In learning about the fundamental theorem of calculus, I relied on both inductive and deductive learning. The activity that we worked on before beginning the lesson was a little confusing and unclear to me at first, but after completing it and coming to a conclusion, I think that it contributed to helping me develop a deeper understanding of the theorem. The lesson really helped to clearly show me the concept and rules involved and solidified my learning from the activity. I feel that inductive reasoning, or the type of learning used when we worked on the activity, is a more challenging way of learning a new concept at first, but after coming to a conclusion it helps me to better understand the concept because I came to the conclusion in my own way and in a way that makes sense to me. I think that deductive learning, or the type of learning used when working through the lessons, is a more effective way of learning for me because every concept is covered clearly and I know that I am not missing anything. Sometimes this can be hard if I don't understand the basis or the details of how the conclusions in the lesson are reached, but through deductive learning we can start with the big concepts and analyze them to learn the smaller concepts that build up to them. I believe that a combination of inductive and deductive reasoning is the best way to develop the strongest understanding of a concept.
I think that the fundamental theorem of calculus is so fundamental because it is the basis of everything that we have learned so far in calculus. The theorem tells us that the derivative of the integral from a to x is equal to the integrand, or the function with respect to x. This means that derivatives and integrals, or differentiation and integration, are inverses of one another. The theorem also tells us that every continuous function is a derivative of some other function, and every continuous function has an anti-derivative. I think that the fundamental theorem of calculus fits into the context of calculus broadly because it is based on those basic concepts that are at the center of all calculus. It allows us to see the big picture of how derivatives, integrals, functions, and areas are all related and fit together.
I think that the fundamental theorem of calculus is so fundamental because it is the basis of everything that we have learned so far in calculus. The theorem tells us that the derivative of the integral from a to x is equal to the integrand, or the function with respect to x. This means that derivatives and integrals, or differentiation and integration, are inverses of one another. The theorem also tells us that every continuous function is a derivative of some other function, and every continuous function has an anti-derivative. I think that the fundamental theorem of calculus fits into the context of calculus broadly because it is based on those basic concepts that are at the center of all calculus. It allows us to see the big picture of how derivatives, integrals, functions, and areas are all related and fit together.