內容大鋼
The first edition was intended to be a synthesis of reform and traditional approaches to calculus instruction.In this second edition I continue to follow that path by empha-sizing conceptual understanding through visual,numerical,and algebraic approaches.The principal way in which this book differs from my more traditional calculus textbooks is that it is more streamlined. For instance,there is no complete chapter on techniques of integration;I don't prove as many theorems;and the material on transcendental functions and on parametric equations is interwoven throughout the book instead of being treated in separate chapters.Instruc-tors who prefer fuller coverage of traditional calculus topics should look at my books Calculus,Fourth Edition and Calculus: Early Transcendentals,Fourth Edition.
目錄
A Preview of Calculus
1 Functions and Models
1.1 Four Ways to Represent a Function
1.2 Mathematical Models
1.3 New Functions from Old Functions
1.4 Graphing Calculators and Computers
1.5 Exponential Functions
1.6 Inverse Functions and Logarithms
1.7 Parametric Curves
Laboratory Project Running Circles around Circles
Review
Principles of Problem Solving
2 Limits and Derivatives
2.1 The Tangent and Velocity Problems
2.2 The Limit of a Function
2.3 Calculating Limits Using the Limit Laws
2.4 Continuity
2.5 Limits Involving Infinity
2.6 Tangents, Velocities, and Other Rates of Change
2.7 Derivatives
Writing Project: Early Methods for Finding Tangents
2.8 The Derivative as a Function
2.9 Linear Approximations
2.10 What Does f' Say about f?
Review
Focus on Problem Solving
3 Differentiation Rules
3.1 Derivatives of Polynomials and Exponential Functions
Applied Project Building a Better Roller Coaster
3.2 The Product and Quotient Rules
3.3 Rates of Change in the Natural and Social Sciences
3.4 Derivatives of Trigonometric Functions
3.5 The Chain Rule
Laboratory Project Bezier Curves
Applied Project Where Should a Pilot Start Descent
3.6 Implicit Differentiation
3.7 Derivatives of Logarithmic Functions
Discovery Project Q Hyperbolic Functions
3.8 Linear Approximations and Differentials
Laboratory Project ~ Taylor Polynomials
Review
Focus on Problem Solving
4 Applications of Differentiation
4.1 Related Rates
4.2 Maximum and Minimum Values
Applied Project The Calculus of Rainbows
4.3 Derivatives and the Shapes of Curves
4.4 Graphing with Calculus and Calculators
4.5 Indeterminate Forms and l'Hospital's Rule
Writing Project The Origins of l'Hospital's Rule
4.6 Optimization Problems
Applied Project The Shape of a Can
4.7 Applications to Economics
4.8 Newton's Method
4.9 Antiderivatives
Review
Focus on Problem Solving
5 Integrals
5.1 Areas and Distances
5.2 The Definite Integral
5.3 Evaluating Definite Integrals
Discovery Project Area Functions
5.4 The Fundamental Theorem of Calculus
Writing Project Newton, Leibniz, and the Invention of Calculus
5.5 The Substitution Rule
5.6 Integration by Parts
5.7 Additional Techniques of Integration
5.8 Integration Using Tables and Computer Algebra Systems
Discovery Project ~ Patterns in Integrals
5.9 Approximate Integration
5.10 Improper Integrals
Review
Focus on Problem Solving
6 Applications of Integration
6.1 More about Areas
6.2 Volumes
Discovery Project o Rotating on a Slant
6.3 Arc Length
Discovery Project Arc Lengh Contest
6.4 Average Value of a Function
Applied Project Where to Sit at the Movies
6.5 Applications to Physics and Engineering
6.6 Applications to Economics and Biology
6.7 Probability
Review
Focus on Problem Solving
7 Differential Equations
7.1 Modeling with Differential Equations
7.2 Direction Fields and Euler's Method
7.3 Separable Equations
Applied Project o Which Is Faster, Going Up or Coming Down
7.4 Exponential Growth and Decay
Applied Project o Calculus and Baseball
7.5 The Logistic Equation
7.6 Predator-Prey Systems
Review
Focus on Problem Solving
8 Infinite Sequences and Series
8.1 Sequences
Laboratory Project Logistic Sequences
8.2 Series
8.3 The Integral and Comparison Tests; Estimating Sums
8.4 Other Convergence Tests
8.5 Power Series
8.6 Representations of Functions as Power Series
8.7 Taylor and Maclaurin Series
8.8 The Binomial Series
Writing Project How Newton Discovered the Binomial Series
8.9 Applications of Taylor Polynomials
Applied Projeet Radiation from the Stars
8.10 Using Series to Solve Differential Equations
Review
Focus on Problem Solving
9 Vectors and the Geometry of Space
9.1 Three-Dimensional Coordinate Systems
9.2 Vectors
9.3 The Dot Product
9.4 The Cross Product
Discovery Projeet The Geometry of a Tetrahedron
9.5 Equations of Lines and Planes
9.6 Functions and Surfaces
9.7 Cylindrical and Spherical Coordinates
Laboratory Project Families of Surfaces
Review
Focus oil Problem Solving
10 Vector Functions
10.1 Vector Functions and Space Curves
10.2 Derivatives and Integrals of Vector Functions
10.3 Arc Length and Curvature
10.4 Motion in Space
Applied Project Kepler's Laws
10.5 Parametric Surfaces
Review
Focus on Problem Solving
11 Partial Derivatives
11.1 Functions of Several Variables
11.2 Limits and Continuity
11.3 Partial Derivatives
11.4 Tangent Planes and Linear Approximations
11.5 The Chain Rule
11.6 Directional Derivatives and the Gradient Vector
11.7 Maximum and Minimum Values
Applied Project Designing a Dumpster
Discovery Project Quadratic Approximations and Critical Points
11.8 Lagrange Multipliers
Applied Project Rocket Science
Apldied Project Hydro-Turbine Optimization
Review
Focus on Problem Solving
12 Multiple Integrals
12.1 Double Integrals over Rectangles
12.2 Iterated Integrals
12.3 Double Integrals over General Regions
12.4 Double Integrals in Polar Coordinates
12.5 Applications of Double Integrals
12.6 Surface Area
12.7 Triple Integrals
Discovery Project Volumes of Hyperspheres
12.8 Triple Integrals in Cylindrical and Spherical Coordinates
Apldied Project Roller Derby
Discovery Project The Intersection of Three Cylinders
12.9 Change of Variables in Multiple Integrals
Review
Focus on Problem Solving
13 Vector Calculus
13.1 Vector Fields
13.2 Line Integrals
13.3 The Fundamental Theorem for Line Integrals
13.4 Green's Theorem
13.5 Curl and Divergence
13.6 Surface Integrals
13.7 Stokes' Theorem
Writing Project o Three Men and Two Theorems
13.8 The Divergence Theorem
13.9 Summary
Review
Foeus on Problem Solving
Appendixes
A Intervals, Inequalities, and Absolute Values
B Coordinate Geometry
C Trigonometry
D Precise Definitions of Limits
E A Few Proofs
F Sigma Notation
G Integration of Rational Functions by Partial Fractions
H Polar Coordinates
I Complex Numbers
J Answers to Odd-Numbered Exercises
Index
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