Additional Topics: Theorems, Analysis & Optimizing - Videos & Lessons

About This Chapter

Expand your knowledge of math concepts by watching the video lessons in this chapter. They were created to teach you about analytic functions and math theorems. You could also learn the rules for solving optimization problems.

Additional Topics - Theorems, Analysis and Optimizing - Chapter Summary

The material in this chapter's online video lessons focuses on the necessary steps for optimizing simple and complex systems. The lessons also emphasize the benefits of visualizing optimization problems in order to solve them. You could learn how to use Newton's Method to solve equations and discover how to estimate speed using linearization. The information in this chapter can also teach you how to:

Provide an example of a Cauchy-Riemann equation
Discuss the applications of basic theorems and analytic functions
Describe the Taylor series and discuss its use
Use singularities, poles and the Laurent series
Define the Residue Theorem and discuss its equation
Describe Newton's Method along with the bisection and iterative methods
Provide examples of linearization
Find the roots of an equation with Newton's Method
Discuss optimization techniques
Explore the five steps for solving optimization problems

The video lessons in this chapter are brief and engaging. They are designed to present topics on analysis, theorems and optimizing in ways that are fun and easy to understand. You can use the timeline to skip between topics in the videos. Your lesson instructors are experienced in the subject matter, so feel free to submit your questions. Consult the corresponding text transcripts if you'd like to clarify any of the information in the video lessons.

8 Lessons in Chapter 26: Additional Topics: Theorems, Analysis & Optimizing

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1. Cauchy-Riemann Equations: Definition & Examples

The derivative of a complex function may or may not exist. To find out, we use test equations. In this lesson, we will derive and use the Cauchy-Riemann equations and then apply these tests to several examples.

2. Taylor Series for Functions of a Complex Variable

This lesson explores how analytic functions can be expressed as Taylor series. We use the concepts of complex differentiable functions and Cauchy-Riemann equations.

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 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.

5. 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.

6. 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.

7. 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.

8. 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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