# UTPA STEM/CBI Courses/Calculus/Taylor Series

Course Title: Calculus II

Lecture Topic: Taylor Series

Instructor: Barnabas Bede

Institution: UTPA

## Backwards Design[edit | edit source]

**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.

- In Class
**Summative Assessment**- Test has Taylor series
- One student goes public by solving a problem at the board

## Legacy Cycle[edit | edit source]

**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[edit | edit source]

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.