About This Chapter
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- Find out how to use motion to define the rate of change.
- Learn to relate slopes and tangents to motion and rate of change.
- Describe the Mean Value Theorem and represent it graphically.
- Show how Rolle's Theorem is related to the Mean Value Theorem.
- Understand the derivative and graph it.
- Describe what it means to be differentiable.
1. Velocity and the Rate of Change
The rate of change refers to how one variable changes based on another variable. Learn about velocity and rate of change by reading an example of the velocity of my drive to work. Then, learn about velocity and inconstant slopes.
2. Slopes and Rate of Change
The rate of change is shown through one variable as it changes the function of another variable and can be seen furthermore as location changes as a function of time. Learn more about slopes, rates of change, and the rate of velocity.
3. What is the Mean Value Theorem?
In physics, the Mean Value Theorem is an important aspect of working with rates of change. In this lesson, take a look at the average rate of change of three drivers and review instantaneous rate of change to better understand the Mean Value Theorem.
4. Rolle's Theorem: A Special Case of the Mean Value Theorem
Rolle's theorem is based on the ideas of the mean value theorem, where objects in motion eventually travel at their average velocity speed. Learn the concept behind Rolle's theorem through how it appears in both equations, and graphs.
5. Derivatives: The Formal Definition
The derivative in calculus is the rate of change of a function. In this lesson, explore this definition in greater depth and learn how to write derivatives.
6. Derivatives: Graphical Representations
The derivative of a point can be found using the graph of a function. Learn how to find the tangent of a curve at a point from a graphical representation of a function.
7. What It Means To Be 'Differentiable'
Functions with smooth graphs that allow us to calculate derivatives are considered differentiable. Learn more about the definition of 'differentiable' through examples.
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