# 4-2 Mean Value Theorem

## Learning Targets

You will be able to

## Concepts / Definitions

### Mean Value Theorem for Derivatives

If $y = f(x)$ is continuous at every point of the closed intervale $(a,b)$ and *differentiable* at every point of its interior $(a,b)$, then there is at least one point $c$ in $(a, b)$ at which the instantaneous rate of change equals the mean rate of change.

\[f'(c) = \frac{f(b) - f(a)}{b-a}\]
### Increasing Function and Decreasing Function

Let $f$ be a function defined on an interval $I$ and let $x_1$ and $x_2$ be any two points in $I$.

- $f$ increases on $I$ if $x_1 < x_2 = f(x_1) < f(x_2)$
- $f$ decreases on $I$ if $x_1 < x_2 = f(x_1) < f(x_2)$

Note: This is wrong if $x_1$ and $x_2$ can be arbitrarily set.

### Increasing and Decreasing Functions

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$

- If $f’>0$ at each point of $(a,b)$, then $f$ increases on $[a,b]$
- If $f’<0$ at each point of $(a,b)$, then $f$ decreases on $[a,b]$

### Functions with the Same Derivative differ by a constant

If $f’(x) = g’(x)$ at each point on an interval $I$, then there is a constant $C$ such that $f(x) = g(x) + C$ for all $x$ in $I$.

### Antiderivative

A function $F(x)$ is an antiderivative of a function $f(x)$ if $F’(x) = f(x)$ for all $x$ in the domain of $f$. The process is called antidifferentiation.

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