of beginning calculus
TRANSCRIPT
SCHAUM'S OUTLINE OF
THEORY AND PROBLEMS
OF
BEGINNING CALCULUS Second Edition
•
ELLIOTT MENDELSON, Ph.D. Professor of Mathematics
Queens College City University ofNew York
SCHAUM'S OUTLINE SERIES McGRAW-HILL
New York St. Louis San Francisco A uckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi
San Juan Singapore Sydney Tokyo Toronto
Contents
Chapter / COORDINATE SYSTEMS ON A LINE 1 1.1 The Coordinates of a Point 1 1.2 Absolute Value 2
Chapter 2 COORDINATE SYSTEMS INA PLANE 8 2.1 The Coordinates of a Point 8 2.2 The Distance Formula 9 2.3 The Midpoint Formulas 10
Chapter 3 GRAPHS OF EQUATIONS 14
Chapter 4 STRAIGHT LINES 24 4.1 Slope 24 4.2 Equations of a Line 27 4.3 Parallel Lines 28 4.4 Perpendicular Lines 29
Chapter 5 INTERSECTIONS OF GRAPHS 36
Chaptertf SYMMETRY 41 6.1 Symmetry about a Line 41 6.2 Symmetry about a Point 42
Chapter 7 FUNCTIONS AND THEIR GRAPHS 46 7.1 The Notion of a Function 46 7.2 Intervals 48 7.3 Even and Odd Functions 50 7.4 Algebra Review: Zeros of Polynomials 51
Chapter 8 LIMITS 59 8.1 Introduction 59 8.2 Properties of Limits 59 8.3 Existence or Nonexistence of the Limit 61
v
vi CONTENTS
Chapter 9 SPECIAL LIMITS 67 9.1 One-Sided Limits 67 9.2 Infinite Limits: Vertical Asymptotes 68 9.3 Limits at Infinity: Horizontal Asymptotes 70
Chapter 10 CONTINUITY 78 10.1 Definition and Properties 78 10.2 One-Sided Continuity , 79 10.3 Continuity over a Closed Interval 80
Chapter// THE SLOPE OF A TANGENT LINE 86
Chapter 12 THE DERIVATIVE 92
Chapter 13 MORE ON THE DERIVATIVE 99 13.1 Differentiability and Continuity 99 13.2 Further Rules for Derivatives 100
Chapter 14 MAXIMUM AND MINIMUM PROBLEMS 104 14.1 Relative Extrema 104 14.2 Absolute Extrema 105
Chapter 15 THE CHAIN RULE 116 15.1 Composite Functions 116 15.2 Differentiation of Composite Functions 117
Chapter 16 IMPLICIT DIFFERENTIATION 126
Chapter 17 THE MEAN-VALUE THEOREM AND THE SIGNOF THE DERIVATIVE 129 17.1 Rolle's Theorem and the Mean-Value Theorem 129 17.2 The Sign of the Derivative 130
CONTENTS vn
Chapter 18 RECTILINEAR MOTION AND INSTANTANEOUS VELOCITY 136
Chapter 19 INSTANTANEOUS RATE OF CHANGE 143
Chapter 20 RELATED RATES 147
Chapter 21 APPROXIMATION BY DIFFERENTIALS; NEWTON'S METHOD .. 155 21.1 Estimating the Value of a Function 155 21.2 The Differential 155 21.3 Newton's Method 156
Chapter 22 HIGHER-ORDER DERIVATIVES 161
Chapter 23 APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING 167 23.1 Concavity 167 23.2 Test for Relative Extrema 169 23.3 Graph Sketching 171
Chapter 24 MORE MAXIMUM AND MINIMUM PROBLEMS 179
Chapter 25 ANGLE MEASURE 185 25.1 Are Length and Radian Measure 185 26.2 Directed Angles 186
Chapter 26 SINE AND COSINE FUNCTIONS 190 26.1 General Definition 190 26.2 Properties 192
Chapter 27 GRAPHS AND DERIVATIVES OF SINE AND COSINE FUNCTIONS 202 27.1 Graphs 202 27.2 Derivatives 205
CONTENTS
Chapter 28 THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS ... 214
Chapter 29 ANTIDERIVATIVES 221 29.1 Definition and Notation 221 29.2 Rules for Antiderivatives 222
Chapter 30 THE DEFINITE INTEGRAL 229 30.1 Sigma Notation 229 30.2 Area under a Curve 229 30.3 Properties of the Definite Integral 232
Chapter 31 THE FUNDAMENTAL THEOREM OF CALCULUS 238 31.1 Calculation of the Definite Integral 238 31.2 Average Value of a Function 239 31.3 Change of Variable in a Definite Integral 240
Chapter 32 APPLICATIONS OF INTEGRATION I: AREA AND ARC LENGTH 249 32.1 Area between a Curve and the y-axis 249 32.2 Area between Two Curves 250 32.3 Are Length 251
Chapter 33 APPLICATIONS OF INTEGRATION II: VOLUME 257 33.1 Solids of Revolution 257 33.2 Volume Based on Cross Sections 259
Chapter 34 THE NATURAL LOGARITHM 268 34.1 Definition 268 34.2 Properties 268
Chapter 35 EXPONENTIAL FUNCTIONS 275 35.1 Introduction 275 35.2 Properties of ax 275 35.3 The Function ex 275
Chapter 36 L'HÖPITAL'S RULE; EXPONENTIAL GROWTH AND DECAY 284 36.1 L'Höpital's Rule 284 36.2 Exponential Growth and Decay 285
CONTENTS ix
Chapter 37 INVERSE TRIGONOMETRIC FUNCTIONS 292 37.1 One-One Functions 292 37.2 Inverses of Restricted Trigonometrie Functions 293
Chapter 38 INTEGRATION BY PARTS 305
Chapter 39 TRIGONOMETRIC INTEGRANDS AND TRIGONOMETRIC SUBSTITUTIONS 311 39.1 Integration of Trigonometrie Functions 311 39.2 Trigonometrie Substitutions 313
Chapter 40 INTEGRATION OF RATIONAL FUNCTIONS; THE METHOD OF PARTIAL FRACTIONS 320
Appendix^ TRIGONOMETRIC FORMULAS 329
Appendixfi BASIC INTEGRATION FORMULAS 330
AppendixC GEOMETRlC FORMULAS 331
AppendixZ) TRIGONOMETRIC FUNCTIONS 332
Appendix^ NATURAL LOGARITHMS 333
Appendix/" EXPONENTIAL FUNCTIONS 334
ANSWERS TO SUPPLEMENTARY PROBLEMS 335
INDEX 371