# Ch 11: Applications of Derivatives

### About This Chapter

## Applications of Derivatives

If you've been studying calculus and learned what derivatives are, now you can learn what to do with them. How can they be useful in solving math problems and, perhaps more importantly, in real life? Derivatives have a wide variety of useful purposes that you'll learn about in these lessons.

For example, in these lessons on the applications of derivatives, you'll learn about the linearization of functions. Essentially, this boils down to finding the linear approximation of a function at a particular point. You'll delve deeper in a related lesson, where you'll learn about estimating the value of a function using linearization.

Then there's Isaac Newton and the tool that's come to be known as Newton's method. In some situations, you'll want to know the root of an equation. In other words, you'll be trying to find f(*x*) = 0. Newton's method is the key to solving these problems.

Finally, you'll study optimization. In calculus, optimization is a method for determining the largest or smallest value of a function. Let's say that you want to know the largest possible area you can enclose with a fence based on the amount of fencing materials you have. This is where derivatives can save the day. We promised that calculus can be helpful in the real world.

Video | Objectives |
---|---|

Linearization of Functions | Learn how to define the linearization of a function and use linearization to estimate travel time. |

How to Estimate Function Values Using Linearization | Explore how to estimate the value of a function using linearization. |

What Is Newton's Method? | See an explanation and an example of Newton's method at work. |

How to Use Newton's Method to Find Roots of Equations | Learn to use Newton's Method instead of a calculator to find the root of any equation |

Optimization and Differentiation | Learn what optimization means from a mathematical standpoint and how it relates to differentiation. |

Optimizing Simple Systems | Discover how to use a handy five-step formula to tackle to simple optimization problems. |

Optimizing Complex Systems | Learn a five-step process to solve complex optimization problems. |

### 1. Linearization of Functions

Over the river and through the woods to Grandmother's house we go ... Are we there yet? In this lesson, apply linearization to estimate when we will finally get to Grandma's house!

### 2. How to Estimate Function Values Using Linearization

Sometimes landing on Mars isn't that easy. You might need to use linearization to estimate if you will crash into the planet, or miss it entirely. Learn how to do just that in this lesson.

### 3. What is Newton's Method?

Hang gliding can be perilous, especially if you think you might land in a stinky pigpen. In this lesson, find out how Newton's Method might help you determine whether you'll be covered in mud when you land.

### 4. How to Use Newton's Method to Find Roots of Equations

Finding the roots of equations usually requires the use of a calculator. However, in this lesson you'll use Newton's Method to find the root of any equation, even when you can't solve for it explicitly.

### 5. Optimization and Differentiation

In this lesson, you can learn what optimization means from a mathematical standpoint. Using the techniques taught in this lesson, you can use the five steps to optimization to figure out practical things, like how much sleep you need to get before an exam.

### 6. Optimizing Simple Systems

Optimization problems may seem overwhelming, but they can actually be quite simple. In this lesson, learn how to use a handy five-step formula to tackle these daunting problems.

### 7. Optimizing Complex Systems

In this lesson, you'll learn how to follow a five-step process to solve complex optimization problems by visualizing, defining, writing an equation, finding the minimum or maximum, and answering the question.

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### Other Chapters

Other chapters within the Math 104: Calculus course

- Graphing and Functions
- Continuity
- Vectors in Calculus
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Series
- Limits
- Rate of Change
- Calculating Derivatives and Derivative Rules
- Graphing Derivatives and L'Hopital's Rule
- Area Under the Curve and Integrals
- Integration and Integration Techniques
- Integration Applications
- Differential Equations
- Studying for Math 104