# Ch 26: Additional Topics: Theorems, Analysis & Optimizing

### About This Chapter

## 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.

### 1. Cauchy-Riemann Equations: Definition & Examples

Cauchy-Riemann equations are the conditions that are present in complex derivatives. Learn how to identify the derivative of a complex function, and use provided examples to understand where Cauchy-Riemann equations appear.

### 2. Taylor Series for Functions of a Complex Variable

In mathematics, the Taylor series can be used to solve complicated functions. Learn how to use the Taylor series for functions of a complex variable. Review complex differentiability and analytic functions, and understand how to express these using a Taylor series.

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

Newton's method uses information about derivatives to find roots. In this lesson, explore this idea in depth, including using linearization to estimate where functions equal zero.

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

Linearization is the process of using a delta along with partial information, to infer and estimate other information about the equation. See how linearization is useful in estimating speeds, and examples of how to apply this concept to estimate unknown information.

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

Newton's Method of identifying the roots of an equation involves a simple algorithm starting with a guess. Learn the steps involved in systematically approaching the answer, and how organizing this method can help find the roots of equations in no time.

### 6. Optimization and Differentiation

Optimization is the process of applying mathematical principles to real-world problems to identify an ideal, or optimal, outcome. Learn to apply the five steps in optimization: visualizing, definition, writing equations, finding minimum/maximums, and concluding an answer.

### 7. Optimizing Simple Systems

Most optimization problems involving simple systems can be solved using the five-step method. Explore how to solve optimization problems using these five steps: visualizing, defining the problem, writing the equation, finding the minimum or maximum, and answering the question.

### 8. Optimizing Complex Systems

Optimizing complex system starts by using a five-step system that is taught in mathematics; define the problem, write an equation for it, find the minimum or maximum for the problem and answer the questions. Learn more on the five steps to optimizing complex systems.

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### Other Chapters

Other chapters within the GRE Math: Study Guide & Test Prep course

- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Definite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Analytic Geometry in 3-Dimensions
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Determinants & Transformations
- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- GRE Math Flashcards