Calculus : basic concepts and applications

Here is a textbook of intuitive calculus. The material is presented in a concrete setting with many examples and problems chosen from the social, physical, behavioural and life sciences. Chapters include core material and more advanced optional sections. The book begins with a review of algebra and graphing. Cover......Page 1 CALCULUS: Basic concepts and applications......Page 4 9780521095907......Page 5 Contents......Page 6 Preface......Page 12 How this book is organized, and how it can be used......Page 14 Some study hints......Page 16 0.1 Fundamental operations; parentheses......Page 18 0.2 Zero and negatives......Page 20 0.3 Fractions and rational numbers......Page 22 0.4 Integral exponents......Page 25 0.5 Radicals, fractional exponents, and real numbers......Page 26 0.7 Equalities......Page 28 0.8 Inequalities......Page 30 0.9 Linear equations......Page 33 0.10 Quadratic equations......Page 35 0.11 Higher-degree equations......Page 37 0.12 Progressions......Page 41 0.13 Logarithms......Page 44 0.14 Keeping track of units......Page 46 0.15 Mensuration formulas......Page 47 1.2 An example......Page 50 1.3 Variation of one quantity with another; graphical interpolation......Page 51 1.4 More on graphing, interpolation, and extrapolation......Page 54 1.5 Linear interpolation......Page 58 1.6 Relations expressed by formulas......Page 61 * 1.7 Formulas (continued)......Page 70 1.8 Relationships between science and mathematics......Page 72 1.9 Functions......Page 73 1.10 Further discussion of functions: notation and natural domains......Page 76 1.11 Inverse functions......Page 79 1.13 Summary......Page 82 2.1 Average speed and average velocity......Page 89 2.2 Instantaneous velocity and limits......Page 91 2.3 Theorems on limits......Page 98 * 2.4 Proofs of some results on limits......Page 101 2.5 Average slope in an interval and slope at a point......Page 104 2.6 Tangent to a curve......Page 108 2.7 The derivative......Page 110 * 2.8 Guessing limits with a calculator......Page 118 2.9 Review......Page 120 3.1 The Mean-Value Theorem......Page 127 3.2 Increasing and decreasing functions......Page 129 3.3 Approximate increments......Page 131 3.4 Applications to economics: marginal cost and unit cost......Page 136 3.5 Maxima and minima: the basic idea......Page 140 3.6 How do we know whether we have a maximum or a minimum?......Page 142 3.7 Further questions about maxima and minima......Page 146 3.8 Applied maxima and minima......Page 147 3.9 Maxima and minima in some problems in economics......Page 153 3.10 Approximate solution of equations: the Newton-Raphson Contents method and the bisection method......Page 156 3.11 Review......Page 161 4.1 Repeated differentiation and derived curves......Page 165 4.2 Points of inflection and third test for maxima and minima......Page 169 4.3 Extreme rates......Page 173 4.4 Derivative of a function of a function: the Chain Rule......Page 174 4.5 Continuity......Page 178 * 4.6 Proof that differentiability implies continuity and proof of the Chain Rule......Page 181 4.7 Notation......Page 182 4.8 Related rates......Page 185 4.9 Functions in implicit form and implicit differentiation......Page 188 4.10 Derivatives of fractional powers......Page 190 4.11 Implicit differentiation applied to related rates......Page 191 4.12 Differentials......Page 193 4.13 Formulas for derivatives of products and quotients......Page 195 4.14 Marginal cost, marginal revenue, and optimal production levels......Page 199 * 4.15 Maxima and minima using implicit differentiation......Page 202 4.16 Summary......Page 203 5.1 The reverse of differentiation......Page 211 5.2 The antiderivatives of a given function differ by at most a constant......Page 213 5.3 Formulas for antiderivatives......Page 214 5.4 Repeated antidifferentiation: projectiles thrown vertically......Page 219 5.5 The limit of a sum......Page 222 5.6 Further limits of sums......Page 226 5.7 The Fundamental Theorem......Page 230 5.8 Applications of the Fundamental Theorem......Page 234 5.9 Use of the Chain Rule in integration (antidifferentiation)......Page 239 5.10 The indefinite integral......Page 241 5.11 Summary......Page 242 6.1 Introduction to exponential functions......Page 248 6.2 The rate of change: preliminary remarks......Page 252 6.3 Compound interest......Page 254 6.4 Continuous compounding......Page 257 6.5 The derivative of the exponential function......Page 260 6.6 Relative errors and relative rates......Page 263 6.7 Antiderivatives of the exponential......Page 266 6.8 e^u: derivative and antiderivative......Page 270 6.9 Summary......Page 272 7.2 Inverse functions and the inverse of the exponential......Page 274 7.3 Laws of logarithms......Page 276 7.4 The derivative of the log function......Page 281 7.5 Antiderivatives of 1/x......Page 283 7.6 Derivatives of b^x and log_bx......Page 285 7.7 Log-log and semilog graphs......Page 286 7.8 Summary......Page 291 8.1 Introduction......Page 299 8.2 An approximate solution of a differential equation......Page 301 8.3 Variables separable......Page 303 8.4 Comparison of approximate and exact solutions......Page 304 8.5 Population changes......Page 306 8.6 The logistic equation......Page 307 8.7 The method of partial fractions......Page 308 8.8 The logistic equation (continued)......Page 309 8.9 Linear differential equations with constant coefficients......Page 311 8.10 Linear differential equations with constant coefficients (continued)......Page 316 * 8.11 Approximating the solutions of a pair of simultaneous differential equations......Page 319 9.2 Review of the use of the Chain Rule in integration (antidifferentiation)......Page 325 9.3 Force of attraction......Page 327 9.4 Loads......Page 329 9.5 Moment of a force......Page 331 9.6 Consumers' and producers' surpluses......Page 332 9.7 Horizontal rectangular strips and circular strips......Page 334 9.8 The idea of an average......Page 337 9.9 Average velocity......Page 338 9.10 The average of a function defined on an interval......Page 339 * 9.11 Further averages......Page 342 9.12 Summary......Page 346 9.13 Quadrature......Page 351 9.14 More on quadrature: the trapezoidal rule and its adjustment......Page 353 10.1 Introduction......Page 360 10.2 Angle measure......Page 363 10.3 The sine and cosine functions......Page 366 10.4 The tangent function, and application of the basic functions to triangles......Page 371 10.5 Differentiation of the trigonometric functions......Page 375 10.6 Antidifferentiation and integration of trigonometric functions......Page 380 10.7 Inverse trigonometric functions......Page 382 10.8 Further integration involving trigonometric functions......Page 388 * 10.9 Other periodic functions......Page 392 10.10 A return to differential equations......Page 397 10.11 Summary......Page 403 Answers to selected problems......Page 408 A Compound interest: (1 + r)^n......Page 428 B_1 Values of e^x and e^{-x}......Page 429 B_2 Natural logarithms (In x)......Page 430 C Logarithms, base 10......Page 432 D Trigonometric functions......Page 434 Index......Page 438 Cover 1 CALCULUS: Basic concepts and applications 4 Copyright 5 9780521250122 5 9780521095907 5 Contents 6 Preface 12 How this book is organized, and how it can be used 14 Some study hints 16 0 Prerequisites 18 0.1 Fundamental operations; parentheses 18 0.2 Zero and negatives 20 0.3 Fractions and rational numbers 22 0.4 Integral exponents 25 0.5 Radicals, fractional exponents, and real numbers 26 0.6 Notation for implication 28 0.7 Equalities 28 0.8 Inequalities 30 0.9 Linear equations 33 0.10 Quadratic equations 35 0.11 Higher-degree equations 37 0.12 Progressions 41 0.13 Logarithms 44 0.14 Keeping track of units 46 0.15 Mensuration formulas 47 1 Functional relationships 50 1.1 Introduction 50 1.2 An example 50 1.3 Variation of one quantity with another; graphical interpolation 51 1.4 More on graphing, interpolation, and extrapolation 54 1.5 Linear interpolation 58 1.6 Relations expressed by formulas 61 * 1.7 Formulas (continued) 70 1.8 Relationships between science and mathematics 72 1.9 Functions 73 1.10 Further discussion of functions: notation and natural domains 76 1.11 Inverse functions 79 * 1.12 Absolute values 82 1.13 Summary 82 2 Rate of change 89 2.1 Average speed and average velocity 89 2.2 Instantaneous velocity and limits 91 2.3 Theorems on limits 98 * 2.4 Proofs of some results on limits 101 2.5 Average slope in an interval and slope at a point 104 2.6 Tangent to a curve 108 2.7 The derivative 110 * 2.8 Guessing limits with a calculator 118 2.9 Review 120 3 Applications of the derivative 127 3.1 The Mean-Value Theorem 127 3.2 Increasing and decreasing functions 129 3.3 Approximate increments 131 3.4 Applications to economics: marginal cost and unit cost 136 3.5 Maxima and minima: the basic idea 140 3.6 How do we know whether we have a maximum or a minimum? 142 3.7 Further questions about maxima and minima 146 3.8 Applied maxima and minima 147 3.9 Maxima and minima in some problems in economics 153 3.10 Approximate solution of equations: the Newton-Raphson Contents method and the bisection method 156 3.11 Review 161 4 Further differentiation 165 4.1 Repeated differentiation and derived curves 165 4.2 Points of inflection and third test for maxima and minima 169 4.3 Extreme rates 173 4.4 Derivative of a function of a function: the Chain Rule 174 4.5 Continuity 178 * 4.6 Proof that differentiability implies continuity and proof of the Chain Rule 181 4.7 Notation 182 4.8 Related rates 185 4.9 Functions in implicit form and implicit differentiation 188 4.10 Derivatives of fractional powers 190 4.11 Implicit differentiation applied to related rates 191 4.12 Differentials 193 4.13 Formulas for derivatives of products and quotients 195 4.14 Marginal cost, marginal revenue, and optimal production levels 199 * 4.15 Maxima and minima using implicit differentiation 202 4.16 Summary 203 5 Antidifferentiation and integration 211 5.1 The reverse of differentiation 211 5.2 The antiderivatives of a given function differ by at most a constant 213 5.3 Formulas for antiderivatives 214 5.4 Repeated antidifferentiation: projectiles thrown vertically 219 5.5 The limit of a sum 222 5.6 Further limits of sums 226 5.7 The Fundamental Theorem 230 5.8 Applications of the Fundamental Theorem 234 5.9 Use of the Chain Rule in integration (antidifferentiation) 239 5.10 The indefinite integral 241 5.11 Summary 242 6 Exponential functions 248 6.1 Introduction to exponential functions 248 6.2 The rate of change: preliminary remarks 252 6.3 Compound interest 254 6.4 Continuous compounding 257 6.5 The derivative of the exponential function 260 6.6 Relative errors and relative rates 263 6.7 Antiderivatives of the exponential 266 6.8 e^u: derivative and antiderivative 270 6.9 Summary 272 7 Logarithmic functions 274 7.1 Introduction 274 7.2 Inverse functions and the inverse of the exponential 274 7.3 Laws of logarithms 276 7.4 The derivative of the log function 281 7.5 Antiderivatives of 1/x 283 7.6 Derivatives of b^x and log_bx 285 7.7 Log-log and semilog graphs 286 7.8 Summary 291 8 Differential equations 299 8.1 Introduction 299 8.2 An approximate solution of a differential equation 301 8.3 Variables separable 303 8.4 Comparison of approximate and exact solutions 304 8.5 Population changes 306 8.6 The logistic equation 307 8.7 The method of partial fractions 308 8.8 The logistic equation (continued) 309 8.9 Linear differential equations with constant coefficients 311 8.10 Linear differential equations with constant coefficients (continued) 316 * 8.11 Approximating the solutions of a pair of simultaneous differential equations 319 9 Further integration 325 9.1 Introduction 325 9.2 Review of the use of the Chain Rule in integration (antidifferentiation) 325 9.3 Force of attraction 327 9.4 Loads 329 9.5 Moment of a force 331 9.6 Consumers' and producers' surpluses 332 9.7 Horizontal rectangular strips and circular strips 334 9.8 The idea of an average 337 9.9 Average velocity 338 9.10 The average of a function defined on an interval 339 * 9.11 Further averages 342 9.12 Summary 346 9.13 Quadrature 351 9.14 More on quadrature: the trapezoidal rule and its adjustment 353 10 Trigonometric functions 360 10.1 Introduction 360 10.2 Angle measure 363 10.3 The sine and cosine functions 366 10.4 The tangent function, and application of the basic functions to triangles 371 10.5 Differentiation of the trigonometric functions 375 10.6 Antidifferentiation and integration of trigonometric functions 380 10.7 Inverse trigonometric functions 382 10.8 Further integration involving trigonometric functions 388 * 10.9 Other periodic functions 392 10.10 A return to differential equations 397 10.11 Summary 403 Answers to selected problems 408 Appendix: Tables 428 A Compound interest: (1 + r)^n 428 B_1 Values of e^x and e^{-x} 429 B_2 Natural logarithms (In x) 430 C Logarithms, base 10 432 D Trigonometric functions 434 Index 438 9780521250122,9780521095907 The core material is presented with a minimum of arithmetic or algebraic complexity, and no symbolism is introduced unless of pedagogic value. The optional sections involve deeper ideas and more sophisticated operations than the core. Thus the book is suited to students with diverse abilities and backgrounds. Over 2500 problems are given in the expository material and at the end of sections and chapters. Answers to about half are provided in the book. Some of the problems provide direct practice in the techniques of differentiation and integration, others are word problems of a realistic sort, using genuine data. Consequently, the student is introduced to and learns modeling and the formulation of mathematical descriptions of "real-world" situations. Indeed, in some of the problems, the information supplied is deliberately incomplete and the student is asked to make reasonable assumptions as a first step in obtaining solutions Because of their significance in the social, behavioral, and life sciences, special attention is paid to the exponential and logarithm functions. The novel approach to the derivatives of exponentials is both elegant and simple. Considerable attention is paid to log and semilog plotting. There is extensive material on numerical methods - for instance, the Newton-Raphson method, quadrature, and approximate solutions of differential equations - using calculators, programmable or otherwise, or computers. That such methods produce approximate results is emphasized and attention is paid to their accuracy. The instructor can spend as much time on numerical methods as desired, with whatever hardware is appropriate. (Publisher) The book includes several major features. It emphasizes concepts as well as techniques and applications so it is not merely a how-to book but one that presents mathematics as a science and an art, thereby contributing to the intellectual development of the student. It is designed to encourage the reader to play an active role in learning calculus, and is suitable for independent study. All statements are expressed carefully but without technical rigor, and are explained by plausibility arguments and illustrated with "real-world" examples. The student therefore has a clear grasp of the concepts and will not have to unlearn anything in more advanced courses Here is a textbook in intuitive calculus. The material is presented in a concrete setting with many examples and problems chosen from the social, physical, behavioral, and life sciences. Chapters include core material and more advanced optional sections. The book begins with a review of algebra and graphing. The core material in Chapters 1 through 7 contains the basis for a one-semester, three-hour course for an average class, with optional sections on numerical methods. The last three chapters may be studied in any order, allowing flexible use