# Ch 12: Using Derivatives

### About This Chapter

## Using Derivatives - Chapter Summary and Learning Objectives

The Using Derivatives chapter teaches the use of differentiation and related rates. You can learn the steps for graphing maximum and minimum values. The video lessons also teach the rules for solving the population growth problem with the population growth formula. Assess your retention of the material by taking the lesson quizzes.

Video | Objective |
---|---|

Using Monotonicity and Concavity to Analyze Curves | Learn how to evaluate curves using monotonicity and concavity. |

How to Determine Maximum and Minimum Values of a Graph | Learn to understand extrema and solve examples involving maximum and minimum values. |

Using Differentiation to Find Maximum and Minimum Values | Discover how to find the maxima and minima of a function using the properties of a derivative. |

Separation of Variables to Solve System Differential Equations | Learn to solve equations by separating variables. |

Calculating Rate and Exponential Growth: The Population Dynamics Problem | Examine the use of exponential equations to predict population growth. |

Related Rates: The Draining Tank Problem | Learn the steps for solving the draining tank problem. |

Related Rates: The Distance Between Moving Points Problem | Become familiar with problems that determine the distance between moving points and discover how to use variables to solve differential equations. |

### 1. Using Monotonicity & Concavity to Analyze Curves

There are all kinds of graphs, and we have various ways of describing our curves. We have words such as monotonic, concave up, and concave down. Watch this video lesson to learn how to identify these kinds of graphs.

### 2. 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.

### 3. 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.

### 4. Separation of Variables to Solve System Differential Equations

In this lesson, we discuss how to solve some types of differential equations using the separation of variables technique. We'll ponder the dastardly deeds of a mad scientist, using his chemical concoction as an example for how to use separation of variables.

### 5. Calculating Rate and Exponential Growth: The Population Dynamics Problem

You know how the world population keeps increasing? It's increasing faster now than it was 100 or 1,000 years ago. In this lesson, learn how differential equations predict this type of exponential growth.

### 6. Related Rates: The Draining Tank Problem

Grab an empty cup and pour some water into it. In this lesson we will watch how the height of the water changes as we learn about related rates of change and learn how to solve the draining tank problem.

### 7. Related Rates: The Distance Between Moving Points Problem

Remember the classic problem of math horror stories everywhere? You know, where one train leaves Kentucky at 2 p.m. and another leaves Sacramento at 4 p. m.? In this lesson, tame the horror and learn how to solve these problems using differentiation and related rates.

### 8. What is L'Hopital's Rule?

A Swiss mathematician and a French mathematician walk into a bar ... and they walk out with the famous L'Hopital's rule for finding limits. In this lesson, learn what these two mathematicians came up with and how to use it to avoid the limit of zero divided by zero!

### 9. Applying L'Hopital's Rule in Simple Cases

L'Hôpital's rule may have disputed origins, but in this lesson you will use it for finding the limits of a range of functions, from trigonometric to polynomials and for limits of infinity/infinity and 0/0.

### 10. Applying L'Hopital's Rule in Complex Cases

L'Hôpital's rule is great for finding limits, but what happens when you end up with exactly what you started with? Find out how to use L'Hôpital's rule in this and other advanced situations in this lesson.

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### Other Chapters

Other chapters within the AP Calculus AB: Exam Prep course

- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Continuity of Functions
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Computing Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Understanding & Applying Integration Techniques
- Approximating Definite Integrals
- Using a Scientific Calculator for Calculus
- About the AP Calculus Exam
- AP Calculus AB Flashcards