About This Chapter
Optimization in Calculus - Chapter Summary
Optimization in calculus refers to the minimum or maximum values a mathematical function, or the expression of a relationship between input and output, can hold. In this chapter, you'll learn about optimization and differentiation, which calculates how a function can change according to its numerical input. You'll also learn how to optimize simple and complex systems, such as those requiring more advanced calculations or that have a significant number of mathematical constraints.
This chapter also includes a description of related rates, measurements that are used to determine how a quantity is changing in relationship to other quantities. In addition to a general explanation, the instructor will take you through The Draining Tank Problem and show you how to calculate the distance between moving points. At the end of the lesson, you'll learn how to use technological tools to solve problems that include differentiation. When you complete the chapter, you should know how to:
- Use differential and integral calculus to model and solve a variety of problems, including those related to acceleration, mass, work and velocity
- Analyze how technology can be used to solve mathematical problems in the real world
- Illustrate concepts related to differential and integral calculus.
Our easy-to-understand video lessons on optimization in calculus are professionally taught by experienced and knowledgeable instructors. Designed to help you make sense of this complex mathematical subject, each tutorial is accompanied by a transcript, where key concepts and terms are linked to text-based lessons for further information. Additional features include an online self-assessment quiz to help you determine how well you grasped the material. If needed, you can also use the video tags to review specific parts of the presentation, without having to re-watch the entire video.
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
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Other chapters within the TExES Mathematics 7-12: Practice and Study Guide course
- Real Numbers
- Mathematical Models
- Complex Numbers & the Complex Plane
- Number Theory
- Number Patterns
- Functions and Graphs
- Linear Functions
- Quadratic Functions & Polynomials
- Evaluating Piecewise & Composite Functions
- Rational and Radical Functions
- Inequalities and Absolute Values
- Exponentials & Logs
- The Unit Circle
- Trigonometric Functions
- Using a Scientific Calculator for Calculus
- Limits of Functions
- Understanding Rate of Change
- Calculating Derivatives of Functions
- Derivatives and Graphs
- Definite Integrals and Sums
- Indefinite Integrals in Calculus
- Integration Applications in Calculus
- Working with Measurement
- Geometrical Figures
- Introduction to Proofs and Constructions
- Congruence and Similarity
- Conic Sections
- Coordinate Geometry
- Finding Volume, Area & Perimeter
- Descriptive Statistics
- Sampling in Statistics
- Normal Distribution in Statistics
- Inference About a Mean
- Categorical Data
- Regression and Correlation
- Finding Probability
- Probability Distributions and Statistical Inference
- Experiments and Surveys
- Teaching Strategies & Activities for the Math Classroom
- Differentiated Instructional Strategies for the Math Classroom
- Using Student Assessments in the Math Classroom