SageMathTM Advice For Calculus - Rowan University

SageMathTM Advice For Calculus

Tuan A. Le and Hieu D. Nguyen Rowan University

Copyright 2016

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Contents

1 Introduction

7

1.1 SageMath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Creating an Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.3 Help Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.4 Sharing Sage Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Sage Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1 Naming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2 Delimiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Lists, Tables, and Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.4 Commenting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.2 Useful Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Graphs, Limits, and Continuity of Functions

25

2.1 Plotting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Basics Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.2 Plot Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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2.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Evaluating Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 Limits Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . 40 2.2.3 Limits Involving Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Differentiation

55

3.1 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.1 Slope of Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.2 Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2 Higher-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Chain Rule and Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Derivatives of Inverse, Exponential and Logarithmic Functions . . . . . . . . . . . 67

3.4.1 Inverse Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.2 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . 71

4 Applications of the Derivative

75

4.1 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.1 Traffic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.2 Minimum Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.3 Packaging (Minimum Surface Area) . . . . . . . . . . . . . . . . . . . . . 84

4.3.4 Maximize Revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4.1 Programing Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4.3 Slow Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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5 Integration

93

5.1 Antiderivatives (Indefinite Integral) . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Riemann Sums and the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.1 Riemann Sum Using Left Endpoints . . . . . . . . . . . . . . . . . . . . . 95

5.2.2 Riemann Sum Using Right Endpoints . . . . . . . . . . . . . . . . . . . . 98

5.2.3 Riemann Sum Using Midpoints . . . . . . . . . . . . . . . . . . . . . . . 101

5.3 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . 103

5.4 Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Applications of the Integral

113

6.1 Area Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Average Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Volume of Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.1 The Methods of Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.2 The Method of Washers . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.3.3 The Method of Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . 124

Bibliography

131

Appendices

133

A Common Mathematical Operations

135

B Useful Commands for Plotting and Algebra

137

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Chapter 1

Introduction

1.1 SageMath

Welcome to SageMath! This tutorial manual is intended as a supplement to Rogawski's Calculus textbook and aimed at students looking to quickly learn Sage through examples. It also includes a brief summary of each calculus topic to emphasize important concepts. Students should refer to their textbook for a further explanation of each topic.

1.1.1 Creating an Account

SageMath is a powerful computer algebra system (CAS) whose capabilities and features can be overwhelming for new users. Thus, to make your experience in using Sage as easy as possible, we recommend that you read this introductory chapter carefully. We will discuss basic syntax and frequently used commands. There are two ways to use Sage, you can run Sage on it server (cloud) or install Sage and run it on your computer: SageMath Cloud: To use SageMath on the cloud, go to cloud. and create an account. After logging in, you will see all of your projects will be listed. Since it's the first time, click on NewProject . . . to create one. Give the project a name and click on CreateProject. Your

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CHAPTER 1. INTRODUCTION

project now is created and listed under ShowingProject. For example, I have create a new project name "Testing Sage Manual" among other projects. The screen will look like this:

Click on the project you want to work on, click Create or upload files...

where we can create a file of upload a file from our computer. Since we want to run Sage on cloud, we create a new file name StartingWithSage, and select the type as SageMath Worksheet

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