Algebraic Thinking Definition
Curious about the Fundamental Theorem of Calculus – Understanding a geometric definition?
I’ve been a mathematics instructor for 21 years. I’ve only recently discovered Yahoo! Answers & I love this! I’ve been answering questions & asking a few and it just dawned on me that I could ask all of you a question that’s bugged me for years!
So here goes….
Of course I understand the “algebraic” mechanics of the Fund Theorem of Calculus. However the physical or geometric interpretation has always seemed to elude me.
What I’m referring to is this train of thought:
If the Integral represents the AREA under the curve and derivatives represent the SLOPES of lines, how can an area be equal to the difference of slopes?
I’d really like to hear all of your thoughts on this! Fire away! 
I don;t think you are thinking of this correctly. As, when we find area, we are finding the anti-derivaitve not= slope
Here is a physical interpretation, which if I could draw would be easier. here is my attempt(I use this in AP Calculus class lessons for the last 15 years)
Draw the representation of integral of a to b of f(t)
this is the shaded area. Now, when we do derivaitve, we want to know rate of change of the area.
so, what we are saying is how fast is the area changing as the upper limit moves(becomes x)?
So add just a tiny, tiny piece of area, the thinnist piece you can draw. What is that value??? it is the line going from the x-axis to the curve , which is the y-value: Hence the rate of change of the area = the function!
Wish I could draw it.. but do not know how yet.
Let me know if you want more! I’ve been teaching AP Calc for 15 years.
Math60-Positive and Negative Thinking
