The Calculus Tutoring Book

Preface.

FUNCTIONS.

Introduction.

The Graph of a Function.

The Trigonometric Functions.

Inverse Functions and the Inverse Trigonometric Functions.

Exponential and Logarithm Functions.

Solving Inequalities Involving Elementary Functions.

Graphs of Translations, Reflections, Expansions and Sums.

Review of Problems for Chapter 1.

LIMITS.

Introduction.

Finding Limits of Combinations of Functions.

Indeterminate Limits.

Review Problems for Chapter 2.

THE DERIVATIVE PART I.

Preview.

Definition and Some Applications of the Derivative.

Derivatives of the Basic Functions.

Nondifferentiable Functions.

Derivatives of Constant Multiples, Sums, Products and Quotients.

The Derivative of a Composition.

Implicit Differentiation and Logarithmic Differentiation.

Antidifferentiation.

Review Problems for Chapter 3.

THE DERIVATIVE PART II.

Relative Maxima and Minima.

Absolute Maxima and Minima.

L'Hôpital's Rule and Orders of Magnitude.

Indeterminate Products, Differences and Exponential Forms.

Drawing Graphs of Functions.

Related Rates.

Newton's Method.

Differentials.

Separable Differential Equations.

Review Problems for Chapter 4.

THE INTEGRAL PART I.

Preview.

Definition and Some Applications of the Integral.

The Fundamental Theorem of Calculus.

Numerical Integration.

Nonintegrable Functions.

Improper Integals.

Review Problems for Chapter 5.

THE INTEGRAL PART II.

Further Applications of the Integral.

The Centroid of a Solid Hemisphere.

Area and Arc Length.

The Surface Area of a Cone and a Sphere.

Integrals with a Variable Upper Limit.

Review Problems for Chapter 6.

ANTIDIFFERENTIATION.

Introduction.

Substitution.

Pre-Table Algebra I.

Pre-Table Algebra II: Partial Faction Decomposition.

Integration by Parts.

Recursion Formulas.

Trigonometric Substitution.

Choosing a Method.

Combining Techniques of Antidifferentiation with the Fundamental Theorem.

Review Problems for Chapter 7.

SERIES.

Introduction.

Geometric Series.

Convergence Tests for Positive Series I.

Convergence Tests for Positive Series II.

Alternating Series.

Power Series Functions.

Power Series Representations for Elementary Functions I.

Power Series Representations for Elementary Functions II (Maclaurin Series).

The Taylor Reminder Formula and an Estimate for the Number e.

Power Series in Powers of x - b (Taylor Series).

Review Problems for Chapter 8.

VECTORS.

Introduction.

Vector Addition, Subtraction, Scalar Multiplication and Norms.

The Dot Product.

The Cross Product.

The Scalar Triple Product.

The Velocity Vector.

The Acceleration Vector.

Review Problems for Chapter 9.

TOPICS IN THREE-DIMENSIONAL ANALYTIC GEOMETRY.

Spheres.

Planes.

Lines.

Cylindrical and Quadric Surfaces.

Cylindrical and Spherical Coordinates.

Review Problems for Chapter 10.

PARTIAL DERIVATIVES.

Graphs and Level Sets.

Partial Derivatives.

Chain Rules for First-Order Partial Derivatives.

Chain Rules for Second-Order Partial Derivatives.

Maxima and Minima.

The Gradient.

Differentials and Exact Differential Equations.

Review Problems for Chapter 11.

MULTIPLE INTEGRALS.

Definition and Some Applications of the Double Integral.

Computing Double Integrals.

Double Integration in Polar Coordinates.

Area and Volume.

Further Applications of the Double Integral.

Triple Integrals.

Triple Integration in Spherical Coordinates.

Center of Mass.

Review Problems for Chapter 12.

APPENDIX.

A1: Distance and Slope.

A2: Equations of Lines.

A3: Circles, Ellipses, Hyperbolas and Parabolas.

A4: The Binomial Theorem.

A5: Determinants.

A6: Polar Coordinates.

Solutions to the Problems.

Abbreviations Used in the Solutions.

List of Symbols.

Index.

Authors' Biographies.