Karl's Calculus Tutor Table of Contents © 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004 by Karl Hahn last update 1-Mar-2007

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Section Index 1) Number Systems 2) Limits 3) Continuity 4) Derivatives 5) Applications of Derivatives 6) Exponentials and Logs 7) Trig Functions 8) More Tricks with Derivatives 9) Practice Midterm 10) Integrals 11) Methods of Integration 12) Applications of Integrals Interactive Features Special Interest Pages Online Calculator From B. Cherkas at CUNY:

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)

• prependix a: Math Notation over the Web
• prependix b: Why Bother to Learn Calculus
• prependix c: Stuff You Should Already Know
• prependix d: Study Tips
• Karl's Calculus Tutor: Interactive Features
• Special Interest and Optional Pages
• Karl's Calculus Forum where you can discuss calculus topics with other visitors to Karl's Calculus Tutor.
• 0 Greetings and Welcome
• 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 Derivatives
• 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
• 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)
• 6.2 The Farther We Go The Faster We Get There (derivatives of exponentials)
• 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)
• 7.2 As the World Turns (derivatives of trig functions)
• Box: Common Trig Identities and their Derivations
• 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)
• 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 Still under construction
• 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
• 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
• Sections Coming to Karl's Calculus Tutor in the months ahead:
• Improper Integrals
• Areas and Volumes
• Convergent Series
• More on Taylor & Maclaurin Series
• Sample Exam Questions on Integral Calculus
• PC Users: To aid your studies you can download useful calculator programs that can do complex-number arithmetic and much more.

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Special Interest and Optional Pages Quadratic Polynomials || Cubic Polynomials || Quartic Polynomials || Hero's Formula || Einstein's Formula || Log Base 10 Tricks || Measuring the Earth || Long Ago in a Galaxy Far Away || An Object Falling through Air || Proof of The Intermediate Value Theorem || Proof of The Extreme Value Theorem || Rational Unit Circle Points/Pythagorean Triples || Multi-variable Polynomial Long Division

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