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E-raamat: Make: Calculus

4.00/5 (12 hinnangut Goodreads-ist)

Rich Cameron, Joan Horvath

Formaat: 328 pages
Ilmumisaeg: 09-Aug-2022
Kirjastus: Make Community, LLC
Keel: eng
ISBN-13: 9781680457377

Teised raamatud teemal:

Calculus

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Formaat: 328 pages
Ilmumisaeg: 09-Aug-2022
Kirjastus: Make Community, LLC
Keel: eng
ISBN-13: 9781680457377

Teised raamatud teemal:

Calculus

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When Isaac Newton developed calculus in the 1600s, he was trying to tie together math and physics in an intuitive, geometrical way. But over time math and physics teaching became heavily weighted toward algebra, and less toward geometrical problem solving. However, many practicing mathematicians and physicists will get their intuition geometrically first and do the algebra later.

Make:Calculus imagines how Newton might have used 3D printed models, construction toys, programming, craft materials, and an Arduino or two to teach calculus concepts in an intuitive way. The book uses as little reliance on algebra as possible while still retaining enough to allow comparison with a traditional curriculum.

This book is not a traditional Calculus I textbook. Rather, it will take the reader on a tour of key concepts in calculus that lend themselves to hands-on projects. This book also defines terms and common symbols for them so that self-learners can learn more on their own.

Preface: Who This Book Is For

xiv

What We Assume You Know Already

Teaching And Learning With This Book

xvi

Developing A Hands-On Calculus Course

xvi

3D Printable Models

xvi

Chapter Layout

xvii

Acknowledgments

xviii

About The Authors

xix

Chapter 1 The Fundamental Theorem

Building Calculus

(8)

The Steadily-Increasing Wall

(5)

The Curved Wall

(1)

Negative Changes

(1)

Examples To Try

(1)

Measuring Real-World Change

(2)

Instantaneous Slope

(1)

Looking Ahead

(1)

Second Fundamental Theorem

(1)

Chapter Key Points

(1)

Terminology And Symbols

(1)

Solutions

(3)

Chapter 2 Calculus And Its Limits

What Is Calculus?

(2)

Functions

(1)

When Our Brick Models Fail

(5)

Limits

(4)

Derivatives And Curves

(3)

Fundamental Theorem Model

(3)

Dimensional Analysis

(2)

Equal, But Not The Same

(1)

Chapter Key Points

(1)

Terminology And Symbols

(1)

References

(3)

Chapter 3 3D Printed Models

Openscad

(5)

Openscad Workflow

(2)

Idiosyncrasies Of Openscad

(2)

Navigating On The Screen

(1)

Comments

(1)

The Models

(6)

Example 1 Changing A Parameter

(3)

Example 2 Changing A Model With The Customizer

(1)

Some Models Have Small Parts

(1)

Downloading The Models: Github

(1)

3D Printing

(25)

3D Printing Workflow

(2)

Materials

(1)

Printing Tips

(2)

If You Do Not Have A 3D Printer

(1)

Chapter Key Points

(1)

Terminology And Symbols

(1)

Learning More

(3)

Chapter 4 Derivatives: The Basics

The Derivative-Integral Model

(1)

Model Parameters

(2)

Using Other Lego Bricks

(1)

Testing Your Derivatives

(2)

Customizer Workarounds

(1)

Plotting Curves And Derivatives Not In The Customizer

(1)

Paper Models

(1)

Instantaneous Slope

(1)

Tangent Lines

(1)

The Mean Value Theorem

(1)

Examples

(1)

Derivatives Of Other Powers Of X

(3)

Sines And Cosines

(2)

Degrees, Radians, And Pi

(1)

Exponential Growth

(2)

Offset Calculation

(1)

Euler's Number, E

(1)

Logarithms

(2)

Exponential Curve Offset

(2)

Experiments To Try

(1)

Chapter Key Points

(1)

Terminology And Symbols

(1)

References

(3)

Chapter 5 Using And Calculating Derivatives

Maxima, Minima, Inflection Points

(6)

Second Derivatives

(1)

Inflection Points

(2)

Other Inflection Point Situations

(1)

Sketching A Curve From Its Derivatives

(1)

Calculating Derivatives

(7)

The Chain Rule

(2)

Derivatives Of Products And Quotients

(1)

Derivative Of A Product

(1)

Derivative Of A Quotient

(1)

L'Hopitals Rule

(3)

Other Ways Of Writing Derivatives

(1)

Partial Derivatives

(4)

Modeling The Surface

(1)

Modeling The Partial Derivatives

(1)

Higher-Order Partial Derivatives

(1)

Chapter Key Points

(1)

Terminology And Symbols

(1)

Exercise Answers

(3)

References

101

(2)

Chapter 6 Integrals: The Basics

What Is An Integral?

103

(1)

Assembling An Integral

104

(3)

The Second Part Of The Fundamental Theorem Of Calculus

107

(1)

Computing Integrals

108

(5)

Indefinite Integrals Lantiderivatives)

108

(4)

Area Under A Curve

112

(1)

Area Of A Region

112

(1)

Computing An Average

112

(1)

The Mean Value Theorem, Reprised

113

(1)

3D Printing Integrals

114

(1)

Integrals Of Powers Of X

115

(2)

Integrals Of Sine And Cosine

117

(1)

Integrals Of Exponentials

118

(1)

Application: Pid Controllers

119

(4)

Experiments To Try

123

(1)

Chapter Key Points

123

(1)

Terminology And Symbols

124

(1)

References

124

(4)

Chapter 7 Integrals And Volume

3D Coordinates

128

(1)

Volumes Of Revolution

129

(7)

Volume Of A Cone

129

(1)

Method Of Disks

129

(1)

Cavalieri's Principle

130

(2)

Calculating With Method Of Disks

132

(1)

Volumes Of Other Solids Of Revolution

132

(1)

Revolution Models

133

(2)

Surfaces Of Revolution

135

(1)

Computing Volume Of More General Solids

136

(5)

Calculating Volume

136

(3)

Checking Our Results

139

(1)

Printing This Model

140

(1)

Integral Of A Product Or Quotient

141

(3)

Integral Of A Quotient

143

(1)

Doing The Algebra

143

(1)

Printing And Experimenting With The Model

144

(1)

Volume Under A Surface

144

(2)

Chapter Key Points

146

(1)

Terminology And Symbols

147

(1)

References

147

(2)

Chapter 8 Modeling Exponential Growth And Decay

Ordinary Differential Equations

149

(10)

Exponential Growth Or Decay Equation

150

(2)

Radioactive Decay

152

(2)

Other Exponentials

154

(2)

The Logistic Equation

156

(1)

Math Of Epidemics

157

(2)

Difference Equations

159

(5)

Brick Model Reprise

159

(1)

Numerical Models Of Derivatives

160

(1)

Numerical Models Of Higher Derivatives

161

(1)

Error In Numerical Solutions

161

(1)

Error, Exponential Equation

161

(2)

Error, Logistic Equation

163

(1)

Numerical Models Of Integrals

164

(1)

Working With Real Data

165

(1)

Chapter Key Points

166

(1)

Terminology And Symbols

166

(1)

References

167

(2)

Chapter 9 Modeling Periodic Systems

Going Around In Circles

169

(6)

Phase Shifts

170

(1)

Sine And Cosine Derivative Relationships

171

(2)

Approximating Sine And Cosine

173

(2)

Simple Harmonic Motion

175

(7)

Second Order Ordinary Differential Equations

177

(3)

Spring Experiment

180

(1)

Pendulum Experiment

181

(1)

Systems Of Differential Equations

182

(12)

Reprising The Logistic Equation

182

(1)

The Lotka-Volterra Equations

182

(1)

Population Behavior Over Time

183

(2)

Exploring The Lotka-Volterra Equations

185

(1)

Creating The Models

186

(1)

Phase Space

187

(2)

Phase-Space Model

189

(1)

Slope Fields

190

(1)

Stable Point

191

(2)

Changing Population Ratios

193

(1)

Attof Ox Problem

194

(1)

Separation Of Variables

194

(1)

Chapter Key Points

195

(1)

Terminology And Symbols

196

(1)

References

196

(4)

Chapter 10 Calculus, Circuits, And Code

Calculus Models Of Circuits

200

(1)

Simulating Circuits

200

(1)

Definitions And Units Of Electrical Components

201

(5)

Resistor, Capacitor, And Inductor Circuits

206

(8)

Rc Circuits

206

(2)

Capacitive Touch Sensing

208

(2)

Lc Circuits

210

(3)

Rland RLC Circuits

213

(1)

Filters

214

(1)

Accelerometers And Gyroscopes

214

(1)

Accelerometer Mouse

215

(10)

Setting Up A Circuit Playground Classic Or Express

216

(2)

Arduino Sketch Structure

218

(1)

Algorithm For The Accelerometer Mouse

218

(2)

Circuit Playground Sketch For Accelerometer Mouse

220

(3)

Setting Up The Mouse

223

(1)

Testing Out The Mouse

223

(2)

Light-Up Pendulum

225

(5)

Making The Led Pendulum

225

(3)

Led Pendulum Sketch

228

(2)

Other Circuit Playground Accelerometer Project Ideas

230

(1)

Pid Controllers

231

(2)

Temperature Control

231

(1)

Ball And Beam

232

(1)

Inverted Pendulums

232

(1)

Chapter Key Points

233

(1)

Terminology And Symbols

233

(1)

References

233

(2)

Chapter 11 Coordinate Systems And Vectors

Cartesian, Polar, Cylindrical, And Spherical Coordinates

235

(9)

Creating The Models

239

(2)

Integrals And Derivatives In Polar Coordinates

241

(3)

Vector Basics

244

(3)

Vector Addition

244

(1)

Method Of Shells

245

(1)

Multiplying A Vector By A Scalar

246

(1)

Complex Numbers

247

(3)

The Complex Plane

248

(1)

Raising Complex Numbers To A Power

248

(2)

Vectors Meet Calculus

250

(8)

Vector Multiplication: Dot Product

250

(1)

Applying The Dot Product: Work

251

(1)

Vector Multiplication: Cross Product

252

(2)

Applying The Cross Product: Torque

254

(1)

Vector Fields

255

(1)

Grad, Div, And Curl

256

(2)

Chapter Key Points

258

(1)

Terminology And Symbols

259

(1)

References

259

(2)

Chapter 12 Series

Sequences Vs. Series

261

(1)

Series

262

(1)

Infinite Series

263

(3)

Series Expansions Of Functions

266

(2)

Power Series

264

(1)

Taylor And Maclaurin Series

265

(1)

Maclaurin Series Of Sine, Cosine, And Exponential

266

(2)

Modeling Convergence

268

(6)

Sinusoid Models

268

(4)

Exponential Model

272

(1)

Printing The Models

273

(1)

Broader Applications

273

(2)

Euler's Equation

275

(1)

De Moivre's Theorem

276

(1)

Proving Euler's Equation

276

Limits And Series

274

(4)

Chapter Key Points

278

(1)

Terminology And Symbols

278

(1)

References

278

(3)

Chapter 13 Your Toolbox

Calculating Integrals And Derivatives

281

(4)

Integration By Parts

285

(2)

Trigonometric Identities

287

(2)

Cofunctions

287

(1)

Double Angles And Sums Of Angles

288

(1)

Squared Functions

288

(1)

Trigonometric Substitution

289

(2)

Math Modeling In Real Life

291

(1)

Chapter Key Points

292

(1)

Terminology And Symbols

293

(1)

Resources For Further Study

293

(1)

Useful Websites And Search Suggestions

293

(1)

Calculation Resources

294

(1)

Books

294

Joan Horvath is an MIT alumna, a recovering rocket scientist, and educator. She is a cofounder of Nonscriptum LLC. Rich Cameron is an open-source 3D printer hacker who designed the RepRapWallace and Bukito printers. He is a cofounder of Nonscriptum LLC.

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