- "In learning about the fundamental theorem of calculus, what type of learning did you primarily rely on, deductive or inductive? Or did you rely on both? Be specific! Also, explain why you believe the fundamental theorem of calculus is so fundamental? In your mind, what does it mean, what are it’s implications, and how does it fit in the context of calculus broadly?”
When I began learning about the fundamental theorem of calculus, I primarily relied on my knowledge of differential calculus. Having a good understanding of differential calculus definitely helped when learning about integral calculus and the fundamental theorem of calculus. I think it learned both inductively and deductively, but it was easier when thinking of the big idea and slowly getting down to the specifics. Learning for my self first and experimenting with the activities was always a new way of learning, especially in math... So to do this with a huge important fundamental theorem was a little intimidating but I believe it can be one of the best ways to learn.
I believe that this theorem for calculus is so important because it solidified a connection between integration and differentiation. It started a mathematical development that propelled the scientific revolution for the next 200 years. Essentially, the FTofC says that the definite integral of a function is a differentiable function of its upper limit of integration and it tells us what the derivative is. Also the definite integral of a continuous function from a to b can be found from any one of the function's anti-derivatives F as the number F(b) - F(a).