The slope of the area S(x) under the graph of f(x) is again f(x). Indeed, the slope of the area is the speed with which the area increases: (S(x+h) - S(x))/h. But S(x+h) ≈ S(x) + f(x) * h, so (S(x+h) - S(x))/h ≈ f(x).
Hi MHF!
I'm wondering...Can you tell me why a definite integral is the opposite of a derivative? How can an area be the opposite of a slope? Are we going from first to 2nd dimension somehow...? If so, why?
Thanks!
The slope of the area S(x) under the graph of f(x) is again f(x). Indeed, the slope of the area is the speed with which the area increases: (S(x+h) - S(x))/h. But S(x+h) ≈ S(x) + f(x) * h, so (S(x+h) - S(x))/h ≈ f(x).