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.
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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. http://tutorial.math.lamar.edu/Classes/CalcI/ComputingDefiniteIntegrals.aspx |
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January 2018
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