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In this article

Part 1: Proof of fundamental theorem of calculus Part 2: Intuition for second part of fundamental theorem of calculus

Part 1: Proof of fundamental theorem of calculus

The first part of the fundamental theorem of calculus tells us that if we define 𝘍(𝘹) to be the definite integral of function ƒ from some constant 𝘢 to 𝘹, then 𝘍 is an antiderivative of ƒ. In other words, 𝘍'(𝘹)=ƒ(𝘹). See why this is so.

When studying the proof of the Fundamental Theorem of Calculus (FTC), focus on the following key points:

Concept of Integration and Differentiation: Understand the relationship between the two processes; integration is the inverse of differentiation.
Continuous Functions: The FTC applies primarily to continuous functions, so grasping the mean value theorem and properties of continuous functions is vital.
Defining the Integral: Know how to define the definite integral of a function over an interval using Riemann sums.
The First Part of the FTC: This states that if $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$ , then:
$\int_a^b f(x) \, dx = F(b) - F(a)$
Understand how to use this to evaluate definite integrals.
The Second Part of the FTC: This establishes that if $f$ is continuous on $[a, b]$ , then the function $F(x) = \int_a^x f(t) \, dt$ is differentiable, and $F'(x) = f(x)$ .
Differentiability and Integrability: Recognize that continuity of $f$ leads to its integrability and the differentiability of the integral function $F$ .
Application of Limits: Be familiar with limits and how they relate to both the definition of the derivative and the properties of definite integrals.
Examples and Applications: Review examples that illustrate the theorem’s application to solve problems involving area under curves and rates of change.
Graphical Intuition: Understanding the geometric interpretation helps in visualizing how the integral and derivative connect.

Mastering these points will provide a solid foundation for comprehending the proof and implications of the Fundamental Theorem of Calculus.

Part 2: Intuition for second part of fundamental theorem of calculus

The second part of the fundamental theorem of calculus tells us that to find the definite integral of a function ƒ from 𝘢 to 𝘣, we need to take an antiderivative of ƒ, call it 𝘍, and calculate 𝘍(𝘣)-𝘍(𝘢). Get some intuition into why this is true.

The second part of the Fundamental Theorem of Calculus connects differentiation and integration, establishing that:

Definition of Antiderivative: If $f$ is continuous on $[a, b]$ , and $F$ is an antiderivative of $f$ (i.e., $F' = f$ ), then the integral of $f$ from $a$ to $b$ can be computed as:
$\int_a^b f(x) \, dx = F(b) - F(a)$
Evaluation of Definite Integrals: This theorem allows for the easy computation of definite integrals by finding an antiderivative and evaluating it at the bounds.
Relationship between Integration and Differentiation: It emphasizes that differentiation and integration are inverse processes. Understanding this relationship helps in solving problems involving rates of change and area under curves.
Continuity Requirement: The continuity of $f$ over the interval ensures the existence of an antiderivative, making the theorem applicable.
Practical Applications: The theorem is foundational for calculating areas, solving problems in physics (like motion), and integrating functions in various fields.

By grasping these key points, students can gain a deeper understanding of how integration and differentiation are interrelated through the Fundamental Theorem of Calculus.

Part 1: Proof of fundamental theorem of calculus

Part 2: Intuition for second part of fundamental theorem of calculus

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