The Fundamental Theorem of Calculus connects the two main concepts of calculus: differentiation and integration. It tells us that differentiation and integration are inverse operations. In simple terms, it states that the process of taking the derivative (differentiation) and the process of finding the area under a curve (integration) are related.

Key Points:

• First Part of the Theorem: If you have a continuous function f(x) and you find its integral over a range from a to b, the result of that integration can be used to find the original function F(x), where F'(x) = f(x). In simpler terms, you can recover the original function by differentiating the result of an integral.
• Second Part of the Theorem: If you start with a function F(x) (which is the antiderivative of f(x)), and you integrate f(x) over an interval from a to b, you get F(b) - F(a), which is the total area under the curve between x = a and x = b.

Example 1: The First Part of the Fundamental Theorem of Calculus

Suppose you have a function f(x) = 2x. The integral of f(x) = 2x is F(x) = x² + C, where C is a constant.
Now, differentiate F(x) = x² + C:
F'(x) = 2x, which is the original function f(x).
This shows that taking the integral of a function and then differentiating it brings us back to the original function.

Example 2: The Second Part of the Fundamental Theorem of Calculus

Let's find the area under the curve of the function f(x) = 2x from x = 1 to x = 3.
First, find the antiderivative of f(x) = 2x, which is F(x) = x².
Now, evaluate the definite integral:
∫_1^3 2x dx = F(3) - F(1) = 3² - 1² = 9 - 1 = 8.
This means the area under the curve from x = 1 to x = 3 is 8 square units.

Key Takeaways:

• The Fundamental Theorem of Calculus shows the connection between differentiation and integration.
• The first part of the theorem tells us that differentiating an integral of a function gives us the original function.
• The second part of the theorem allows us to compute the area under a curve (definite integral) by using the antiderivative of the function.

Summary:

The Fundamental Theorem of Calculus is a critical idea in calculus that connects the operations of differentiation and integration. It shows that they are two sides of the same coin. By understanding this theorem, we can calculate areas under curves and understand the relationship between a function and its accumulated quantities.

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