About This Chapter
Differentiable Functions & Min-Max Problems - Chapter Summary
If your memory of the Mean Value Theorem is a little rusty, brush up on it and Rolle's Theorem here. These videos start off with some basic definitions and give you a firm grasp on the ideas of average and instantaneous rates of change. Cap it off with a discussion of slopes and velocity.
Change may not be easy for us, but finding rates of change will be for you once you review the lessons in this chapter. Whether you need to refresh the calculus lessons you took years ago or learn these topics anew, we're ready to get you on track with these lessons on differentiation and working with minimums and maximums. The topics we cover in these videos include:
- The Mean Value Theorem
- Rolle's Theorem
- Determining maximum and minimum values of a graph
- Finding maximum and minimum values with differentiation
- Rate of change with velocity and slopes
Each lesson comes with an engaging video taught by a math expert. The instructors use real-life scenarios to enhance your understanding and give you good mind-pictures to draw on when you need to remember these functions. Print out the lesson transcript to see the problems worked out in full with key terms set in bold for easy study. Complete the self-assessments to check understanding and determine which areas you may need to return to for review and practice. If you are still struggling, use the instructor contact function to ask for help from our instructors (not available for basic accounts).
1. 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.
2. 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.
3. How to Determine Maximum and Minimum Values of a Graph
What is the highest point on a roller coaster? Most roller coasters have a lot of peaks, but only one is really the highest. In this lesson, learn the difference between the little bumps and the mother of all peaks on your favorite ride.
4. Using Differentiation to Find Maximum and Minimum Values
If you are shot out of a cannon, how do you know when you've reached your maximum height? When walking through a valley, how do you know when you are at the bottom? In this lesson, use the properties of the derivative to find the maxima and minima of a function.
5. 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.
6. 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.
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- Limits of Sequences & Functions
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