Make: Calculus

Make: Calculus

by Joan Horvath, Rich Cameron
Released August 2022
Publisher(s): Make: Community
ISBN: 9781680457391

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Book description

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.

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Table of contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Dedication
  6. Preface
    1. Who this Book Is For
    2. What We Assume you Know Already
    3. Teaching and Learning With This Book
      1. Developing a Hands-on Calculus Course
      2. 3D Printable Models
      3. Chapter Layout
    4. Acknowledgments
    5. About the Authors
  7. Chapter 1: The Fundamental Theorem
    1. Building Calculus
      1. The Steadily-Increasing Wall
      2. The Curved Wall
      3. Negative Changes
      4. Examples to Try
    2. Measuring Real-World Change
      1. Instantaneous Slope
      2. Looking Ahead
    3. Second Fundamental Theorem
    4. Chapter Key Points
    5. Terminology and Symbols
    6. Solutions
  8. Chapter 2: Calculus and its Limits
    1. What is Calculus?
    2. Functions
    3. When Our Brick Models Fail
      1. Limits
    4. Derivatives and Curves
      1. Fundamental Theorem Model
    5. Dimensional Analysis
    6. Equal, but Not the Same
    7. Chapter Key Points
    8. Terminology and Symbols
    9. References
  9. Chapter 3: 3D Printed Models
    1. Openscad
      1. OpenSCAD Workflow
      2. Idiosyncrasies of OpenSCAD
      3. Navigating on the Screen
      4. Comments
    2. The Models
      1. Example 1: Changing a Parameter
      2. Example 2: Changing a Model With the Customizer
      3. Some Models Have Small Parts
      4. Downloading the Models: Github
    3. 3D Printing
      1. 3D Printing Workflow
      2. Materials
      3. Printing Tips
    4. If you do not Have a 3D Printer
    5. Chapter Key Points
    6. Terminology and Symbols
    7. Learning More
  10. Chapter 4: Derivatives: The Basics
    1. The Derivative-Integral Model
      1. Model Parameters
      2. Using Other LEGO Bricks
      3. Testing Your Derivatives
      4. Customizer Workarounds
      5. Plotting Curves and Derivatives Not in the Customizer
      6. Paper Models
    2. Instantaneous Slope
      1. Tangent Lines
      2. The Mean Value Theorem
    3. Examples
      1. Derivatives of Other Powers of x
      2. Sines and Cosines
      3. Degrees, Radians, and Pi
      4. Exponential Growth
      5. Offset Calculation
      6. Euler’s Number, e
      7. Logarithms
      8. Exponential Curve Offset
      9. Experiments to Try
    4. Chapter Key Points
    5. Terminology and Symbols
    6. References
  11. Chapter 5: Using and Calculating Derivatives
    1. Maxima, Minima, Inflection Points
      1. Second Derivatives
      2. Inflection Points
      3. Other Inflection Point Situations
      4. Sketching a Curve From Its Derivatives
    2. Calculating Derivatives
      1. The Chain Rule
      2. Derivatives of Products and Quotients
        1. Derivative of a Product
        2. Derivative of a Quotient
      3. L’Hôpital’s Rule
    3. Other Ways of Writing Derivatives
    4. Partial Derivatives
      1. Modeling the Surface
      2. Modeling the Partial Derivatives
      3. Higher-Order Partial Derivatives
    5. Chapter Key Points
    6. Terminology and Symbols
    7. Exercise Answers
    8. References
  12. Chapter 6: Integrals: the Basics
    1. What is an Integral?
    2. Assembling an Integral
    3. The Second Part of the Fundamental Theorem of Calculus
    4. Computing Integrals
      1. Indefinite Integrals (Antiderivatives)
      2. Area Under a Curve
      3. Area of a Region
      4. Computing an Average
    5. The Mean Value Theorem, Reprised
    6. 3D Printing Integrals
    7. Integrals of Powers of X
    8. Integrals of Sine and Cosine
    9. Integrals of Exponentials
    10. Application: PID Controllers
    11. Experiments to Try
    12. Chapter Key Points
    13. Terminology and Symbols
    14. References
  13. Chapter 7: Integrals and Volume
    1. 3D Coordinates
    2. Volumes of Revolution
      1. Volume of a Cone
      2. Method of Disks
      3. Cavalieri’s Principle
      4. Calculating With Method of Disks
      5. Volumes of Other Solids of Revolution
      6. Revolution Models
      7. Surfaces of Revolution
    3. Computing Volume of More General Solids
      1. Calculating Volume
      2. Checking Our Results
      3. Printing This Model
    4. Integral of a Product or Quotient
      1. Integral of a Quotient
      2. Doing the Algebra
      3. Printing and Experimenting With the Model
    5. Volume Under a Surface
    6. Chapter Key Points
    7. Terminology and Symbols
    8. References
  14. Chapter 8: Modeling Exponential Growth and Decay
    1. Ordinary Differential Equations
      1. Exponential Growth or Decay Equation
      2. Radioactive Decay
      3. Other Exponentials
      4. The Logistic Equation
      5. Math of Epidemics
    2. Difference Equations
      1. Brick Model Reprise
      2. Numerical Models of Derivatives
      3. Numerical Models of Higher Derivatives
      4. Error in Numerical Solutions
      5. Error, Exponential Equation
      6. Error, Logistic Equation
    3. Numerical Models of Integrals
    4. Working with Real Data
    5. Chapter Key Points
    6. Terminology and Symbols
    7. References
  15. Chapter 9: Modeling Periodic Systems
    1. Going Around in Circles
      1. Phase Shifts
      2. Sine and Cosine Derivative Relationships
      3. Approximating Sine and Cosine
    2. Simple Harmonic Motion
      1. Second Order Ordinary Differential Equations
      2. Spring Experiment
      3. Pendulum Experiment
    3. Systems of Differential Equations
      1. Reprising the Logistic Equation
      2. The Lotka-Volterra Equations
      3. Population Behavior Over Time
      4. Exploring the Lotka-Volterra Equations
      5. Creating the Models
      6. Phase Space
      7. Phase-Space Model
      8. Slope Fields
      9. Stable Point
      10. Changing Population Ratios
      11. Attofox Problem
    4. Separation of Variables
    5. Chapter Key Points
    6. Terminology and Symbols
    7. References
  16. Chapter 10: Calculus, Circuits, and Code
    1. Calculus Models of Circuits
      1. Simulating Circuits
    2. Definitions and Units of Electrical Components
    3. Resistor, Capacitor, and Inductor Circuits
      1. RC Circuits
      2. Capacitive Touch Sensing
      3. LC Circuits
      4. RL and RLC Circuits
      5. Filters
    4. Accelerometers and Gyroscopes
    5. Accelerometer Mouse
      1. Setting up a Circuit Playground Classic or Express
      2. Arduino Sketch Structure
      3. Algorithm for the Accelerometer Mouse
      4. Circuit Playground Sketch for Accelerometer Mouse
      5. Setting up the Mouse
      6. Testing Out the Mouse
    6. Light-Up Pendulum
      1. Making the LED Pendulum
      2. LED Pendulum Sketch
    7. Other Circuit Playground Accelerometer Project Ideas
    8. PID Controllers
      1. Temperature Control
      2. Ball and Beam
      3. Inverted Pendulums
    9. Chapter Key Points
    10. Terminology and Symbols
    11. References
  17. Chapter 11: Coordinate Systems and Vectors
    1. Cartesian, Polar, Cylindrical, and Spherical Coordinates
      1. Creating the Models
      2. Integrals and Derivatives in Polar Coordinates
    2. Vector Basics
      1. Vector Addition
      2. Method of Shells
      3. Multiplying a Vector by a Scalar
    3. Complex Numbers
      1. The Complex Plane
      2. Raising Complex Numbers to a Power
    4. Vectors Meet Calculus
      1. Vector Multiplication: Dot Product
      2. Applying the Dot Product: Work
      3. Vector Multiplication: Cross Product
      4. Applying the Cross Product: Torque
      5. Vector Fields
      6. Grad, Div, and Curl
    5. Chapter Key Points
    6. Terminology and Symbols
    7. References
  18. Chapter 12: Series
    1. Sequences vs. Series
    2. Series
    3. Infinite Series
    4. Series Expansions of Functions
      1. Power Series
      2. Taylor and Maclaurin Series
      3. Maclaurin Series of Sine, Cosine, and Exponential
    5. Modeling Convergence
      1. Sinusoid Models
      2. Exponential Model
      3. Printing the Models
      4. Broader Applications
    6. Limits and Series
      1. Euler’s Equation
      2. de Moivre’s Theorem
      3. Proving Euler’s Equation
    7. Chapter Key Points
    8. Terminology and Symbols
    9. References
  19. Chapter 13: Your Toolbox
    1. Calculating Integrals and Derivatives
    2. Integration by Parts
    3. Trigonometric Identities
      1. Cofunctions
      2. Double Angles and Sums of Angles
      3. Squared Functions
    4. Trigonometric Substitution
    5. Math Modeling in Real Life
    6. Chapter Key Points
    7. Terminology and Symbols
    8. Resources for Further Study
      1. Useful Websites and Search Suggestions
    9. Calculation Resources
    10. Books
  20. Index
  21. Also from the authors:

Product information

  • Title: Make: Calculus
  • Author(s): Joan Horvath, Rich Cameron
  • Release date: August 2022
  • Publisher(s): Make: Community
  • ISBN: 9781680457391