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
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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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
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