Fundamental Theorem of Calculus, Part 1, Visual Proof, Definite Integral - Calculus

The Fundamental Theorem of Calculus, Part 1, states that if a function f is continuous on an interval [a, b], then the function G(x) defined by the integral of f(t) from a to x (G(x) = ∫_a^x f(t)dt) is continuous on [a, b], differentiable on (a, b), and its derivative, G'(x), is equal to f(x). Essentially, taking the derivative of this integral "undoes" the integration, resulting in the original function f.

💡Key Aspects
• Inverse Operations: This part of the theorem demonstrates that differentiation and integration are inverse operations.
• Antiderivative: The integral G(x) defined in this way is an antiderivative of f(x).
• Conditions: The theorem requires the function f to be continuous on the specified interval.
• The Integral as a Function: The definite integral of f(t) from a constant a to a variable x results in a new function G(x), not just a single numerical value.

💡In simpler terms:
If you define a function G(x) by finding the area under the curve of another function f(t) from a fixed point a up to a variable point x, then the rate at which that area changes (the derivative of G(x)) is simply the height of the curve f(x) at that point x.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Fundamental theorem of calculus, part 1, visual proof

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