Calculus : Concepts and Applications

by Foerster, Paul

• ISBN13: 9781559531177
• ISBN10: 1559531177
• Format: Hardcover
• Copyright: 1998-11-01
• Publisher: Springer Verlag

Table of Contents

Experiments and Projects	xi
A Note to the Student	xiii
Limits, Derivatives, Integrals, and Integrals	1	(36)
The Concept of Instantaneous Rate	3	(3)
Rate of Change by Equation, Graph, or Table	6	(7)
One Type of Integral of a Function	13	(5)
Definite Integrals by Trapezoids, from Equations and Data	18	(7)
Limit of a Function	25	(6)
Calculus Journal	31	(2)
Chapter Review and Test	33	(4)
Properties of Limits	37	(40)
Numerical Approach to the Definition of Limit	39	(1)
Graphical and Algebraic Approaches to the Definition of Limit	40	(5)
The Limit Theorems	45	(7)
Continuity	52	(8)
Limits Involving Infinity	60	(7)
The Intermediate Value Theorem and Its Consequences	67	(4)
Chapter Review and Test	71	(6)
Derivatives, Antiderivatives, and Indefinite Integrals	77	(52)
Graphical Interpretation of Derivative	79	(1)
Difference Quotients and One Definition of Derivative	80	(4)
Derivative Functions, Numerically and Graphically	84	(6)
Derivative of the Power Function and Another Definition of Derivative	90	(8)
Displacement, Velocity, and Acceleration	98	(7)
Introduction to Sine, Cosine, and Composite Functions	105	(2)
Derivatives of Composite Functions---The Chain Rule	107	(4)
Proof and Application of Sine and Cosine Derivatives	111	(8)
Antiderivatives and Indefinite Integrals	119	(4)
Chapter Review and Test	123	(6)
Products, Quotients, and Parametric Functions	129	(50)
Combinations of Two Functions	131	(1)
Derivative of a Product of Two Functions	132	(4)
Derivative of a Quotient of Two Functions	136	(5)
Derivatives of the Other Trigonometric Functions	141	(4)
Derivatives of Inverse Trigonometric Functions	145	(8)
Differentiability and Continuity	153	(7)
Derivative of a Parametric Function	160	(8)
Graphs and Derivatives of Implicit Relations	168	(4)
Chapter Review and Test	172	(7)
Definite and Indefinite Integrals	179	(70)
A Definite Integral Problem	181	(1)
Review of Antiderivatives	182	(1)
Linear Approximations and Differentials	183	(6)
Formal Definition of Antiderivative and Indefinite Integral	189	(6)
Riemann Sums and the Definition of Definite Integral	195	(7)
The Mean Value Theorem and Rolle's Theorem	202	(11)
Some Very Special Riemann Sums	213	(2)
The Fundamental Theorem of Calculus	215	(4)
Definite Integral Properties and Practice	219	(6)
A Way to Apply Definite Integrals	225	(5)
Numerical Integration by Simpson's Rule and a Grapher	230	(10)
Chapter Review and Test	240	(9)
The Calculus of Exponential and Logarithmic Functions	249	(58)
Integral of the Reciprocal Function: A Population Growth Problem	251	(1)
Antiderivative of the Reciprocal Function	252	(1)
Natural Logarithms, and Another Form of the Fundamental Theorem	253	(10)
ln x Really Is a Logarithmic Function	263	(5)
Derivatives of Exponential Functions---Logarithmic Differentiation	268	(4)
The Number e, and the Derivative of Base b Logarithm Functions	272	(7)
The Natural Exponential Function: The Inverse of ln	279	(6)
Limits of Indeterminate Forms: l'Hospital's Rule	285	(5)
Derivative and Integral Practice for Transcendental Functions	290	(6)
Chapter Review and Test	296	(6)
Cumulative Review: Chapters 1-6	302	(5)
The Calculus of Growth and Decay	307	(44)
Direct Proportion Property of Exponential Functions	309	(1)
Exponential Growth and Decay	310	(6)
Other Differential Equations for Real-World Applications	316	(10)
Graphical Solution of Differential Equations by Using Slope Fields	326	(7)
Numerical Solution of Differential Equations by Using Euler's Method	333	(5)
Predator-Prey Population Problems	338	(3)
Chapter Review and Test	341	(5)
Cumulative Review: Chapters 1-7	346	(5)
The Calculus of Plane and Solid Figures	351	(82)
Cubic Functions and Their Derivatives	353	(1)
Critical Points and Points of Inflection	354	(15)
Maxima and Minima in Plane and Solid Figures	369	(11)
Area of a Plane Region	380	(5)
Volume of a Solid by Plane Slicing	385	(11)
Volume of a Solid of Revolution by Cylindrical Shells	396	(7)
Length of a Plane Curve---Arc Length	403	(7)
Area of a Surface of Revolution	410	(7)
Lengths and Areas for Polar Coordinates	417	(9)
Chapter Review and Test	426	(7)
Algebraic Calculus Techniques for the Elementary Functions	433	(70)
Introduction to the Integral of a Product of Two Functions	435	(1)
Integration by Parts---A Way to Integrate Products	436	(4)
Rapid Repeated Integration by Parts	440	(6)
Reduction Formulas and Computer Software	446	(5)
Integrating Special Powers of Trigonometric Functions	451	(5)
Integration by Trigonometric Substitution	456	(6)
Integration of Rational Functions by Partial Fractions	462	(7)
Integrals of the Inverse Trigonometric Functions	469	(4)
Calculus of the Hyperbolic and Inverse Hyperbolic Functions	473	(12)
Improper Integrals	485	(8)
Miscellaneous Integrals and Derivatives	493	(5)
Integrals in Journal	498	(1)
Chapter Review and Test	499	(4)
The Calculus of Motion---Averages, Extremes, and Vectors	503	(52)
Introduction to Distance and Displacement for Motion Along a Line	505	(1)
Distance, Displacement, and Acceleration for Linear Motion	506	(7)
Average Value Problems in Motion and Elsewhere	513	(4)
Related Rates	517	(6)
Minimal Path Problems	523	(6)
Maximum and Minimum Problems in Motion and Elsewhere	529	(3)
Vector Functions for Motion in a Plane	532	(16)
Chapter Review and Test	548	(7)
The Calculus of Variable-Factor Products	555	(42)
Review of Work---Force Times Displacement	557	(1)
Work Done by a Variable Force	557	(6)
Mass of a Variable-Density Object	563	(5)
Moments, Centroids, Center of Mass, and the Theorem of Pappus	568	(10)
Force Exerted by a Variable Pressure---Center of Pressure	578	(6)
Other Variable-Factor Products	584	(8)
Chapter Review and Test	592	(5)
The Calculus of Functions Defined by Power Series	597	(64)
Introduction to Power Series	599	(1)
Geometric Sequences and Series as Mathematical Models	600	(7)
Power Series for an Exponential Function	607	(2)
Power Series for Other Elementary Functions	609	(6)
Taylor and Maclaurin Series, and Operations on These Series	615	(7)
Interval of Convergence for a Series---The Ratio Technique	622	(8)
Convergence of Series at the Ends of the Convergence Interval	630	(12)
Error Analysis for Series	642	(7)
Chapter Review and Test	649	(5)
Cumulative Reviews	654	(7)
Final Examination: ``A Guided Tour Through Calculus''	661	(4)
Appendix A: Summary of Properties of Trigonometric Functions	665	(2)
Appendix B: Answers to Selected Problems	667	(94)
Glossary	761	(4)
Index of Problem Titles	765	(4)
General Index	769