# Ch 6: Rate of Change

### About This Chapter

Motion relates to the rate of change, which is how fast a position changes over time, or in more of a mathematical sense, how fast one variable changes as a function of another variable. To better understand how this works, check out our lessons on the rate of change. This particular chapter will help you learn about the following:

- How motion is related to slopes and tangents
- The mean value theorem
- Rolle's Theorem
- Derivatives, the definitive elements of calculus
- Derivative graphing

Video | Objectives |
---|---|

Velocity and the Rate of Change | Discover how to use motion to define the rate of change. See how fast one variable changes as a function of another variable. |

Slopes and Rate of Change | Learn how slopes and tangents relate to motion and rate of change. |

What Is the Mean Value Theorem? | Get an explanation of the Mean Value Theorem and learn to demonstrate it graphically. |

Rolle's Theorem: A Special Case of the Mean Value Theorem | Explore Rolle's Theorem in terms of the Mean Value Theorem. See Rolle's Theorem as a graph and as an equation. |

Derivatives: The Formal Definition | Learn to formally define the derivative and discover how the derivative is related to the instantaneous rate of change. |

Derivatives: Graphical Representations | Take a graphical look at the definitive element of calculus: the derivative. |

What It Means To Be 'Differentiable' | Define what it means to be differentiable. |

### 1. Velocity and the Rate of Change

Running from your little sister or just window-shopping, your speed is just a measure of how fast you move, or how your position is changing over time. In this lesson, learn about how velocity is a rate of change.

### 2. Slopes and Rate of Change

If you throw a ball straight up, there will be a point when it stops moving for an instant before coming back down. Consider this as we study the rate of change of human cannonballs in this lesson.

### 3. What is the Mean Value Theorem?

Three people set off on a car trip. They all start at the same time and end at the same time. Learn what calculus says about how fast they traveled along the way as you study the Mean Value Theorem in this lesson.

### 4. Rolle's Theorem: A Special Case of the Mean Value Theorem

Super C, the human cannonball, is shot into the air at 35 mph, but his average vertical velocity is zero. In this lesson, you will use Rolle's theorem to explain what this means about Super C's flight.

### 5. Derivatives: The Formal Definition

The derivative defines calculus. In this lesson, learn how the derivative is related to the instantaneous rate of change with Super C, the cannonball man.

### 6. Derivatives: Graphical Representations

Take a graphical look at the definitive element of calculus: the derivative. The slope of a function is the derivative, as you will see in this lesson.

### 7. What It Means To Be 'Differentiable'

Lots of jets can go from zero to 300 mph quickly, but super-jets can do this instantaneously. In this lesson, learn what that means for differentiability.

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### Other Chapters

Other chapters within the Math 104: Calculus course

- Graphing and Functions
- Continuity
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Limits
- Calculating Derivatives and Derivative Rules
- Graphing Derivatives and L'Hopital's Rule
- Applications of Derivatives
- Area Under the Curve and Integrals
- Integration and Integration Techniques
- Integration Applications
- Differential Equations
- Studying for Math 104