So far in the year, everything we have learned has made sense to me. But rolling into these past assignments...yikes. I really don't understand it and it is awful. Finally! The picture I have chosen for the cover of this page is actually a realistic representation of myself. What few problems I do know how to do, I feel like I am just going through the motions, and do I really understand the material? (refer to meme at this moment in time). We have a quiz tomorrow and I am pretty stressed out. I think I am just missing something, something hasn't clicked yet. I am starting to understand the "u" substitution thing for anti deriving composite functions, but the latest stuff has been a little harder for me to grasp. What really confuses me is d/dx vs. dy/dx vs du/dx vs idk any other permutation of any letters of the alphabet. In the book examples they are using different ones to solve ant- differentiation problems and it really confuses me. I have never understood the dependent and independent variable thing and really it's all catching up now. I hope some one can catch me up in all of this madness. I'm sure I will understand it soon, so I will try not to stress about it too much.
Given another calculus exploration paper, we explored how changes to a function relate to its tangent line. We were given the function f(x)=sqrt.x. At the point x=1, the tangent line is y=1/2x+½. We changed the original function, and looked at how this change affected the derivative at the point x=1. The change to the original function was -f(x). The equation of the tangent line at x=1 of this function is -1/2x-½. The change to the function was -f(x). By multiplying the original tangent line by -1, we got the new tangent line as well. The purpose of the first part of this sheet was to allow us to realize that a point’s tangent line will have the same transformation as the function itself.
The second section of the worksheet focused on the derivative of an entire function, as opposed to a single point. Much like the first section, we transformed an original function and studied how how this transformation effected the derivative function. It turns out that the same rules apply to entire derivative functions as they do to single points. We only tested this for square root functions. I have tested this rule on the function x^3 -2x^2+1=f(x). The derivative of f is 3x^2-4x. If we transform the graph of f by a vertical stretch of 3, (3f(x)), it's new derivative becomes 9x^2-12x. Now if we multiply f' by 3, we get the same function. This is useful and saves us an entire step on fishing derivatives of functions. This week was really cool. We had very little homework, and we spent most of our time just learning. The kinds of classes that I like are the ones where you interactively learn things, and it isn't really material based like tons of tests and homework etc. Being a math oriented person, I often feel like I understand the material well, and that having a 45 minute assignment over it is preposterous. I realize that some people are stupid and need lots of homework to help them understand what the heck is going on, but I feel that I am not one of those people. So, I really appreciate that we often don't have homework. And on top of that, I learned just as much as I normally would.
The subject of this week was all about derivatives. I totally understand these and I feel awesome about them. Derivatives are all about the slope of the tangent line at a point in a function, and this is super visual. It really makes a lot of sense taking a graph and drawing the tangent lines on it, and seeing how it all works. I think the whole concept of finding a function just to see the slope of the tangent line any point of another function is a bit arbitrary. While it is incredibly interesting to me, it just seems like something not very important. Graphing is all about x’s and y’s. I don’t understand how suddenly is all about x’s y’s and (f(x+h)-f(x))/h’s. Mr. C knows what he is doing however, and I suppose that I will just trust his judgment. We did a laboratory this week. It focused on the graphs of f’(x) and how they relate to the original. This lab was super helpful and jam-packed with plenty of light bulb moments. When going through the packet, everything seemed to start making sense, like how the f’x graph is negative as f is decreasing and f’x is above the axis when f is increasing. I can now just draw an f’x graph looking at the graph of the original. |