SageMath Advice For Calculus

136 Pages·2016·1.79 MB·English

SageMathTM Advice
For Calculus
Tuan A. Le and Hieu D. Nguyen
Rowan University
Copyright 2016
2
Contents
1 Introduction 7
1.1 SageMath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 CreatinganAccount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 GettingStarted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.3 HelpMenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.4 SharingSageFiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 SageCommands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Naming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Delimiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.3 Lists,Tables,andArrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.4 Commenting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 SolvingEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.2 UsefulCommands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Graphs,Limits,andContinuityofFunctions 25
2.1 PlottingGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 BasicsPlot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 PlotOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3
4 CONTENTS
2.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.1 EvaluatingLimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 LimitsInvolvingTrigonometricFunctions . . . . . . . . . . . . . . . . . . 40
2.2.3 LimitsInvolvingInfinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Differentiation 55
3.1 TheDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 SlopeofTangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.2 DerivativeasaFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Higher-OrderDerivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 ChainRuleandImplicitDifferentiation . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 DerivativesofInverse,ExponentialandLogarithmicFunctions . . . . . . . . . . . 67
3.4.1 InverseFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.2 ExponentialandLogarithmicFunctions . . . . . . . . . . . . . . . . . . . 71
4 ApplicationsoftheDerivative 75
4.1 RelatedRates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 TrafficFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.2 MinimumCost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.3 Packaging(MinimumSurfaceArea) . . . . . . . . . . . . . . . . . . . . . 84
4.3.4 MaximizeRevenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Newton’sMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.1 ProgramingNewton’sMethod . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4.3 SlowConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
CONTENTS 5
5 Integration 93
5.1 Antiderivatives(IndefiniteIntegral) . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 RiemannSumsandtheDefiniteIntegral . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 RiemannSumUsingLeftEndpoints . . . . . . . . . . . . . . . . . . . . . 95
5.2.2 RiemannSumUsingRightEndpoints . . . . . . . . . . . . . . . . . . . . 98
5.2.3 RiemannSumUsingMidpoints . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 TheFundamentalTheoremofCalculus . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 IntegrationTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 ApplicationsoftheIntegral 113
6.1 AreaBetweenCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 AverageValue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 VolumeofSolidsofRevolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.1 TheMethodsofDiscs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.2 TheMethodofWashers . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3.3 TheMethodofCylindricalShells . . . . . . . . . . . . . . . . . . . . . . 124
Bibliography 131
Appendices 133
A CommonMathematicalOperations 135
B UsefulCommandsforPlottingandAlgebra 137
6 CONTENTS
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
theirtextbookforafurtherexplanationofeachtopic.
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
frequentlyusedcommands.
There are two ways to use Sage, you can run Sage on it server (cloud) or install Sage and run it
onyourcomputer:
SageMathCloud: To use SageMath on the cloud, go to www.cloud.sagemath.com and create
anaccount. Afterloggingin,youwillseeallofyourprojectswillbelisted. Sinceit’sthefirsttime,
click on NewProject... to create one. Give the project a name and click on CreateProject. Your
7
8 CHAPTER1. INTRODUCTION
projectnowiscreatedandlistedunderShowingProject. Forexample,Ihavecreateanewproject
name"TestingSageManual"amongotherprojects. Thescreenwilllooklikethis:
Clickontheprojectyouwanttoworkon,clickCreateoruploadfiles...
wherewecancreateafileofuploadafilefromourcomputer. SincewewanttorunSageoncloud,
wecreateanewfilenameStartingWithSage,andselectthetypeasSageMathWorksheet
1.1. SAGEMATH 9
We are now in a Sage file and ready to use it. Pay attention that we are now viewing the file
StartingWithSage.sagewslocatedinsideoftheTestingSageManualproject.
Localhost: The other way to run Sage is to download it and install it on your computer. Go to
www.sagemath.org/download and download Sage package. Install it and restart you computer.
NowrunSage(doubleclickonitsicon),Sagewillautomaticallyopenupyourbrownser
and your Sage notebook on your localhost, displayed all worksheets that you have been working
on
10 CHAPTER1. INTRODUCTION
Click on any worksheet that you want to continue work with or create a new worksheet. To create
a new worksheet, click on New Worksheet, give it a name a click on Rename. For example, let
createanewworksheetcalledSageExample
Each horizontal rectangle is called a cell. Click on that and you are now ready to start learning
Sage.