UTPA STEM/CBI Courses/Calculus/Taylor Series

Course Title: Calculus II

Lecture Topic: Taylor Series

Instructor: Barnabas Bede

Institution: UTPA

Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
  • Use Taylor series to calculate numerical values of a function

• Sub Objectives- The objectives will require that students be able to:
  • Calculate the coefficients of the Taylor series
  • Find an error estimate for the Taylor series

• Difficulties- Students may have difficulty:
  • With finding the error bound
  • To know when the Taylor series diverges

• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
  • Numerical calculations and implementation of numerical method.

Model of Knowledge

• Concept Map
  • Power Series
  • Taylor Series
  • Error estimate

• Content Priorities
  • Enduring Understanding
    • Taylor series
  • Important to Do and Know
    • Using Taylor series for numerical calculations
  • Worth Being Familiar with
    • Error estimates

Assessment of Learning

• Formative Assessment
  • In Class
    • Short quiz to find the Taylor series of the cos x function and its error estimate.
  • Homework (individual)
    • Short quiz to find the Taylor series of the exponential function
    • Write the Taylor series of cos x, ln(1+x) etc.
• Summative Assessment
  • Test has Taylor series
  • One student goes public by solving a problem at the board

Legacy Cycle

OBJECTIVE

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

• Use Taylor series for numerical computation.

The objectives will require that students be able to:

• Calculate Taylor series

THE CHALLENGE

You are working for a Calculator Manufacturer. The hardware was recently improved and there is a new scrolling screen available. The CEO of the company offers a $2000 bonus to the mathematician who can design a method to calculate sin x with 100 decimals.

The calculator is already equipped with software to use the scrolling screen feature for elementary operations.

GENERATE IDEAS

Explain the challenge and ask for opinions.

MULTIPLE PERSPECTIVES

The students work in small groups for 10 min and come up with ideas.

RESEARCH & REVISE

• The professor introduces the Taylor series.
• The students calculate the Taylor series of sin X.
• The professor asks the students how we can insure that the solutions to 100 decimals are exact.
• The students revise their ideas. The error needs to be estimated.
• The professor teaches the error estimate.

TEST YOUR METTLE

Short formative assessment follows.

GO PUBLIC

One student (a volunteer) solves the challenge at the board and finds the number of terms needed to get the precision of 100 decimals.

Test Your Mettle Quiz

1) Find the third term of the Taylor series associated to the function f(x)=cos x

2) Find the error estimate for the Taylor series of cos x.