Single and Multivariable Calculus: Early Transcendentals

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Contents Introduction 1. Analytic Geometry 1. Lines 2. Distance Between Two Points; Circles 3. Functions 4. Shifts and Dilations 2. Instantaneous Rate of Change: The Derivative 1. The slope of a function 2. An example 3. Limits 4. The Derivative Function 5. Adjectives For Functions 3. Rules for Finding Derivatives 1. The Power Rule 2. Linearity of the Derivative 3. The Product Rule 4. The Quotient Rule 5. The Chain Rule 4. Transcendental Functions 1. Trigonometric Functions 2. The Derivative of the sine 3. A hard limit 4. The Derivative of the sine, continued 5. Derivatives of the Trigonometric Functions 6. Exponential and Logarithmic functions 7. Derivatives of the exponential and logarithmic functions 8. Implicit Differentiation 9. Inverse Trigonometric Functions 10. Limits revisited 11. Hyperbolic Functions 5. Curve Sketching 1. Maxima and Minima 2. The first derivative test 3. The second derivative test 4. Concavity and inflection points 5. Asymptotes and Other Things to Look For 6. Applications of the Derivative 1. Optimization 2. Related Rates 3. Newton's Method 4. Linear Approximations 5. The Mean Value Theorem 7. Integration 1. Two examples 2. The Fundamental Theorem of Calculus 3. Some Properties of Integrals 8. Techniques of Integration 1. Substitution 2. Powers of sine and cosine 3. Trigonometric Substitutions 4. Integration by Parts 5. Rational Functions 6. Numerical Integration 7. Additional exercises 9. Applications of Integration 1. Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9. Arc Length 10. Surface Area 10. Polar Coordinates, Parametric Equations 1. Polar Coordinates 2. Slopes in polar coordinates 3. Areas in polar coordinates 4. Parametric Equations 5. Calculus with Parametric Equations 11. Sequences and Series 1. Sequences 2. Series 3. The Integral Test 4. Alternating Series 5. Comparison Tests 6. Absolute Convergence 7. The Ratio and Root Tests 8. Power Series 9. Calculus with Power Series 10. Taylor Series 11. Taylor's Theorem 12. Additional exercises 12. Three Dimensions 1. The Coordinate System 2. Vectors 3. The Dot Product 4. The Cross Product 5. Lines and Planes 6. Other Coordinate Systems 13. Vector Functions 1. Space Curves 2. Calculus with vector functions 3. Arc length and curvature 4. Motion along a curve 14. Partial Differentiation 1. Functions of Several Variables 2. Limits and Continuity 3. Partial Differentiation 4. The Chain Rule 5. Directional Derivatives 6. Higher order derivatives 7. Maxima and minima 8. Lagrange Multipliers 15. Multiple Integration 1. Volume and Average Height 2. Double Integrals in Cylindrical Coordinates 3. Moment and Center of Mass 4. Surface Area 5. Triple Integrals 6. Cylindrical and Spherical Coordinates 7. Change of Variables 16. Vector Calculus 1. Vector Fields 2. Line Integrals 3. The Fundamental Theorem of Line Integrals 4. Green's Theorem 5. Divergence and Curl 6. Vector Functions for Surfaces 7. Surface Integrals 8. Stokes's Theorem 9. The Divergence Theorem 17. Differential Equations 1. First Order Differential Equations 2. First Order Homogeneous Linear Equations 3. First Order Linear Equations 4. Approximation 5. Second Order Homogeneous Equations 6. Second Order Linear Equations 7. Second Order Linear Equations, take two A. Selected Answers B. Useful Formulas Index