This week, I learned about derivatives, the inverse derivatives of trigonometric functions, and a lot of other things. This relates to previous topics because it continues on our understanding of derivatives. This matters because I am sure we are going to continue with derivatives at least until the trimester is over. That means we will have to remember all this information since each test is composed of all the previous topics as well as the current topic. I am sure all of the information we have learned about derivatives will help us with the next unit of information. I understood implicit differentiation really well this week. I struggled with everything at first, but then my grasp of all the concepts strengthened over time. Implicit differentiation is where you find the derivative of y with respect to x, when it is not expressed explicitly. This is y^2 + x^2 = 5, not y = x + 5. You take the derivative of each term like usual, but you tack on a dy/dx when you take the derivative of y. Then you solve and simplify. I thought my participation went very well this week with everything we did. I need to review for the upcoming test on Monday still.
I looked at functions, their derivative, transformations of functions, and the derivative of the transformed function. From this, I can see that when you derive the transformation of a function, it gives you the transformation of the derivative. I think that this concept would apply to all functions. We looked at a radical like root(x), which gives us a derivative of 1/2root(x). Making the function -root(x) simply makes the derivative negative. Looking at a regular function like y=x yields a derivative of 1. Making y=-x gives a derivative of -1. Therefore, my previous statement would appear to be true. I would think the composition of functions does not change this fact. This will work because the composition of functions is just the outside derivative multiplied by the derivative of the inside. The root(x^3+1) will just be 1/2root(x)(3x^2). Therefore, any derivative of a transformation will affect the transformation of the derivative the same way. This week, I learned about taking derivatives with respect to y, as well as with respect to x. I also learned about taking higher order derivatives using these same concepts. All these concepts have been compounded with other concepts, such as taking the antiderivatives and using u substitution. This all goes back to learning about the derivative of a composition of functions. All these concepts could be leading into other ways to figure out derivatives. So far, we've used algebraic skills to determine the derivative. Maybe there are other ways or shortcuts to discovering what the derivative of a function is. I struggled with antiderivatives. I understand how to take the derivative of y with respect to x pretty well. You use rules such as the product or quotient rule and figure out the derivative normally. What you have to know is that the derivative of y is dy/dx. After plugging all this information in and combining like terms, you want to isolate dy/dx. You simplify, and this should give you the derivative. If you're given a point, you can plug it in and find the slope of a point at the tangent line. I thought my participation this week went well during the homework time. I still need to work on reviewing for the quiz that we have on Monday. This week in AP Calculus, I learned more about derivatives and how to solve trigonometric derivatives. This matters because we have been learning about derivatives and now we are continuing with the material. We learned how to take the derivative of sine, cosine, tangent lines, etc. We used the concept of the composition of functions to figure out the derivative. This might lead us into more about derivatives. I understood how to break a function into its parts and multiply to find the derivative. I didn't really struggle with anything this week. To find the derivative of one of these functions, you need to multiply the derivative of the inside of the function by the derivative of the outside of the function. For example, sin 4x would be the derivative of the inside, 4, multiplied by the derivative of the outside, cosine 4x, which yields 4 cos 4x. I thought that my participation this week went well. I participated in all activities, such as answering problems In class and on the whiteboard. I still need to work on getting all the notes from last week because I missed class on Friday. Other than that, I think I understood the material pretty well. the quiz we took last week went pretty well. I didn't do too badly, but I think I could still improve. I need to figure out the material I missed and piece that together with the rest of the week's lessons. I also need to review the concepts we went over on the quiz and figure out what it is that I missed. Using that information, and figuring out the right way to do these problems should help me do well on the upcoming test. All in all, I think I will be fine for this section. The first image depicts root(x) in red and its derivative in green. The second image shows -root(x) in light blue and its derivative in purple. the transformation applied to the function is making it negative. Making the function negative is the common transformation which flips the function over the x-axis. The fact that the derivative of the function follows suit proves that the derivative of a transformation is the transformation of the derivative.This week in AP Calculus, I continued the exploration of derivatives. The class took a quiz on derivatives and is having a test over derivatives next week. I did alright on the derivatives quiz, but I could've done better. Hopefully my knowledge of derivatives will piece together for next week's test, which is composed of all things derivative related. I further explored the rules of limits, including the quotient and product rule, which are not extremely simple. I also took a look at anti-derivatives which are calculated by taking the power, adding one, and dividing the power by the coefficient. This is different from the derivative in that it is calculated by multiplying the coefficient by the power and subtracting one from the power. These extra rules and processes have been given to us to further explain derivatives and determine derivatives in different situations. This could be leading us into more information about rates of change. I really understood how to find derivatives using the power rule. I didn't really struggle with much, but I was gone from school Friday, so I am not sure what we learned. Like I said before, to find a derivative, us the power rule, y'=xn^x-1. I think my participation was good. I participated during whiteboard time and helped an absent student with notes that they missed. I need to work on whatever it is I missed on Friday, since I have an upcoming test. I also need to review a little bit and go over derivatives to refresh my mind. I think I need to look at past concepts, since they compile on every test. I also need to look at the homework I have completed this week. This test should be fun. http://nie.wichitaeagle.com/learning-tools/list.php?m=0&grade=Middle%20School This week in AP Calculus, I learned about derivatives. This is important because it is going to be a big part of calculus. Everyone I know always talks about derivatives that had calculus in previous years. Derivatives help us understand the function. Discussing limits have led us into derivatives. Derivatives are further exploring the detail of a function. I understood how to differentiate a function pretty well this week. I haven't struggled with anything so far pertaining to derivatives. I made some stupid errors on the last limits test, but that's all I've recently struggled with. To differentiate a function you have to plug in f(x+h), then calculate f(x+h) - f(x). Then, you have to make sure all terms have an h and simplify the terms. Divide everything by h, then cancel and simplify. I thought that my participation this week in class was very good. Every time we checked answers I, was participating as well as collaborating in all the labs that we did. I still need to work on derivatives because I just started them, but I am sure that we will start to learn a lot more about them next week. I also need to work on demonstrating my mastery for the limits test because I made a few dumb mistakes that I can easily fix. Other than that, I think that I will be good. So far, I think that I understand everything we know about derivatives, but it could get harder. Hopefully it doesn't though. I haven't done all of the homework yet, but if I don't understand it, I will ask questions about it on Monday when we discuss the questions in class. http://2014matholympics.blogspot.com/?m=1 |
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February 2015
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