List of Symbols | p. viii |

Introduction | p. ix |

The Slope of the Tangent Line | p. 1 |

The Graph of the Train's Position | p. 2 |

Function Machines | p. 3 |

Slope of a Line | p. 6 |

Slope of a Secant Line | p. 7 |

Slope of a Tangent Line | p. 8 |

Tangent Slope for y = x[superscript 2] | p. 10 |

Worksheet: Slope of a Secant Line | p. 11 |

Exercises | p. 11 |

Calculating Derivatives | p. 13 |

Definition of Derivative | p. 14 |

Derivative of a Constant Function | p. 15 |

Derivative of a Straight-Line Function | p. 17 |

Derivative of y = cx[superscript 2] | p. 18 |

Derivative of a Sum | p. 21 |

Derivative of y = x[superscript 3] | p. 22 |

Derivative of y = x[superscript n] | p. 23 |

Worksheet: Derivatives of Polynomials | p. 26 |

Exercises | p. 28 |

Drawing Curves with Derivatives | p. 31 |

Horizontal Tangents | p. 32 |

The Derivative of the Derivative | p. 35 |

The Professor's Bug and the Meaning of the Second Derivative | p. 36 |

Concave-up and Concave-down Curves | p. 37 |

The Spilled Water | p. 38 |

Local Maximum Points | p. 39 |

Worksheet: Maximum and Minimum Points | p. 40 |

Points of Inflection | p. 41 |

Worksheet: Local Maximum and Minimum Points | p. 42 |

Exercises | p. 47 |

Derivatives of Complicated Functions | p. 49 |

Multiplied Functions | p. 51 |

The Product Rule | p. 55 |

Embedded Functions | p. 55 |

The Chain Rule | p. 58 |

Fractional Exponents | p. 58 |

Implicit Functions | p. 59 |

The Power Rule | p. 61 |

Worksheet: The Product Rule | p. 61 |

Exercises | p. 64 |

Derivatives of Trigonometric Functions | p. 66 |

The Gremlin's Horrible Oscillating Chicken-Scaring Machine | p. 67 |

Trigonometeris' Sine Function | p. 68 |

The Derivative of the Sine Function | p. 73 |

The Derivative of the Cosine Function | p. 74 |

Derivatives of Other Trigonometric Functions | p. 75 |

Worksheet: Derivatives of Trigonometric Functions | p. 76 |

Exercises | p. 77 |

Optimum Values and Related Rates | p. 79 |

Differentiation and the Get-Rich-Quick Scheme | p. 80 |

The Optimum-size Box | p. 81 |

Carmorra Magazine and the Optimum Subscription Price | p. 82 |

The Birthday Party Balloon | p. 85 |

The National Park Beach Lifeguard and the Racing Shadow | p. 86 |

Exercises | p. 88 |

The Integral: A Backward Derivative | p. 91 |

Recordis' Exhaustion and the Story of Rutherford | p. 91 |

Differentiating Backwards | p. 92 |

The Antiderivative or the Integral | p. 93 |

Discovering the Indefiniteness of an Indefinite Integral | p. 93 |

Using an Initial Condition to Track Down Rutherford | p. 94 |

Differentials | p. 96 |

The Integral Sign | p. 96 |

Sum Rule for Integrals | p. 98 |

Multiplication Rule for Integrals | p. 99 |

Perfect Integral Rule | p. 100 |

Power Rule for Integrals | p. 101 |

Worksheet: Integrals of Polynomials | p. 102 |

Worksheet: Integration by Substitution | p. 104 |

Exercises | p. 105 |

Finding Areas with Integrals | p. 108 |

Recordis' Pools and the Magic Crystal Water Rate Increase | p. 108 |

Summation Notation | p. 110 |

The Curve's Area Defined as a Limit | p. 112 |

The Gremlin's Terrible Fire-and-Water Threat | p. 112 |

The Mysterious Function A(x) | p. 113 |

The Derivative of A(x) | p. 114 |

Fundamental Theorem of Integral Calculus | p. 115 |

Discovering the Definiteness of Definite Integrals | p. 116 |

Worksheet: Definite Integrals | p. 117 |

Exercises | p. 119 |

Natural Logarithms | p. 122 |

The Unfortunate Accident with Some Beads | p. 123 |

The Power Rule Breakdown: n = -1 | p. 124 |

The Mysterious Function L(a) | p. 125 |

Some Properties of L(a) | p. 126 |

Substitution Method for Evaluating Definite Integrals | p. 128 |

Remembering Logarithms | p. 128 |

The Derivative of the Logarithm Function | p. 129 |

The Fundamental Number e | p. 130 |

Worksheet: Derivatives and Integrals with Logarithms | p. 132 |

Exercises | p. 135 |

Exponential Functions and Integration by Parts | p. 137 |

The Graph of the Logarithm Function | p. 138 |

Pal's Stumble and the Inverse Function | p. 138 |

The Exponential Function and the Professor's Amazing Income | p. 139 |

The Indestructible Function e[superscript x] | p. 141 |

Differentiation of Exponential Functions | p. 142 |

The Method of Logarithmic Implicit Differentiation | p. 142 |

The Integral of the Logarithm Function | p. 144 |

The Method of Integration by Parts | p. 146 |

Worksheet: Integration by Parts | p. 148 |

Exercises | p. 151 |

Integration by Trigonometric Substitution | p. 153 |

The Elliptical Rose Garden | p. 153 |

The Ellipse Area Integral | p. 154 |

Trying a Trigonometric Substitution | p. 155 |

The Area of the Ellipse | p. 158 |

The Method of Trigonometric Substitution | p. 158 |

Derivatives of Inverse Trigonometric Functions | p. 160 |

Worksheet: Integration by Trigonometric Substitution | p. 163 |

Exercises | p. 164 |

Integration by Partial Fractions | p. 167 |

The Red-and-Yellow Fireworks Problem | p. 169 |

The Integral of the Secant Function | p. 174 |

Worksheet: Partial Fractions | p. 175 |

Partial Fractions with Quadratic Denominators | p. 179 |

The Method of Partial Fractions | p. 185 |

Exercises | p. 185 |

Finding Volumes with Integrals | p. 190 |

The Pancake Method of Approximating Volume | p. 192 |

The Amazing Resemblance Between the Continuous Sum and the Definite Integral | p. 193 |

The Volume of the Paraboloid | p. 194 |

Finding Volumes with Cylindrical Shells | p. 197 |

Worksheet: Volume | p. 199 |

Exercises | p. 203 |

Arc Lengths, Surface Areas, and the Center of Mass | p. 205 |

The Straight-line Approximation for a Curve | p. 206 |

The Integral for Arc Lengths | p. 208 |

The Frustum Method of Finding Surface Areas | p. 213 |

The Center of Mass of the Concert Hall Stage | p. 215 |

Exercises | p. 219 |

Introduction to Differential Equations | p. 221 |

The Oscillating Ride and the Ordinary Differential Equation | p. 222 |

Linear Differential Equations | p. 224 |

The Force of Friction and the Damped Sine Wave | p. 231 |

Solution Method for Second-order Linear Homogeneous Constant-Coefficient Differential Equations | p. 233 |

The Driving Force and the Nonhomogeneous Equation | p. 234 |

Resonance and the Infinite Amplitude Ride | p. 240 |

Exercises | p. 241 |

Partial Derivatives and Vectors | p. 243 |

The Two-variable Magazine Subscription Problem | p. 243 |

Partial Derivatives | p. 245 |

The Dome | p. 246 |

Graphs of Functions of Two Variables | p. 247 |

The Gradient Vector | p. 248 |

The Sleigh Ride and the Falling Basket Ride | p. 250 |

Work, Kinetic Energy, and Potential Energy | p. 251 |

Describing Motion: The Position Vector as a Function of Time | p. 252 |

The Derivative of a Vector: Velocity | p. 254 |

The Dot Product of Two Vectors | p. 255 |

Circular Motion | p. 256 |

The Force Vector as the Gradient of the Potential Energy Function | p. 256 |

The Chain Rule for Vector Functions | p. 257 |

The Potential Energy for Gravity | p. 258 |

Exercises | p. 262 |

Numerical Methods, Taylor Series, and Limits | p. 268 |

The Problem Closet Break-in and the Gremlin's Secret Plan | p. 268 |

The Intractable Elliptic Integral | p. 272 |

The Height of the Children and the Area Under the Bell-shaped Normal Curve | p. 272 |

Approximating the Area with Rectangles | p. 273 |

Numerical Integration | p. 278 |

Approximating a Function with an Infinite Series of Derivatives | p. 281 |

Taylor Series | p. 283 |

Visualizing Differential Equations with Slope Fields | p. 284 |

The Ups and Downs of the Space Probe | p. 285 |

Euler's Method for Finding Numerical Solutions to Differential Equations | p. 287 |

The Formal Definition of Limit | p. 288 |

Continuous Curves | p. 289 |

L'Hopital's Rule for Finding Otherwise Unobtainable Limits | p. 291 |

Exercises | p. 293 |

Comprehensive Test of Calculus Problems | p. 296 |

The Return of the Gremlin | p. 296 |

The 45 Problems | p. 297 |

The Solutions | p. 301 |

Stanislavsky Guide to Calculus | p. 322 |

Answers to Worksheets and Exercises | p. 326 |

Summary of Trigonometric Formulas | p. 391 |

Brief Table of Integrals | p. 396 |

Glossary | p. 401 |

Index | p. 402 |

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