So this week we have been working with finding the volumes of objects created by rotating lines around axis or other lines. Basically what must be done to be able to solve for these volumes isn't too complicated, you start by determining whether the shape will be revolved around a line parallel to the X-axis or the Y-axis and account for this by manipulating your equation so that it is either in terms of X or Y. If the line that you are rotating around is below the axis (in the negatives) add whatever negative value to your integral, and if the line is above the axis subtract its value. This being said, I would like to point out that the line of rotation must be parallel to an axis for this to be able to work, and if the line is not parallel to either axis I'm sure that there is a way to determine the volume of the resulting shape even though I am not yet aware of such a method and I'm sure that the method will be rather difficult to learn. Anyway, after determining your line put the equation of your shape into an integral bounded by the ends of the shape and from there I just plug it into NINT.
This week we continued working with u substitution for definite integrals but also moved on to solving indefinite integrals. We also began some work with slope fields. These new topics build off of everything that we have been working on with U substitution and integrals and I expect to go more in depth with both of these topics next week.
This week I understood everything fairly well, although it is hard for me to work backwards with slope fields to find which function that it came from. I think I participated in class this week and I still need to work on slope fields. I would like to explain how to create a slope field for this week's topic: To draw a slope field you simply test your function at different X and Y values and use the value that it gave you as the slope of the line segment through that point. This week we began learning about u substitution with integrals. This section connects our work with Integrals and the work we did last trimester with u substitution. The derivatives made sense to me
While learning the Fundamental Theorem of Calculus I believe that I relied mainly on inductive reasoning. Eventually I was able to recognize the pattern in solving for the derivatives of the integrals.
I think that the fundamental theorem of calculus is fundamental because it kind of came full circle for me, connecting everything that we worked on in the past few weeks to the first derivatives that we were doing in the beginning of the year. It seems like it is kind of a short cut, and includes everything that I have worked on in my high school mathematics career. This week we began chapter 5, The Definite Integral. So far we have covered estimating the area under a graph using the finite areas of rectangles as well as definite integrals and solving by the use of the anti-derivative of the function. These new topics require everything that we have learned in the past about derivatives and anti-derivatives. I imagine that our current topic will lead into more complicated and more accurate ways to find the area under a curve.
This week I think that I have understood how to estimate the area under a curve using rectangles very well. I think that I have a good grasp on our work with integrals, although not completely perfect, and if there is one thing that I need to work on it would be algebraically finding average values using anti-derivatives. I think that this week I had good participation in the learning, more-so than I did before break. The topic I would like to provide an explanation for this week is estimating the area under a curve using Left, Right, or Midpoint Rectangle Approximation Methods, (LRAM, RRAM, MRAM). To begin, start placing rectangles over the graph of the function so that their furthest left point, furthest right point, or midpoint (depending o method being used) leaves the x-axis and meets the graph of the function. The width of the rectangles must be consistent and small enough to provide a semi-accurate representation of the area beneath the curve. Finally, find the area of each rectangle and add their areas. This week we started curve sketching, it puts everything that we had been working on with derivatives to use, for example, finding the second derivative to be able to solve the concavity of a function. Thursday and Friday were spent on a packet that was an introduction to optimization, and I assume that curve sketching, or at least some of the work we did in it, will be used as we continue to work with optimization.
Also, this week I think that I got a good grasp on curve sketching, and there really wasn't anything that I struggled with. I think that I had good participation in class this week, and if there were one thing that I needed to work on more, it would be analytically finding the intercepts of a function. This week I am going to explain how to work through curve sketching problems, how to solve for: Domain- Find the first derivative of the function and solve for critical points, if they do not exist then the domain does not exist on those intervals. Intercepts- can be found using the original function and entering zero for either X or Y depending on which intercept you are solving for. Symmetry-Solve the equation f(-x), if it is equal to f(x) the function has Y-axis symmetry, if it is equal to -f(x) the function has origin symmetry, if it does not have either the function doesn't have symmetry. Asymptotes- can be found by plugging in points growing closer to critical points Intervals of increasing and decreasing and Local extrema- are found using the first derivative and plugging in points to either side of the critical points Intervals of concavity and points of inflection- find the critical points of the second derivative and test points to either side of them. the new critical points are also the inflection points. This week we talked about Implicit Differentiation, Derivatives of Inverse Trig Functions, and Derivatives of Exponential and Logarithmic Functions. These new topics build off of the other topics on derivatives that we have covered, chain rule, Trig functions, Differentition, etc. I would guess that the topics are leading into more that can be done with derivatives.
This week I really understood the derivatives of inverse trig functions lesson but I am not 100% confident with my understanding of the 3.9 derivatives of exponentials and logarithmic functions lesson. I dont think that my participation learning in class this week was much different from any other week. If there was somthing that I needed to work on from this week it would be finding the derivatives of Logarithmic and exponential functions. This week i am going to solve an example from 3.7 Implicit Differentiation: 2y=x^2+sin(y) To begin, take the derivative of 2y, =2dy/dx. then take the derivative of x^2, =2x. and finally take the derivative of sin(y), =cos(y) X dy/dx. Then solve for dy/dx. dy/dx=2x/2-cosy. This week we continued to work with derivatives, anti-derivatives, and the chain rule, most of which was just building on to what we have already been working with. It builds off of section 3.6 that we had been working with last week. I think that I now have a good grasp on the chain rule, but if there is one thing that I need to improve on, it would be solving trig. function derivatives with fractional exponents. This week I solved problem number 35 from page 338 in our book. To begin to solve the problem take the derivative of the outside and multiply the whole of that by the derivative of the inside.
This week we have begun to go through the derivatives of trig functions and applying the chain rule to solving these functions. I believe that learning both of these topics is going to be important in the future because they will be a foundation for the rest of the new topics that we will learn. The new topics, especially the chain rule, connect into what we have been learning. I thought it was cool when we learned that the we had been using the chain rule before in finding the derivatives of functions with exponents. This week I think that I understood the chain rule and rules for derivatives of trig functions well, and I don't think that I had any really serious struggles. I think that I participated well except when I got distracted talking about my new car... and I think that if there was anything that I really needed to work on it would be memorizing the derivatives of Sin, Cos, Tan, etc. To solve a function with the chain rule, first take the derivative of the outside ( f(x) ) leaving the inside alone, and then multiply that function by the derivative of the inside ( g(x) ). So here's a picture of my car btw. Its a 1955 Dodge Royal.
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April 2016
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