While learning about the Fundamental Thereom of Calculus, I found that deductive learning was more helpful that inductive learning. I prefer deductive learning because I feel like it's more effective for me to hear a lecture, take notes, then do examples. When the process is reversed, I don't understand the material as well. For me, doing examples and then trying to understand the concept doesn't do much because I don't know what I'm doing during the examples. It's like trying to build a machine without the instructions. However if I'm given the instructions first, I kind of know the general idea of what I'm attempting to do. Deductive learning can be extremely boring at times, but in the end I have solid notes and examples that I can use if I get stuck on the homework.
I believe the Fundamental Thereom of Calculus is important because, just like any other fundamental idea, it's the basic building block of what's to come. This is the central idea of which we base more complicated concepts on. In order to get into more complicated problems, it's essential that the Fundamental Thereom of Calculus is understood. The first part of the thereom shows the relationship between the derivative and the integral. It says that the derivative of f(t) on the integral from a-x = f(x). So basically the orginal function equals the derivative. The second part of the thereom shows us how intergrals and anti-derivatives are related. It says that integral f(t) on the interval a-b = the anti-derivative of b - the anti-derivative of a. An example of this would be the integral of sinxdx on the interval 0 to pi. First you'd anti-derive it so you'd get -cosx. Next, you would subtract the anti-derivative of a from the anti-derivative of b so it would be -cos(pi) - -cos(0) which would essentially be 1 + 1 which would equal 2 as your final answer.
I believe the Fundamental Thereom of Calculus is important because, just like any other fundamental idea, it's the basic building block of what's to come. This is the central idea of which we base more complicated concepts on. In order to get into more complicated problems, it's essential that the Fundamental Thereom of Calculus is understood. The first part of the thereom shows the relationship between the derivative and the integral. It says that the derivative of f(t) on the integral from a-x = f(x). So basically the orginal function equals the derivative. The second part of the thereom shows us how intergrals and anti-derivatives are related. It says that integral f(t) on the interval a-b = the anti-derivative of b - the anti-derivative of a. An example of this would be the integral of sinxdx on the interval 0 to pi. First you'd anti-derive it so you'd get -cosx. Next, you would subtract the anti-derivative of a from the anti-derivative of b so it would be -cos(pi) - -cos(0) which would essentially be 1 + 1 which would equal 2 as your final answer.