What is the fundamental theorem of calculus?

Answer 1

Stormmedic has it pretty much nailed except that, in part (2) "F is any antiderivative of f" is kinda misleading.

Without getting into a deep and long-winded proof, if f meets the requirement of continuity, then F is guaranteed to exist *and* it is unique (unique up to translation by a constant term)

The most amazing part of the FTC is that the area between some curve (say y = f(x)) and the x azis between two points on the curve (say at x=a and x=b with a

Doug

Answer 2

The almighty Fundamental Theorem of Calculus is usually presented to its victim in two parts:

1.) If f is a function that is continuous on [a, b], then the function g(x) = integral f(t) dt is continuous on [a, b] and differentiable on (a, b), and g'(x) = f(x).

2.) If f is a function that is continuous on [a, b], then integral f(x) dx = F(b) - F(a), where F is any antiderivative of f.

Right. Get that all?

To sum it up...it connects differential and integral calculus. It basically says that the integral is the opposite of the derivative, and that the area under the curve from x=a to x=b is the integral[a,b] of the function.

For example...

f(x) = x^3
f'(x) = 3x^2 (By power rule)
So integral 3x^2 dx = x^3 (By using the integration formula that integral f(x) dx = x^n+1/(n+1)).

Or say you are given the function f(x) = 4x and you want to find the area under the curve from x=3 to x=6. Then by the FTC it would be integral[3,6] 4x dx = 2x^2 evaluated on the interval [3,6]...so its 2(6)^2 - 2(3)^2 = 72 - 18 = 54.

Hope this helps some!

Answer 3

The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other.....google it!