Karl's Calculus Tutor

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Table of Contents

Please note that Karl's Calculus Tutor is still a work in progress. Expect a new unit to come on line every month or so. Currently being drafted: Applications of Integration (Areas)

• 1 Number Systems and Their Properties
  • 1.0 Preliminaries
  • 1.1 The Counting Numbers
  • 1.2 The Integers
  • 1.3 The Rationals
  • 1.4 And At the Heart of Calculus is ...
  • 1.5 The Real Numbers
• 2 Limits
  • 2.0 Limits: A Day at the Races
  • 2.1 Limits: How Close to the Edge Dare You Go?
  • 2.2 Combining Limits: Two Tall Tales
  • 2.3 The Sky Is the Limit
  • 2.4 Limits on the Side
  • 2.5 A Most Useful Rule
  • 2.6 Putting the Squeeze on Limits (the Squeeze Theorem)
• 3 Continuity
  • 3.0 Places You Must Visit
  • 3.1 Track to the Future
    4.0 The Trials of the Animals (intro to derivatives)
  • 4.1 The Main Pillar of the Temple (the derivative defined)
  • 4.2 Rules to Live By (sum and product rules)
  • 4.3 More Rules to Live By (quotient and chain rules)
  • 4.4 Chain Rule Applications (including implicit differentiation)
  • 4.5 The Soft Touch (finding tangent lines)
    • Box 4.4x: Summary of Rules for Taking Derivatives (this page also available in printer-friendly PDF)
• 5 Applications of Derivatives
  • 5.1 Hilltops and Valley Floors (maxima-minima)
  • 5.2 Squigglies, Wormtails, and Pointed Little Heads (higher order derivatives)
  • 5.3 The Road to Shangrila (Newton-Raphson approximation)
  • 5.4 Like a Steam Locomotive (Rolle's Theorem)
  • 5.5 Drawing a Non-Blank (graphing problems)
  • 5.6 Implicit Applications: Using Implicit Derivatives
• 6 Exponentials and Logarithms
  • 6.1 Be Fruitful and Multiply (intro to exponentials)(this page also available in printer-friendly PDF)
  • 6.2 The Farther We Go The Faster We Get There (derivatives of exponentials)(this page also available in printer-friendly PDF)
  • 6.3 Be Frugal and Add (logarithms)
    • Box 6.0: Common Exponential and Log Identities
• 7 Trig Functions (sine, cosine, etc.)
  • 7.1 May the Circle Be Unbroken (review of trigonometry)(this page also available in printer-friendly PDF)
  • 7.2 As the World Turns (derivatives of trig functions)(this page also available in printer-friendly PDF)
    • Box: Common Trig Identities and their Derivations (this page also available in printer-friendly PDF)
    • Some Worked Trig Identities
    • Trigonometry Enrichment (PDF only) Trig review, practice problems, and more.
• 8 More Tricks with Derivatives
  • 8.1 The Cheshire Cat's Grin (L'Hopital's Rule)
    • includes Taking Limits of Differences
  • 8.2 The Dance of the Derivatives (Related Rates) (this page also available in printer-friendly PDF)
    • includes Dimensional Checking (sometimes called Units Checking)
  • 8.3 Name That Function (inferring coefficients from clues about critical and inflection points)
  • 8.4 Little Red Riding Hood Goes to Town (approximation & intro to Taylor & Maclaurin Series)
  • 8.5 Power Tools for Taking Derivatives of Products (logarithmic differentiation and Leibniz' Rule)
• 9 Summing Up Derivatives
  • 9 Midterm Practice Exam
• 10 Integrals
  • 10.1 Thoughts for a Wedding Reception (intro to integrals)
  • 10.2 Living Backward -- The Fundamental Theorem of Calculus
  • 10.3 Grist for the Mill -- Definite and Indefinite Integrals
• 11 Methods of Integration
  • 11.1 Intro to Methods of Integration
  • 11.2 Integration Using your Rearview Mirror
  • 11.3 Integration by Simple Substitution
  • 11.4 Integration by Parts
  • 11.5 Trigonometric Substitution
  • 11.6 Hyperbolic Substitution
  • 11.7 Partial Fractions (This page now available in PDF format)
  • 11.8 More Substitutions
  • 12 Applications of Integrals
    • 12.1 Motion Problems Still under construction
    • 12.2 The Area of a Circle
    • 12.3 The Area Between Curves Still under construction
• Differential Equations
  • Link to Differential Equations E-book

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