Proof videos

Here are the key points to focus on when studying the proof of the Fundamental Theorem of Calculus (FTC):

1. Statement of the FTC:

   • Part 1: If $f$ is continuous on $[a, b]$, then the function $F(x) = \int_a^x f(t) \, dt$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $F'(x) = f(x)$.
   • Part 2: If $f$ is integrable on $[a, b]$ and $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.

2. Understanding Continuity and Integrability:

   • Continuous functions are integrable over closed intervals.
   • The properties of continuous functions, such as the Intermediate Value Theorem, can be useful in the context of the theorem.

3. Construction of $F(x)$:

   • Recognize how $F(x)$ is defined and its role as the accumulation of area under the curve of $f$.

4. Application of the Mean Value Theorem:

   • Use the Mean Value Theorem for integration to show that the average value of $f$ over $[a, x]$ corresponds to a point where $F'(x)$ equals $f(c)$.

5. Differentiation of the Integral:

   • State and apply the Leibniz integral rule for differentiation under the integral sign to connect $F'(x)$ and $f(x)$.

6. Connection Between Derivatives and Integrals:

   • Explore the relationship established by the theorem that differentiating the integral yields the original function.

7. Use of Limits:

   • Emphasize how limits are used in the proofs, specifically when transitioning from discrete sums to integrals.

8. Conclusion and Implications:

   • Discuss the significance of the FTC in linking differentiation and integration and its foundational role in calculus.

9. Examples and Applications:

   • Work through specific examples to illustrate each part of the theorem and practice using it to solve problems.

By focusing on these key points, you'll gain a solid understanding of the proof and implications of the Fundamental Theorem of Calculus.

Here are the key points to grasp when studying the second part of the Fundamental Theorem of Calculus:

1. Link Between Differentiation and Integration: Understand that this part connects the concept of the derivative of a function to the integral of its rate of change.

2. Definite Integral and Anti-Derivative: Recognize that if $F$ is an anti-derivative of $f$ on an interval $[a, b]$, then:

   $$\int_a^b f(x) \, dx = F(b) - F(a)$$

   This shows that the definite integral can be evaluated using anti-derivatives.

3. Continuous Functions: The theorem applies to continuous functions, emphasizing the requirement for continuity in guaranteeing the existence of anti-derivatives.

4. Interpretation of the Integral: Grasp the geometric interpretation, where the definite integral represents the net area under the curve of $f(x)$ from $a$ to $b$.

5. Applications: Recognize practical applications of the theorem in calculating areas, determining total quantities from rates of change, and solving problems in physics and engineering.

6. Fundamental Connection: Acknowledge how this part of the theorem solidifies the relationship between the two fundamental concepts of calculus: differentiation and integration.

By mastering these points, you will develop a strong intuition for the second part of the Fundamental Theorem of Calculus.