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The Fundamental Theorem of Calculus connects the seemingly disparate concepts of derivatives and integrals. It essentially states that differentiation and integration are inverse operations. It’s broken down into two parts:
Part 1: The FTC and Accumulation Functions
Part 1 of the FTC deals with the derivative of an integral. It states that if we have a function $ F(x) $ defined as the integral of another function $ f(t) $ from a constant $ a $ to $ x $ :
$ F(x) = \int_a^x f(t) , dt $
then the derivative of $ F(x) $ is simply $ f(x) $ :
$ \frac{d}{dx} \left[ \int_a^x f(t) , dt \right]] = f(x) $
In simpler terms: The rate of change of the accumulated area under a curve is the height of the curve at that point.
This means that if we have a function representing the accumulation of something (like the total distance traveled given a velocity function), its derivative gives the instantaneous rate of that accumulation (like the instantaneous velocity).
Example:
Let $ f(t) = t^2 $ . Then
$ F(x) = \int_1^x t^2 , dt = \left[ \frac{t^3}{3} \right]]_1^x = \frac{x^3}{3} - \frac{1}{3} $
And $ \frac{dF}{dx} = x^2 = f(x) $ , as the FTC Part 1 states.
Part 2: The FTC and Definite Integrals
Part 2 of the FTC provides a method for evaluating definite integrals. If $ F(x) $ is an antiderivative of $ f(x) $ , then:
$ \int_a^b f(x) , dx = F(b) - F(a) $
In simpler terms: To find the definite integral of a function, find its antiderivative, evaluate it at the upper limit of integration, subtract the evaluation at the lower limit of integration.
Example:
To evaluate $ \int_1^3 x^2 , dx $ , we find the antiderivative of $ x^2 $ , which is $ \frac{x^3}{3} $ . Then:
$ \int_1^3 x^2 , dx = \left[ \frac{x^3}{3} \right]]_1^3 = \frac{3^3}{3} - \frac{1^3}{3} = 9 - \frac{1}{3} = \frac{26}{3} $
Connecting the Two Parts
The two parts of the FTC are intimately related. Part 1 shows that differentiation “undoes” integration, while Part 2 uses this fact to provide a practical method for calculating definite integrals. Part 2 relies on the existence of an antiderivative, which is guaranteed (under certain conditions) by Part 1.
Visual Representation
y = x^2
y = \int_0^x t^2 dt
The graph shows $ y=x^2 $ (blue) and its accumulation function from 0 to x (red). Notice how the slope of the red curve at any point x is equal to the value of the blue curve at that point, illustrating FTC Part 1.
Important Note:
The FTC applies to continuous functions on a closed interval $ [a, b]] $ . While many functions encountered in AP Calculus AB satisfy this condition, it’s crucial to be aware of this limitation.
This rundown provides a concise overview of the Fundamental Theorem of Calculus for AP Calculus AB. Remember to practice applying both parts to various functions to solidify your understanding.