we had to cube x+h quite a few times, and luckily Ishan reminded me and Kiegan of Pascal's triangle and that made it much easier to do the cubing. I think that this week was pretty important because we will probably have to use derivatives in the future. It's kind of a calculus thing. And limits came up again this chapter so I'm starting to see how all the things we learn will kind of connect. Math class is kind of like a toolbox that we keep adding tools to every week.