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Calculus I

The Mathematics Foundation

Single-variable calculus — the math of how things change and pile up — earned step by step. Sequences and limits (what “getting arbitrarily close” really means), the derivative as the slope at a single point and the rules for finding it, the strange number e, the Mean Value Theorem, Taylor series (building a whole curve out of its slopes), maxima and minima, Newton's method and gradient descent (how a computer chases an answer it can't solve outright), the integral, the Fundamental Theorem that ties differentiation and integration together, and infinite series.

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12chapters
65pages
47kwords

Contents

  1. Patterns That Go Somewhere
  2. Getting Arbitrarily Close
  3. The Slope of a Single Point
  4. A Toolkit for Slopes
  5. The Function That Is Its Own Slope
  6. The Mean Value Theorem
  7. Building a Function From Its Derivatives
  8. Maxima, Minima, and the Shape of a Function
  9. When You Can’t Solve It, Chase It
  10. Adding Up the Infinite
  11. The Fundamental Theorem of Calculus
  12. Adding Up Forever

Interactive notebooks

12 notebooks to work through alongside the chapters. Each one downloads from this site; "open in Colab" runs it in your browser with a free Google sign-in, nothing to install.

  1. Chapter 1 · Sequences download .ipynb · open in Colab
  2. Chapter 2 · Limits download .ipynb · open in Colab
  3. Chapter 3 · Derivative download .ipynb · open in Colab
  4. Chapter 4 · Rules download .ipynb · open in Colab
  5. Chapter 5 · Exponentials download .ipynb · open in Colab
  6. Chapter 6 · Mvt download .ipynb · open in Colab
  7. Chapter 7 · Taylor download .ipynb · open in Colab
  8. Chapter 8 · Convexity download .ipynb · open in Colab
  9. Chapter 9 · Newton Gradient Descent download .ipynb · open in Colab
  10. Chapter 10 · Integral download .ipynb · open in Colab
  11. Chapter 11 · Fundamental Theorem download .ipynb · open in Colab
  12. Chapter 12 · Series download .ipynb · open in Colab

Solutions (PDF)