# UTPA STEM/CBI Courses/Calculus/Representing Functions as Power Series

Course Title: Calculus II

Lecture Topic: Power Series and Representing Functions as Power Series

Instructor: Dr.Bao-Feng Feng

Institution: UTPA

## Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Understand what is meant by convergence of a power series
• Understand what a power series is and what is meant by radius of convergence
• Understand functions that can be represented as power series
• Sub Objectives- The objectives will require that students be able to:
• Understand the meaning of convergence of a sequence
• Understand the meaning of convergence of a series
• Difficulties- Students may have difficulty:
• Understand why the sum of infinite numbers could be a finite number (convergence)
• Given a function, how can the function be expressed as a power series?
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
• Use for analysis of pendulum problem

Model of Knowledge

• Concept Map
• Limit
• Sequence
• Series
• Function with the integer as independent variables
• Content Priorities
• Enduring Understanding
• A function can be equivalent to a Power series
• Understand the concept of convergence and the radius of the Power series
• Important to Do and Know
• The general conclusion for the power series: the radius of convergence
• The method to determine the radius of convergence for a power series: Ratio Test
• How to modify the nice example 1/(1-x) to get the power series representation for functions related to this nice example.
• Worth Being Familiar with
• The power series is a series with 'x' being a variable
• The power series may be convergent or divergent depending on the values of x

Assessment of Learning

• Formative Assessment
• In Class (groups)
• Assign practice problems right after the example
• Ask for volunteers to come to the board and explain their understanding to the whole class
• Homework (individual)
• Assign homework from the problems in the textbook
• Assign homework from online homework system: Web Work
• Summative Assessment
• Give a quiz right after the completion of the section
• Give one problem during the in-class test

## Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Understand what the convergence of power series means
• Understand what a power series is and what the radius of convergence means
• Understand functions can be represented as power series

The objectives will require that students be able to:

• Find the power series of various functions related to 1/(1-x)
• Find the convergence of these various functions

THE CHALLENGE The derivatives and integrations related to 1/(1-x);

GENERATE IDEAS Interaction between the instructor and students:

• Give some examples of different dimensions yielding different areas
• Determine what to maximize
• Review the formula for computing the area of a rectangle: A=lw

Group activity:

• Use the given information to express the area as a function of one variable
• Determine the constraints
• Solve the problem

MULTIPLE PERSPECTIVES

One group of students will present their solution. The instructor will ask other groups for comments and critique. The instructor will provide comments and present the complete solution, if necessary. Continue with more examples.

RESEARCH & REVISE

The instructor will summarize and present the general method of finding power series of various functions.

Students are required to work on a similar problem in class. Students will work on the quiz and test related to this lecture.

GO PUBLIC

Students in each group will propose an optimization problem and their strategy to solve the problem. Use different types of homework problems that address various situations to reinforce what students have learned.

## Pre-Lesson Quiz

1. Find power series of 1/(1+x)
2. Find power series of 1/(2-x)
3. Find power series of 1/(1-x^2)
4. Find power series of ln(1-x)