The main theorems of calculus are the Fundamental Theorem of Calculus (FTC) and the Mean Value Theorem for Derivatives (MVT). Below are their statements, conditions, and examples of applications:

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1. Fundamental Theorem of Calculus (FTC)

Part 1:

If $f(x)$ is continuous on $[a, b]$, and $F(x)$ is defined as the integral of $f(x)$ from $a$ to $x$, then $F(x)$ is differentiable on $(a, b)$, and: $F'(x) = f(x)$ That is, the derivative of the integral of a function is the original function.

Part 2:

If $f(x)$ is continuous on $[a, b]$, and $F(x)$ is any antiderivative of $f(x)$, then: $\int_a^b f(x) \, dx = F(b) - F(a)$

Applications:

1. Area under a curve: Computing the total area between a function $f(x)$ and the x-axis over a given interval.

   • Example: To find the area under $f(x) = x^2$ from $x = 0$ to $x = 2$, $\int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} - 0 = \frac{8}{3}.$

2. Net displacement: If $v(t)$ is the velocity of an object, then $\int_a^b v(t) \, dt$ gives the net change in position over $[a, b]$.

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2. Mean Value Theorem for Derivatives (MVT)

Statement:

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one $c \in (a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$ This means there is at least one point where the instantaneous rate of change equals the average rate of change.

Applications:

1. Justifying physical behaviors: In physics, MVT can be used to argue that at some point in time, a vehicle traveling over a distance must have a speed equal to its average speed.

   • Example: If a car travels 120 miles in 2 hours, then at some point, its speed must have been $\frac{120}{2} = 60 \, \text{mph}$.

2. Root estimation: For functions satisfying the conditions, MVT can help determine intervals where roots exist or verify their uniqueness.

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3. Rolle’s Theorem

A special case of the Mean Value Theorem. If $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that: $f'(c) = 0.$

Applications:

1. Finding extrema: Used in optimization problems to ensure a critical point exists.
   • Example: If $f(x) = x^3 - 3x + 2$ on $[-2, 2]$, $f(-2) = f(2) = 2$, and by Rolle's Theorem, there is a $c$ where $f'(c) = 0$.

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Conditions for the Theorems:

1. Continuity: The function $f(x)$ must be continuous on the specified interval.
2. Differentiability: For MVT and Rolle’s Theorem, $f(x)$ must be differentiable on the open interval $(a, b)$.
3. Endpoints behavior: For FTC Part 2, $f(x)$ must have antiderivatives on $[a, b]$.

Would you like more examples or further explanation?

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Relative Questions:

1. What are the differences between FTC Part 1 and Part 2?
2. How is the Mean Value Theorem used in error estimation?
3. Can you explain the physical meaning of FTC in real-world scenarios?
4. What is the connection between Rolle's Theorem and optimization?
5. How does the continuity condition affect the application of these theorems?

Tip: When solving problems with calculus theorems, always verify the conditions are met before applying the theorem.