This week we continued with integrals. We were supposed to be putting all of the pieces of the puzzle together, or something similar to that. What was supposed to be the light-bulb/"Oh this is why we are doing this" moment was when we learned the "Fundamental Theory of Calculus". It gives us the bridge between derivatives and integrals. I really still don't understand it.
I believe that I used the deductive reasoning approach. I believe that this entire theorem is based on on deductive reasoning because we start out with general statements like "every function has a derivative, and an antiderivitive", and "the area under the curve is equal to the antiderivitive of the function times the integral", and then from this, we can conclude that with more specific examples, these principles must be true. I really don't see how we could learn about this theorem inductively, or maybe I just don't understand the two properly.
It's not that I don't understand the theorem, I just don't understand the n
otation. All of the a's, b's, x's, S's, d/x's, d/y's, d/t's. t's ect... It is all super confusing to me. This link helped somewhat.
Looking at the graph and stuff, i feel that I understand the relationship between derivitive and integrals pretty well. The antiderivitive of the furthest point from the origin, b, minus the antiderivitive of the other point gives us the integral of the original function. Earlier we defined that the area under the curve of a velocity function is velocity times time. V x T equals distance of the object. So if we wanted to find the area under the curve, or the distance traveled by the object, we could find the area under the curve of the velocity function, or instead we could use velocities antiderivitve, distance. Once we have the distance function, it makes sense that we would use the upper and lower bounds to determine the value of the interval. Once we have the value of the distance function on a specified interval, we know the integral of the original, velocity function. Isn't this basically what the fundamental theorem of calculus tells us? That we can take the antiderivitve of a function, and then find the difference between the upper and lower bounds of this derivative function, and that this will equal the integral of the original function? This is basically an example of the fundamental theorem of calculus and how it works, just without all of the notation hoopla that is somehow relevant to my understanding of calculus.
I believe that I used the deductive reasoning approach. I believe that this entire theorem is based on on deductive reasoning because we start out with general statements like "every function has a derivative, and an antiderivitive", and "the area under the curve is equal to the antiderivitive of the function times the integral", and then from this, we can conclude that with more specific examples, these principles must be true. I really don't see how we could learn about this theorem inductively, or maybe I just don't understand the two properly.
It's not that I don't understand the theorem, I just don't understand the n
otation. All of the a's, b's, x's, S's, d/x's, d/y's, d/t's. t's ect... It is all super confusing to me. This link helped somewhat.
Looking at the graph and stuff, i feel that I understand the relationship between derivitive and integrals pretty well. The antiderivitive of the furthest point from the origin, b, minus the antiderivitive of the other point gives us the integral of the original function. Earlier we defined that the area under the curve of a velocity function is velocity times time. V x T equals distance of the object. So if we wanted to find the area under the curve, or the distance traveled by the object, we could find the area under the curve of the velocity function, or instead we could use velocities antiderivitve, distance. Once we have the distance function, it makes sense that we would use the upper and lower bounds to determine the value of the interval. Once we have the value of the distance function on a specified interval, we know the integral of the original, velocity function. Isn't this basically what the fundamental theorem of calculus tells us? That we can take the antiderivitve of a function, and then find the difference between the upper and lower bounds of this derivative function, and that this will equal the integral of the original function? This is basically an example of the fundamental theorem of calculus and how it works, just without all of the notation hoopla that is somehow relevant to my understanding of calculus.