Calculus and Analytic Geometry
1. An Overview of Calculus.
1.1 The Derivative
1.2 The Integral
1.3 Survey of the Text
2. Functions, Limits, and Continuity.
2.2 Composite Functions
2.3 The Limit of a Function
2.4 Computations of Limits
2.5 Some Tools for Graphing
2.6 A Review of Trigonometry
2.7 The Limit of (sin Ø)/Ø as Ø Approaches 0
2.8 Continuous Functions
2.9 Precise Definitions of "lim(x->infinity)f(x)=infinity" and "lim(x->infinity)f(x)=L"
2.10 Precise Definition of "lim(x->a)f(x)=L"
3. The Derivative.
3.1 Four Problems with One Theme
3.2 The Derivative
3.3 The Derivative and Continuity
3.4 The Derivative of the Sum, Difference, Product, and Quotient
3.5 The Derivatives of the Trigonometric Functions
3.6 The Derivative of a Composite Function
4. Applications of the Derivative.
4.1 Three Theorems about the Derivative
4.2 The First Derivative and Graphing
4.3 Motion and the Second Derivative
4.4 Related Rates
4.5 The Second Derivative and Graphing
4.6 Newton's Method for Solving an Equation
4.7 Applied Maximum and Minimum Problems
4.9 The Differential and Linearization
4.10 The Second Derivative and Growth of a Function
5. The Definite Integral.
5.1 Estimates in Four Problems
5.2 Summation Notation and Approximating Sums
5.3 The Definite Integral
5.4 Estimating the Definite Integral
5.5 Properties of the Antiderivative and the Definite Integral
5.6 Background for the Fundamental Theorems of Calculus
5.7 The Fundamental Theorems of Calculus
6. Topics in Differential Calculus.
6.2 The Number e
6.3 The Derivative of a Logarithmic Function
6.4 One-to-One Functions and Their Inverses
6.5 The Derivative of b^x
6.6 The Derivatives of the Inverse Trigonometric Functions
6.7 The Differential Equation of Natural Growth and Decay
6.8 l'Hopital's Rule
6.9 The Hyperbolic Functions and Their Inverses
7. Computing Antiderivatives.
7.1 Shortcuts, Integral Tables, and Machines
7.2 The Substitution Method
7.3 Integration by Parts
7.4 How to Integrate Certain Rational Functions
7.5 Integration of Rational Functions by Partial Fractions
7.6 Special Techniques
7.7 What to Do in the Face of an Integral
8. Applications of the Definite Integral.
8.1 Computing Area by Parallel Cross Sections
8.2 Some Pointers on Drawing
8.3 Setting Up a Definite Integral
8.4 Computing Volumes
8.5 The Shell Method
8.6 The Centroid of a Plane Region
8.8 Improper Integrals
9. Plane Curves and Polar Coordinates.
9.1 Polar Coordinates
9.2 Area in Polar Coordinates
9.3 Parametric Equations
9.4 Arc Length and Speed on a Curve
9.5 The Area of a Surface of Revolution
9.7 The Reflection Properties of the Conic Sections
10.1 An Informal Introduction to Series
10.4 The Integral Test
10.5 Comparison Tests
10.6 Ratio Tests
10.7 Tests for Series with Both Positive and Negative Terms
11. Power Series and Complex Numbers.
11.1 Taylor Series
11.2 The Error in Taylor Series
11.3 Why the Error in Taylor Series Is Controlled by a Derivative
11.4 Power Series and Radius of Convergence
11.5 Manipulating Power Series
11.6 Complex Numbers
11.7 The Relation between the Exponential and the Trigonometric Functions
12.1 The Algebra of Vectors
12.3 The Dot Product of Two Vectors
12.4 Lines and Planes
12.6 The Cross Product of Two Vectors
12.7 More on Lines and Planes
13. The Derivative of a Vector Function.
13.1 The Derivative of a Vector Function
13.2 Properties of the Derivative of a Vector Function
13.3 The Acceleration Vector
13.4 The Components of Acceleration
13.5 Newton's Law Implies Kepler's Laws
14. Partial Derivatives.
14.2 Quadratic Surfaces
14.3 Functions and Their Level Curves
14.4 Limits and Continuity
14.5 Partial Derivatives
14.6 The Chain Rule
14.7 Directional Derivatives and the Gradient
14.8 Normals and the Tangent Plane
14.9 Critical Points and Extrema
14.10 Lagrange Multipliers
14.11 The Chain Rule Revisited
15. Definite Integrals over Plane and Solid Regions.
15.1 The Definite Integral of a Function over a Region in the Plane
15.2 Computing |R f(P) dA Using Rectangular Coordinates
15.3 Moments and Centers of Mass
15.4 Computing |R f(P) dA Using Polar Coordinates
15.5 The Definite Integral of a Function over a Region in Space
15.6 Computing |R f(P) dV Using Cylindrical Coordinates
15.7 Computing |R f(P) dV Using Spherical Coordinates
16. Green's Theorem.
16.1 Vector and Scalar Fields
16.2 Line Integrals
16.3 Four Applications of Line Integrals
16.4 Green's Theorem
16.5 Applications of Green's Theorem
16.6 Conservative Vector Fields
17. The Divergence Theorem and Stokes' Theorem.
17.1 Surface Integrals
17.2 The Divergence Theorem
17.3 Stokes' Theorem
17.4 Applications of Stokes' Theorem
A. Real Numbers.
B. Graphs and Lines.
C. Topics in Algebra.
E. Mathematical Induction.
F. The Converse of a Statement.
G. Conic Sections.
H. Logarithms and Exponentials Defined through Calculus.
I. The Taylor Series for f(x,y).
J. Theory of Limits.
K. The Interchange of Limits.
L. The Jacobian.
M. Linear Differential Equations with Constant Coefficients.
Answers to Selected Odd-Numbered Problems and to Guide Quizzes
List of Symbols
| About the Authors
- Sherman Stein, received his Ph.D. from Columbia University. After a one-year instructorship at Princeton University, he joined the faculty at the University of California, Davis, where he taught until 1993. His main mathematical interests are in algebra, combinatorics, and pedagogy. He has been the recipient of two MAA awards; the Lester R. Ford Award for Mathematical Exposition, and the Beckenbach Book Prize for Algebra and Tiling (with Sandor Szabo). He also received The Distinguished Teaching Award from the University of California, Davis, and an honorary Doctor of Humane Letters from Marietta College.