1st Fundamental Theorem of Calculus PROOF | Calculus 1 | jensenmath.ca

JensenMath
16 Dec 202309:01
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TLDRThe video script presents a simplified proof of the first part of the Fundamental Theorem of Calculus, which establishes the relationship between integration and differentiation. The theorem is applicable when the function 'f' is continuous on a closed interval between 'a' and 'b', and the variable 'x' lies within this interval. The key concept is that the integral from 'a' to 'x' of 'f(t)dt' is a function of 'x', representing the area under the curve. The script illustrates this with an interactive example in Desmos, showing how the area changes as 'x' moves. The proof involves taking the derivative of the integral function 'a(x)', which leads to the conclusion that 'a' prime of 'x' equals 'f(x)'. This demonstrates that integrals and derivatives are inverse operations. The mean value theorem for integrals is used to approximate the area under the curve with a rectangle, leading to the final result that the derivative of the integral function is the original function itself. The script ends with a teaser for the second part of the theorem and its applications in calculus.

Takeaways
  • 📚 The Fundamental Theorem of Calculus (FTC) has two parts, and the script focuses on the first part, which relates to the relationship between integration and differentiation.
  • 🎯 The function under consideration, typically denoted as f(T), must be continuous on the closed interval between a and b, where T is a dummy variable.
  • 📉 The FTC states that the integral of f(T) from a to a variable X within the interval [a, b] defines a function F(X), which represents the area under the curve between the lower and upper boundaries.
  • 🔄 The integral can be visualized as the area under the curve that changes with the position of X, and this is why it is defined as a function of X.
  • 📈 The derivative of the integral function F(X), denoted as F'(X), is equal to the original function f(X), indicating that integration and differentiation are inverse operations.
  • 🧮 The proof begins by rewriting F'(X) using the first principles of derivatives, which involves taking the limit as H approaches zero of the difference quotient.
  • 📌 The area function at X + H is the area from a to X + H, and subtracting the area function at X gives the area between X and X + H under the curve.
  • 🟢 The Mean Value Theorem for integrals is used to find a point C between X and X + H such that the area of a rectangle with height f(C) and width H equals the area under the curve between X and X + H.
  • 🔵 As H approaches zero, X + H approaches X, and by the Squeeze Theorem, C also approaches X, leading to the conclusion that F'(X) = f(X).
  • 📊 The script uses Desmos to visually demonstrate how the area under the curve changes with the position of X and how the Mean Value Theorem is applied in the proof.
  • 📘 The proof concludes that F'(X) = f(X), which is a key result of the first part of the FTC, and sets the stage for the second part to be discussed in a subsequent video.
Q & A

  • What is the Fundamental Theorem of Calculus?

    -The Fundamental Theorem of Calculus is a central theorem that connects differentiation and integration, showing that these two operations are essentially the inverse of each other. It states that if a function is continuous on a closed interval [a, b], then the integral of the function from a to some variable x is equal to a function F(x), and the derivative of F(x) is equal to the original function f(x).

  • What conditions must be met for the Fundamental Theorem of Calculus to apply?

    -The function must be continuous on the closed interval between a and b, and the variable x must be within this interval.

  • Why is it important to use a dummy variable T instead of X when dealing with the integral of f(T)?

    -The dummy variable T is used to avoid confusion because x is already being used as the variable of integration within the interval. This allows for clarity when differentiating between the variable of integration and the variable representing the upper limit of integration.

  • How does the area function change as the variable X changes within the interval [a, b]?

    -As X moves along the interval [a, b], the area under the curve between the lower boundary a and the upper boundary X changes, reflecting the dependency of the integral on the position of X.

  • What does the mean value theorem for integrals state?

    -The mean value theorem for integrals states that for a continuous function on a closed interval, there exists a point c in the interval such that the area under the curve from a to x plus h can be exactly represented by a rectangle with height equal to the function's value at c and width equal to h.

  • How does the proof of the Fundamental Theorem of Calculus demonstrate the relationship between the derivative and the integral?

    -The proof shows that the derivative of the integral from a to x of f(t) dt, denoted as F'(x), is equal to f(x). This is done by taking the limit as h approaches zero of the difference quotient, which represents the rate of change of the area function as x changes.

  • What is the significance of the area function being dependent on the variable X?

    -The dependency of the area function on X is significant because it allows for the visualization of the integral as the area under the curve between the lower and upper boundaries. This area changes as X changes, which is why the integral is defined as a function of X.

  • How does the proof use the concept of limits to establish the relationship between the derivative of the integral and the original function?

    -The proof uses the concept of limits to show that as h approaches zero, the difference in the area function between x and x plus h approaches the area under the curve at a specific point c. By applying the squeeze theorem, it is shown that as h approaches zero, c also approaches x, leading to F'(x) being equal to f(x).

  • What is the role of the point c in the proof of the Fundamental Theorem of Calculus?

    -The point c is a value between x and x plus h, and it is chosen such that the area of the rectangle with height f(c) and width h is equal to the area under the curve between x and x plus h. As h approaches zero, c also approaches x, which is crucial for establishing the derivative of the integral.

  • What does the term 'squeeze theorem' refer to in the context of the proof?

    -The squeeze theorem is a method used in calculus to find the limit of a function by 'squeezing' it between two other functions that have the same limit. In the proof, it is used to show that as x plus h approaches x, the value of c, which lies between x and x plus h, also approaches x.

  • Why is the Fundamental Theorem of Calculus considered fundamental in the study of calculus?

    -The Fundamental Theorem of Calculus is considered fundamental because it provides a powerful connection between differentiation and integration, two of the main branches of calculus. It allows for the computation of definite integrals using antiderivatives and is essential for solving a wide range of problems in calculus and its applications.

  • What will be covered in the second part of the Fundamental Theorem of Calculus?

    -The second part of the Fundamental Theorem of Calculus deals with the computation of a definite integral from a to b by using the antiderivative of the integrand. It states that if F is an antiderivative of f on [a, b], then the integral from a to b of f(x) dx is equal to F(b) - F(a).

Outlines
00:00
📚 Introduction to the Fundamental Theorem of Calculus

This paragraph introduces the Fundamental Theorem of Calculus, emphasizing the conditions required for a function, typically denoted as f(T), to be applicable. The function must be continuous on the closed interval [a, b], and the variable X must lie within this interval. The theorem states that under these conditions, the integral from a to X of f(T) dT is a function of X, often denoted as F(X), representing the area under the curve between the lower boundary (a) and the upper boundary (variable X). The integral is visualized as varying with X, and the area under the curve changes accordingly. The key takeaway is that the derivative of F(X), denoted as F'(X), is equal to f(X), signifying that integration and differentiation are inverse operations. The proof provided involves rewriting the derivative from first principles and interpreting the area under the curve between X and X+H, leading to the conclusion that F'(X) = f(X).

05:02
📈 Mean Value Theorem and the Derivative of the Integral

The second paragraph delves into the proof of why the derivative of the integral from a to X of f(T) dT is equal to f(X). It begins by discussing the area function at X+H and the difference between the areas at X+H and X. Using the mean value theorem for integrals, a rectangle is constructed with its height based on a point C between X and X+H, such that the area of this rectangle equals the area under the curve between X and X+H. As H approaches zero, the value of C is also squeezed to approach X. The limit of F(C) as H approaches zero is shown to be equal to f(X), which completes the proof that the derivative of the integral from a to X of f(T) dT is indeed f(X). The paragraph concludes with a teaser for the next video, which will cover the second part of the Fundamental Theorem of Calculus and its applications in calculus problems.

Mindmap
Keywords
💡Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) is a foundational result that links the concept of the definite integral to that of the antiderivative. In the video, it is discussed in two parts, with the first part explaining how the integral of a function can be represented as the antiderivative of the function, and the second part (to be discussed in the next video) will deal with differentiation under the integral sign. The FTC is crucial for understanding calculus as it establishes that differentiation and integration are essentially inverse operations.
💡Continuous Function
A continuous function is one that does not have any breaks or gaps in its graph. In the context of the video, it is a condition that must be met for the function f(T) to be integrated from a to x. The continuity of the function on the closed interval [a, b] is a prerequisite for applying the FTC, ensuring that there are no abrupt changes that would disrupt the integral's value.
💡Dummy Variable
A dummy variable, often denoted as T in the video, is a placeholder used in mathematical expressions, particularly in integrals. It is a variable that is used temporarily and does not affect the final result. In the video, T is used to represent the variable of integration while x is the variable within the interval [a, b], highlighting the distinction between the two.
💡Closed Interval
A closed interval in mathematics is an interval that includes its endpoints. In the video, the function f is said to be continuous on the closed interval [a, b], which means it is continuous from the point 'a' to the point 'b' inclusive. This is an essential condition for applying the FTC, as it ensures the function behaves well over the entire interval of integration.
💡Antiderivative
An antiderivative, also known as an indefinite integral or primitive, is a function whose derivative is equal to the original function. The video discusses how, according to the FTC, the integral of a function from a to x can be represented as an antiderivative of the function evaluated at x. This is a key concept that shows the relationship between integration and differentiation.
💡Derivative
The derivative of a function measures the rate at which the function's output changes with respect to changes in its input. In the video, it is shown that the derivative of the integral function a(x), denoted as a'(x), is equal to f(x). This demonstrates the inverse relationship between integration and differentiation, which is a core aspect of the FTC.
💡Mean Value Theorem for Integrals
The Mean Value Theorem for Integrals states that if a function is continuous on a closed interval, then there exists at least one point c in that interval such that the area under the curve from a to x is equal to the function's value at c times the width of the interval. In the video, this theorem is used to approximate the area under the curve between x and x+h, which is crucial for proving the FTC.
💡Squeeze Theorem
The Squeeze Theorem, also known as the Sandwich Theorem, is a method used to determine the limit of a function by 'squeezing' it between two other functions with known limits. In the video, it is used to show that as h approaches zero, the value of c (from the Mean Value Theorem for Integrals) also approaches x, which helps in proving that the derivative of the integral function a(x) is f(x).
💡Desmos
Desmos is an online graphing calculator that allows users to plot functions and manipulate mathematical expressions in a visual manner. In the video, Desmos is used to illustrate the changing area under the curve as x varies within the interval [a, b], providing a visual aid to understand the concept of integration and how it relates to the area function.
💡Area Function
The area function, denoted as a(x) in the video, represents the integral of a function from a to x. It is a function that changes as x varies within the interval [a, b], reflecting the changing area under the curve of the function f(T). The area function is central to the discussion of the FTC as it is shown to be the antiderivative of the function f(x).
💡Limit
In calculus, a limit is a value that a function or sequence approaches as the input approaches some value. The concept of limits is fundamental in defining continuity, derivatives, and integrals. In the video, limits are used to express the derivative of the area function a(x) as h approaches zero, which is a critical step in the proof of the FTC.
Highlights

The Fundamental Theorem of Calculus is introduced with a simplified explanation.

The function f is assumed to be continuous on the closed interval between a and b.

The variable X is within the interval A and B, and represents the area under the curve.

The integral from a to X of f(T)dT is visualized as the area function of X.

Desmos is used to demonstrate how the area under the curve changes with the position of X.

The integral from a to X of f(T)dT is defined as a function a(X).

The derivative of a(X), denoted as a'(X), is shown to be equal to f(X).

The fundamental theorem states that integrals and derivatives are inverse operations.

A proof is provided to show why the derivative of the integral from a to X is f(X).

The derivative from first principles is used to express a'(X).

The area function at X+H and the difference between areas at X+H and X is interpreted.

The mean value theorem for integrals is used to find a rectangle that represents the area under the curve.

A rectangle with height based on f(C) is used to approximate the area between X and X+H.

The squeeze theorem is applied as H approaches zero, causing C to approach X.

The limit as H approaches zero of f(C) is shown to be equal to f(X).

Desmos is used to visually demonstrate the approach of C to X as H approaches zero.

The proof concludes that a'(X) equals f(X), confirming the fundamental theorem of calculus.

A teaser is given for the next video, which will cover the second part of the fundamental theorem of calculus.

Transcripts

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CalculusIntegralsDerivativesProofTheoremMathematicsEducationSqueeze TheoremMean Value TheoremDesmosInverse Operations