This week was probably the first week where I was a little lost. I understood the chain rule and anti-differentiation using “u” substitution and felt I understood it well in time for the mini-quiz, but leading up to it I didn’t understand it very well. The composition activity that we did on Monday I felt was a good lead up to the chain rule, but once we actually got to doing the chain rule with the product and quotient rules I found it a little more difficult to grasp. I think that it was more difficult because there’s so much to keep organized. At one point during class Mr. Cresswell told us that if we were having difficulty understanding what we were doing to take a break, because there’s no point in trying to work with a fried brain. I ended up saving the problems that I had left for homework, and I had a moment at home where it all clicked. That moment that clicked I was working on a chain rule problem with a power rule within it and I figured out the pattern that was used when finding the derivative. This week was a little more difficult than previous weeks, but I did eventually figure out things. I’m actually kind of glad that it was a little difficult because we’ve been doing derivatives for a while now and they were starting to get monotonous. The difficulty changed up the speed, so while I’m dreading working with derivatives for much longer I’m glad for the change in difficulty. How I feel when I find out we are still doing derivatives
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This week we started to learned different rules that will allow us to find the derivative of functions. The 4 rules that we learned in 3.3 are a really quick way to find the derivative, and it makes it a lot faster to find the derivative. In learning the quicker way to find the derivative, we were able to use other rules like the quotient and product rules to find the derivatives. We also learned how to find the original equation when we are given the derivative, without having to guess. This is also known as the anti-derivative. The one thing that I need to remember is to add ‘c’ at the end of the function. This ‘c’ is added because the derivative of a constant is zero, so we don’t see it when finding the anti-derivative. We also learned the derivatives/anti derivatives of trig functions. These are pretty easy because they are set values for each function, so you aren’t as likely to get numbers mixed up. They also simplify nicely, which make them kind of fun to do. We aren’t done with the chapter yet, so I can’t imagine what else we could so with derivatives. I’m not going to complain though, because they aren’t very difficult and I’m doing well in the class. This week was the first full week spent on derivatives. On Friday we learned what exactly a derivative was. I realized the reason that we did limits as our first section was because to find a derivative of a function you have to find the limit as h approaches 0. We also talked about continuity because when finding if a function is differentiable it has to be continuous.
I also noticed that back in my week 2 blog post the picture that I used was what we had been using all week to find the derivative of a function, and at the time I had no clue what the thing had meant besides the fact that it said “I like pushing things to the limits.” So the things we learned are building on each other which is pretty cool. Besides that we are building upon things we actually learned a lot of new concepts this week in terms of derivatives (which I had only heard we were going to be spending a lot of time on from past calculus student). I had no clue there were different ways to write derivatives, what differentiability was, or that there were rules for differentiation. I am actually not to overwhelmed by all these new things and found it to be pretty easy things to grasp. I kind of scared for what is going to come next week because you said this is the point where students start to hate you, but I am ready to learn more about what you can do with derivatives. This week we had an activity where we created three GIFs, the three that I created are these: This week we had an activity where we created three GIFs, the three that I created are these: The first activity helped with the generalization represented in the first GIF by showing that the slope of a secant line will narrow down until you get to the point where the tangent line is. You can generalize the tangent’s slope by finding the slope of the secant lines that slowly get closer to the tangent line. The one thing that my whole table struggled with was not knowing exactly what tangent and secant lines were. We figured out how to do the generalization problem once we knew what they were. When making the first graph the one thing I struggled with doing was keeping the line at one point instead of having it move up and down the graph. I figured it out after talking it over with people. One change that had to be made from the first to second graph was to make the one stationary point a point that also moves. Instead of having the points for the slope be (a, f(a)) and (2,2) they became (a, f(a)) and (b, f(b)) for the slope equation m. I also had to write a separate equation for the line. The analysis of secant lines helps to determine the tangent line of a function by allowing for us to slide the line along the function until it is at the point of tangency and then the slope can be calculated and the equation for the line can be written. |
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March 2017
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