Well, do I really have to say this???Its been said over and over, well at least for my blogs... If I asked you what we did today, I'm pretty sure you know what we did... :D right?right? Yes I know what you're thinking "well get on with it already..." heheheh well ok, after saying this a million times, here I go again, to start the class, we had to do questions regarding yesterday's class. We are again faced with problems specifically made so that each problem will have confusing parts so we learn better. The first and second problem was about what we learned yesterday. We had an accumulation function and we needed to complete the table of values for the first problem, sketch the graph of that function, name critical numbers of that function and determine the intervals where that function is increasing. All of this is not new to us because we did this before and all of it is just a review so I will not bother discussing it.
What I would like to talk about is the second part of the fundamental theorem of calculus which states that:
If F(t) is continuous and A(x) is defined as an accumulation function:
the A'(x) = f(x)
To be able to remember this we have to go back to our roots. We learn addition first then subtraction. And we know that they are inverses of each other. Just like addition and subtraction, Differentiation and integration are inverses of each other. Even though its not straight forward because getting the slope of tangent line and calculating bunch of little squares are not really making sense but the because of the first part of the theorem we know it is true. Because we know that the first theorem is true then we understand that if we get the derivative of an integral then the function is just the underlying function; that simple.