### 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.**

Upon successful completion of this unit, you will be able to:

- state whether a given point on a graph is a global/local maximum/minimum;
- find critical points and extreme values (max/min) of functions by using derivatives;
- determine the values of a function guaranteed to exist by Rolle's Theorem and by the Mean Value Theorem;
- use the graph of to sketch the shape of the graph of ;
- use the values of to sketch the graph of and state whether is increasing or decreasing at a point;
- use the values of to determine the concavity of the graph of ;
- use the graph of to determine if is positive, negative, or zero;
- solve maximum and minimum problems by using derivatives;
- restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer;
- determine asymptotes of a function by using limits;
- determine the values of indeterminate form limits by using derivatives and L'Hopital's Rule.

### 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.

Read this section to learn about maximums, minimums, and extreme values for functions. Work through practice problems 1-5.

Watch this video on how to identify minimum/maximum points of a function.

Work through the odd-numbered problems 1-43. Once you have completed the problem set, check your answers.

### 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.

Read this section to learn about the Mean Value Theorem and its consequences. Work through practice problems 1-3.

Watch this video on local maximums/minimums and proves Rolle's Theorem and the Mean Value Theorem.

Work through the odd-numbered problems 1-35. Once you have completed the problem set, check your answers.

### 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.

Read this section to learn how the first derivative is used to determine the shape of functions. Work through practice problems 1-9.

Watch both parts of this video on the first derivative test.

Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

### 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.

Read this section to learn how the second derivative is used to determine the shape of functions. Work through practice problems 1-9.

Watch this video on the second derivative test. This video describes a way to identify critical points as minima or maxima other than the first derivative test, using the second derivative.

Watch this video, which works through an example of the second derivative test.

Work through the odd-numbered problems 1-17. Once you have completed the problem set, check your answers.

### 4.5: Applied Maximum and Minimum Problems

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

Read this section to learn how to apply previously learned principles to maximum and minimum problems. Work through practice problems 1-3. There is no review for this section; instead, make sure to study the problems carefully to become familiar with applied maximum and minimum problems.

Watch this video on optimization.

Watch this video on optimization.

Work through the odd-numbered problems 1-33. Once you have completed the problem set, check your answers.

### 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.

Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.

Watch this video on finding limits at infinite, and graphing using first and second derivative tests.

Work through the odd-numbered problems 1-59. Once you have completed the problem set, check your answer

### 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.

Read this section to learn how to use and apply L'Hopital's Rule. Work through practice problems 1-3.

Watch this video for an introduction to L'Hopital's Rule.

Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

### Unit 4 Assessment

- Receive a grade
Take this assessment to see how well you understood these concepts.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment