- "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 learning about the fundamental theorem of calculus I relied primarily on deductive learning. It all makes sense when it's broken down and proofed so to speak. But, in a way I relied on both methods to truly understand the theorem. Essentially the fundamental theorem of calculus is:
Basically we now have a way to find the definite integral of a function in its entirety. Before, using rectangles and even when using the trapezoidal rule, we were able to come close to the answer, but could not find the exact answer to the definite integral. It was also helpful learning how to use the fnINT function on the calculator.
So, say I was given the function: 6x^2 + 4 on the interval [0, 4]. I would simply plug 6x^2 + 4 in for f(x) and set the lower bound (a) as 0 and the upper bound (b) as (4). Now I would have to take the anti-derivative of the function (add one to the power and divide the co-efficient by that number), so the anti-derivative of 6x^2 + 4 would be: 2x^3 + 4x. Now, plug the upper bound value (b) into the anti-derivative: 2(4)^3 + 4 which equals 132. Then, plug the lower bound value (a) into the anti-derivative: 2(0)^3 + 4 which equals 4. Now subtract F(a) from F(b): 132 - 4 = 128.
The same process can be followed even if (a) is a larger value than (b), you would simply implement one of the rules of finding definite integrals:
So, say I was given the function: 6x^2 + 4 on the interval [0, 4]. I would simply plug 6x^2 + 4 in for f(x) and set the lower bound (a) as 0 and the upper bound (b) as (4). Now I would have to take the anti-derivative of the function (add one to the power and divide the co-efficient by that number), so the anti-derivative of 6x^2 + 4 would be: 2x^3 + 4x. Now, plug the upper bound value (b) into the anti-derivative: 2(4)^3 + 4 which equals 132. Then, plug the lower bound value (a) into the anti-derivative: 2(0)^3 + 4 which equals 4. Now subtract F(a) from F(b): 132 - 4 = 128.
The same process can be followed even if (a) is a larger value than (b), you would simply implement one of the rules of finding definite integrals:
I also utilized other resources like this video to better understand the fundamental theorem of calculus.