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The Mean Value Theorem (MVT) in calculus is a critical concept that links the average rate of change of a function over an interval to the instantaneous rate of change at a specific point. It requires the function to be continuous on a closed interval and differentiable on an open interval. The MVT has practical applications in physics, economics, and everyday life, such as determining a car's speed during a trip or an object's instantaneous acceleration.

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## Definition and Conditions of the Mean Value Theorem

### Definition of the Mean Value Theorem

The Mean Value Theorem establishes a relationship between the average rate of change and the instantaneous rate of change of a function

### Continuity and Differentiability

Continuity

A function is continuous on an interval if it has no breaks, jumps, or holes in its graph over that interval

Differentiability

A function is differentiable on an interval if it has a defined tangent with a specific slope at every point within the interval

### Critical Conditions for the Mean Value Theorem

The Mean Value Theorem requires a function to be both continuous and differentiable on a closed and open interval, respectively

## Application of the Mean Value Theorem

### Calculating Average and Instantaneous Rates of Change

The Mean Value Theorem can be used to find a point where the derivative of a function equals the average rate of change over an interval

### Applications in Physics

The Mean Value Theorem can be applied in physics to analyze kinematics and determine instantaneous acceleration

### Real-World Implications

The Mean Value Theorem has practical applications in fields such as economics and daily experiences, such as driving a car

## Importance and Proof of the Mean Value Theorem

### Understanding Derivatives

The Mean Value Theorem highlights the significance of derivatives in representing the instantaneous rate of change of a function

### Proving the Mean Value Theorem

Leveraging Rolle's Theorem

The Mean Value Theorem can be proven by using Rolle's Theorem, which is a special case of the MVT

Construction of an Auxiliary Function

The proof of the Mean Value Theorem involves creating a new function that incorporates the conditions of continuity and differentiability

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