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.
Author(s): Joan Horvath, Rich Cameron
Edition: 1
Publisher: Make Community, LLC
Year: 2022
Language: English
Commentary: It's says it's "True EPUB", whatever that means.
Pages: 327
City: Santa Rosa, CA 95407
Tags: Calculus; Fundamental Theorem of Calculus; Limits; Derivatives; Integrals; OpenSCAD; Coordinate Systems; Coordinate Vectors; Complex Numbers; Series
PREFACE
Who This Book Is For
WHAT WE ASSUME YOU KNOW ALREADY
TEACHING AND LEARNING WITH THIS BOOK
Developing a Hands-on Calculus Course
3D Printable Models
Chapter Layout
ACKNOWLEDGMENTS
ABOUT THE AUTHORS
CHAPTER 1: THE FUNDAMENTAL THEOREM
BUILDING CALCULUS
The Steadily-Increasing Wall
The Curved Wall
Negative Changes
Examples to Try
MEASURING REAL-WORLD CHANGE
Instantaneous Slope
Looking Ahead
SECOND FUNDAMENTAL THEOREM
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
SOLUTIONS
CHAPTER 2: CALCULUS AND ITS LIMITS
WHAT IS CALCULUS?
FUNCTIONS
WHEN OUR BRICK MODELS FAIL
Limits
DERIVATIVES AND CURVES
Fundamental Theorem Model
DIMENSIONAL ANALYSIS
EQUAL, BUT NOT THE SAME
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 3: 3D PRINTED MODELS
OPENSCAD
OpenSCAD Workflow
Idiosyncrasies of OpenSCAD
Navigating on the Screen
Comments
THE MODELS
Example 1: Changing a Parameter
Example 2: Changing a Model With the Customizer
Some Models Have Small Parts
Downloading the Models: Github
3D PRINTING
3D Printing Workflow
MATERIALS
Printing Tips
IF YOU DO NOT HAVE A 3D PRINTER
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
LEARNING MORE
CHAPTER 4: DERIVATIVES: THE BASICS
THE DERIVATIVE-INTEGRAL MODEL
Model Parameters
Using Other LEGO Bricks
Testing Your Derivatives
Customizer Workarounds
Plotting Curves and Derivatives Not in the Customizer
Paper Models
INSTANTANEOUS SLOPE
Tangent Lines
The Mean Value Theorem
EXAMPLES
Derivatives of Other Powers of x
Sines and Cosines
Degrees, Radians, and Pi
Exponential Growth
Offset Calculation
Euler’s Number, e
Logarithms
Exponential Curve Offset
Experiments to Try
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 5: USING AND CALCULATING DERIVATIVES
MAXIMA, MINIMA, INFLECTION POINTS
Second Derivatives
Inflection Points
Other Inflection Point Situations
Sketching a Curve From Its Derivatives
CALCULATING DERIVATIVES
The Chain Rule
Derivatives of Products and Quotients
Derivative of a Product
Derivative of a Quotient
L’Hôpital’s Rule
OTHER WAYS OF WRITING DERIVATIVES
PARTIAL DERIVATIVES
Modeling the Surface
Modeling the Partial Derivatives
Higher-Order Partial Derivatives
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
EXERCISE ANSWERS
REFERENCES
CHAPTER 6: INTEGRALS: THE BASICS
WHAT IS AN INTEGRAL?
ASSEMBLING AN INTEGRAL
THE SECOND PART OF THE FUNDAMENTAL THEOREM OF CALCULUS
COMPUTING INTEGRALS
Indefinite Integrals (Antiderivatives)
Area Under a Curve
Area of a Region
Computing an Average
THE MEAN VALUE THEOREM, REPRISED
3D PRINTING INTEGRALS
INTEGRALS OF POWERS OF X
INTEGRALS OF SINE AND COSINE
INTEGRALS OF EXPONENTIALS
APPLICATION: PID CONTROLLERS
EXPERIMENTS TO TRY
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 7: INTEGRALS AND VOLUME
3D Coordinates
VOLUMES OF REVOLUTION
Volume of a Cone
Method of Disks
Cavalieri’s Principle
Calculating With Method of Disks
Volumes of Other Solids of Revolution
Revolution Models
Surfaces of Revolution
COMPUTING VOLUME OF MORE GENERAL SOLIDS
Calculating Volume
Checking Our Results
Printing This Model
INTEGRAL OF A PRODUCT OR QUOTIENT
Integral of a Quotient
Doing the Algebra
Printing and Experimenting With the Model
VOLUME UNDER A SURFACE
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 8: MODELING EXPONENTIAL GROWTH AND DECAY
ORDINARY DIFFERENTIAL EQUATIONS
Exponential Growth or Decay Equation
Radioactive Decay
Other Exponentials
The Logistic Equation
Math of Epidemics
DIFFERENCE EQUATIONS
Brick Model Reprise
Numerical Models of Derivatives
Numerical Models of Higher Derivatives
Error in Numerical Solutions
Error, Exponential Equation
Error, Logistic Equation
NUMERICAL MODELS OF INTEGRALS
WORKING WITH REAL DATA
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 9: MODELING PERIODIC SYSTEMS
GOING AROUND IN CIRCLES
Phase Shifts
Sine and Cosine Derivative Relationships
Approximating Sine and Cosine
SIMPLE HARMONIC MOTION
Second Order Ordinary Differential Equations
Spring Experiment
Pendulum Experiment
SYSTEMS OF DIFFERENTIAL EQUATIONS
Reprising the Logistic Equation
The Lotka-Volterra Equations
Population Behavior Over Time
Exploring the Lotka-Volterra Equations
Creating the Models
Phase Space
Phase-Space Model
Slope Fields
Stable Point
Changing Population Ratios
Attofox Problem
SEPARATION OF VARIABLES
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 10: CALCULUS, CIRCUITS, AND CODE
CALCULUS MODELS OF CIRCUITS
Simulating Circuits
DEFINITIONS AND UNITS OF ELECTRICAL COMPONENTS
RESISTOR, CAPACITOR, AND INDUCTOR CIRCUITS
RC Circuits
Capacitive Touch Sensing
LC Circuits
RL and RLC Circuits
Filters
ACCELEROMETERS AND GYROSCOPES
ACCELEROMETER MOUSE
Setting up a Circuit Playground Classic or Express
Arduino Sketch Structure
Algorithm for the Accelerometer Mouse
Circuit Playground Sketch for Accelerometer Mouse
Setting up the Mouse
Testing Out the Mouse
LIGHT-UP PENDULUM
Making the LED Pendulum
LED Pendulum Sketch
OTHER CIRCUIT PLAYGROUND ACCELEROMETER PROJECT IDEAS
PID CONTROLLERS
Temperature Control
Ball and Beam
Inverted Pendulums
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 11: COORDINATE SYSTEMS AND VECTORS
CARTESIAN, POLAR, CYLINDRICAL, AND SPHERICAL COORDINATES
Creating the Models
Integrals and Derivatives in Polar Coordinates
VECTOR BASICS
Vector Addition
Method of Shells
Multiplying a Vector by a Scalar
COMPLEX NUMBERS
The Complex Plane
Raising Complex Numbers to a Power
VECTORS MEET CALCULUS
Vector Multiplication: Dot Product
Applying the Dot Product: Work
Vector Multiplication: Cross Product
Applying the Cross Product: Torque
Vector Fields
Grad, Div, and Curl
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 12: SERIES
SEQUENCES VS. SERIES
SERIES
INFINITE SERIES
SERIES EXPANSIONS OF FUNCTIONS
Power Series
Taylor and Maclaurin Series
Maclaurin Series of Sine, Cosine, and Exponential
MODELING CONVERGENCE
Sinusoid Models
Exponential Model
Printing the Models
Broader Applications
Euler’s Equation
de Moivre’s Theorem
Proving Euler’s Equation
LIMITS AND SERIES
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
REFERENCES
CHAPTER 13: YOUR TOOLBOX
CALCULATING INTEGRALS AND DERIVATIVES
INTEGRATION BY PARTS
TRIGONOMETRIC IDENTITIES
Cofunctions
Double Angles and Sums of Angles
Squared Functions
TRIGONOMETRIC SUBSTITUTION
MATH MODELING IN REAL LIFE
CHAPTER KEY POINTS
TERMINOLOGY AND SYMBOLS
RESOURCES FOR FURTHER STUDY
Useful Websites and Search Suggestions
CALCULATION RESOURCES
BOOKS
