Hello all blog viewers! This week was a very successful week in Advanced Placement Calculus. During Monday and Tuesday, we reviewed old lessons on U substitution. It makes a lot more sense why we learned anti derivatives the way we did in the first trimester in class. It was so we could later use that previous knowledge to understand integrals more and have new ways to solve for integrals. Using U substitution allows for one to solve harder integrals that you wouldn't be able to solve by simple anti-derivative rules. During Monday and Tuesday we also were able to reflect upon the previous chapter when Mr. Cresswell handed back chapter 5 tests. I did very well on it and feel very confident with most of the material. Later in the week we started to discuss slope fields. These fields allow for one to view what a graph of a unique function looks like by using the different slopes of the derivative. For example, when using dot paper and graphing the function from the derivative of dy/dx=x+y, you plug in different coordinates to find different slopes. The slope at (1,1)=2 and so on. This field is very useful when trying to envision things one would not be able to do with normal analysis tools. We have a quiz coming up on Monday and I plan to do a few homework assignments over the weekend to prepare for it. Go Math!
Throughout most of calculus, I have used the method of deductive reasoning to understand new concepts . This is because it is much easier for me to understand any specific type of proof or theorem by looking at strict definitions and formulas rather than doing explorations and learning through simpler geometry. however, after being introduced to material by formulas it did help solidify it by using the simple explorations. Specifically when doing the fundamental theorem of calculus exploration I was able to use deductive reasoning throughout the worksheet to complete it and then use inductive reasoning to understand the whole theorem. The best of both worlds right? The word fundamental is defined as forming a necessary base or core. Most of calculus application and fundamental analysis is based of the use of derivatives right? Well what a coincidence! There is a theorem that specifically explains that there is a derivative and anti-derivative for any function. This seems pretty "fundamental" to any application of a derivative. Do you know what the best part of the theorem is? It ties directly into what we are learning about in class this chapter. Integrals. Through analytically analyzing of an integral, one is able to find that the integrand is actually f'(x) of an original function f(x). This is what the whole theorem implies. That there is a derivative and anti-derivative to any function! Go math.
Hello everyone, Long time no talk. This week in AP calculus, the class just continued to learn and practice with integrals (the area underneath the curve). On Tuesday we took an AP style quiz. Although the quiz wasn't worth any actual points, it was a good test to see where I personally stood in the class. I think that I am in good shape going into the end of the school year to take the test, but I have a ton of things that I will need to improve my understanding on and practice more. On Wednesday the class took a quiz over integrals. The quiz was over the material from the past two weeks: Raymond sums, integrals, rules of integrals, you name it. Even though it wasn't the most difficult quiz that we've taken in the class, I took 1/2 of the time taking the quiz trying to figure out the extra credit question. It asked you to prove that on an interval a to b of a linear function, that f(c)=(f(a)+f(b)0/2. After long consideration of how to solve the question, i had to give up and ask Mr. Cresswell how to complete it. It turns out that I was a long way away. To solve it, you must use an equation such as y=cx and then integrate this equation, do some careful algebraic analysis, and then end up with the original proof.
hey all blog Viewers! How has your week been? This was our first week back from winter break and oh it was quite a shock. We started to learn about Integrals and the area underneath the curve. We first did this by using Raymond sums (rectangular approximation method). I wasn't a huge fan of this method since it wasn't exact. During these few days we spent on Raymond sums, I figured out how to use integrals. It is actually very simple. You can think of it this way. If you have a graph of velocity with relation to time and you can find distance by multiplying velocity in time. Which is the area underneath the curve! So basically, you can find position by taking the antiderivative of velocity. You can do this with all functions. It's the same as taking the integral. If you were looking at the integral of the function 5x on the interval (a,b) , you would take the anti derivative of it and get 5/2(x)^2. Them you plug in B and subtract that area from a. This gives you the integral. Very easy with some practice. We also took a look at the mean value theorem. This helps you find an average value of any function without much help from the calculus. We are getting ready to quiz and I feel very confident. Go math!
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