The Fundamental Theory of Calculus gives us a shortened version of how integrals relate to derivatives. This theory tells us that every continuous function has a derivative at every point. It also tells us that the derivative of an integral of a function will equal that function, or d/dx S from a to x of f(t)dt = f(x).
In learning about this theory, the type of learning I relied on was inductive reasoning. This is because most of what we looked at was from taking little bits and pieces of information that we knew was true and then turned it into a big broad statement that encompassed it all, like the Fundamental Theory of Calculus. For example, we know how to evaluate integrals, and we know how to find anti-derivatives, and we know that the anti-derivatives of functions will help us to find the integrals and so forth. Putting all of those parts together allow us to make a statement that encompasses it all. I think that that is what makes calculus so interesting; things can be really complicated but with a good understanding of specific details, you can generalize everything to one broad and easily understood statement.
I think that the Fundamental Theory of Calculus is so fundamental because it is the basis of mostly everything that we have been learning in Calculus. Looking back, I realize that a majority of calculus involves derivatives in any sort of shape or form. Really, on the simplest note possible, all the theorem does is tell you that the derivative of the integral of a function from a to b is the same as the function with respect to x. This theorem allows us to see relationships between different graphs and derivatives and how to evaluate their integrals (areas under the curve). To me, this is one of the biggest parts of calculus, which is what makes it so fundamental. What I just stated is exactly what the theorem means. An implication of the theorem are that every continuous function is a derivative of some other function. Other implications are that every continuous function has an anti-derivative, and that integration and differentiation are inverses. It is such a simple way to describe what we have been trying to figure out this entire time. It allows us to see that everything really isn't that complicated, because all you need to do is plug in the x value for the t or whatever the variable is in the original equation. This fits into calculus broadly because it gives us a view of how everything fits together into one, derivatives and graphs and integrals and equations.
In learning about this theory, the type of learning I relied on was inductive reasoning. This is because most of what we looked at was from taking little bits and pieces of information that we knew was true and then turned it into a big broad statement that encompassed it all, like the Fundamental Theory of Calculus. For example, we know how to evaluate integrals, and we know how to find anti-derivatives, and we know that the anti-derivatives of functions will help us to find the integrals and so forth. Putting all of those parts together allow us to make a statement that encompasses it all. I think that that is what makes calculus so interesting; things can be really complicated but with a good understanding of specific details, you can generalize everything to one broad and easily understood statement.
I think that the Fundamental Theory of Calculus is so fundamental because it is the basis of mostly everything that we have been learning in Calculus. Looking back, I realize that a majority of calculus involves derivatives in any sort of shape or form. Really, on the simplest note possible, all the theorem does is tell you that the derivative of the integral of a function from a to b is the same as the function with respect to x. This theorem allows us to see relationships between different graphs and derivatives and how to evaluate their integrals (areas under the curve). To me, this is one of the biggest parts of calculus, which is what makes it so fundamental. What I just stated is exactly what the theorem means. An implication of the theorem are that every continuous function is a derivative of some other function. Other implications are that every continuous function has an anti-derivative, and that integration and differentiation are inverses. It is such a simple way to describe what we have been trying to figure out this entire time. It allows us to see that everything really isn't that complicated, because all you need to do is plug in the x value for the t or whatever the variable is in the original equation. This fits into calculus broadly because it gives us a view of how everything fits together into one, derivatives and graphs and integrals and equations.