• Unit 4: Derivatives and Graphs

If you are a visual person, you should find this unit very helpful for understanding the concepts of calculus. Displaying concepts graphically allows us to see what we have only been able to imagine so far. Graphs help us visualize ideas that are hard to conceptualize, like limits going to infinity but still being finite, or asymptote lines that approach each other but never quite get there. Graphs can also be used "in reverse" by causing us to question what we see, and causing us to take another look at the mathematics behind what we find.

Completing this unit should take you approximately 9 hours.

• 4.1: Finding Maximums and Minimums

Minimizing and maximizing quantities can be found in both theory and many real-life applications. A company may want to minimize costs and maximize revenue. A teacher may want to minimize the time spent in developing a curriculum. In this section, you will learn how to find maximum and minimum values of functions.

• 4.2: The Mean Value Theorem and Its Consequences

In this section, you will learn about the Mean Value Theorem and why it is important in the study of mathematics.

• 4.3: The First Derivative and the Shape of a Function f(x)

In this section, you will learn about the connection between the shape of a function and the behavior of a function. You will find out how you can use the shape of a function to find out if an extreme point is a maximum or minimum.

• 4.4: The Second Derivative and the Shape of a Function f(x)

Earlier, you learned that the first derivative gives information about the shape of the function. In this section, you will learn more about the type of connection between the second derivative and the shape of a function.

• 4.5: Applied Maximum and Minimum Problems

In this section, you will apply the techniques you learned earlier to applied problems.

• 4.6: Infinite Limits and Asymptotes

In this section, you will learn how to apply infinite limits to asymptotes. You will investigate the long-term behavior of functions.

• 4.7: L'Hopital's Rule

In this section, you will learn how to use and apply L'Hopital's Rule. L'Hopital's Rule is a way to simplify the evaluation of limits.