Lecture Notes: Fundamental Theorem of Calculus

Learning Objectives

• Describe the Mean Value Theorem for Integrals.
• State and explain the Fundamental Theorem of Calculus, Parts 1 and 2.
• Use the theorem to evaluate derivatives and definite integrals.
• Understand the relationship between differentiation and integration.

Overview

• Previous sections covered definite integrals and area under curves using geometric and Riemann sums.
• New techniques explored in this section simplify evaluating definite integrals.
• Discover foundational relationship between differentiation and integration, developed by Newton and Leibniz.

Mean Value Theorem for Integrals

• Statement: A continuous function on a closed interval \[a, b\] takes on its average value at some point within the interval.

• Formula: If \ f(x) \ is continuous over \[a, b\], there exists \ c \ in \[a, b\] such that:

  \[ f(c) = \frac{1}{b-a} \int_a^b f(x) , dx \]

• Proof:

  • Uses Extreme Value Theorem (function assumes minimum and maximum values).
  • Application of the Intermediate Value Theorem to find \ c \

Fundamental Theorem of Calculus

Part 1: Integrals and Antiderivatives

• Statement: If \ f(x) \ is continuous over an interval \[a, b\], then \ F(x) = \int_a^x f(t) , dt \ implies \ F'(x) = f(x) \.
• The theorem highlights that integration and differentiation are inverse processes.
• Guarantees that every integrable function has an antiderivative.

Example Problems:

• Use Part 1 to find derivatives of integrals.

Part 2: Evaluation Theorem

• Statement: If \ f \ is continuous over \[a, b\], then \ \int_a^b f(x) , dx = F(b) - F(a) \, where \ F(x) \ is an antiderivative of \ f(x) \.
• Simplifies evaluation of a definite integral by finding an antiderivative.

Proof:

• Involves partitioning and applying the Mean Value Theorem.

Example Problems:

• Evaluate definite integrals using the theorem.

Applications and Examples

• Real-world applications such as calculating distances, solving engineering problems, financial modeling, and analyzing motion.
• Various exercises and checkpoints to practice using the theorem.
• Student projects and additional problems to explore concepts deeply, such as analyzing skydiving velocity and planetary motion.

Exercises

• Practical problems using calculus to find integrals, evaluate functions, and explore theoretical applications.
• Engages with diverse scenarios like roller-skating races, skydiving, and planetary orbits.

Key Takeaways

• The Fundamental Theorem of Calculus bridges between differentiation and integration, offering a powerful tool for solving a wide range of mathematical problems.
• Understanding both parts of the theorem is critical for evaluating definite integrals efficiently.
• Exercises and real-world applications illustrate the theorem's versatility and importance in various disciplines.