TO THE INSTRUCTORXIII
TO THE STUDENTXXV
PRELIMINARIES1(1)
LINES1(9)
FUNCTIONS AND GRAPHS10(14)
EXPONENTIAL FUNCTIONS24(7)
INVERSE FUNCTIONS AND LOGARITHMS31(13)
TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES44(16)
PARAMETRIC EQUATIONS60(7)
MODELING CHANGE67(18)
QUESTIONS TO GUIDE YOUR REVIEW76(1)
PRACTICE EXERCISES77(3)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,80(5)
APPLICATIONS
LIMITS AND CONTINUITY85(62)
RATES OF CHANGE AND LIMITS85(14)
FINDING LIMITS AND ONE-SIDED LIMITS99(13)
LIMITS INVOLVING INFINITY112 (11)
CONTINUITY123 (11)
TANGENT LINES134 (13)
QUESTIONS TO GUIDE YOUR REVIEW141 (1)
PRACTICE EXERCISES142 (1)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,143 (4)
APPLICATIONS
DERIVATIVES147 (78)
THE DERIVATIVE AS A FUNCTION147 (13)
THE DERIVATIVE AS A RATE OF CHANGE160 (13)
DERIVATIVES OF PRODUCTS, QUOTIENTS, AND173 (6)
NEGATIVE POWERS
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS179 (8)
THE CHAIN RULE187 (11)
IMPLICIT DIFFERENTIATION198 (9)
RELATED RATES207 (18)
QUESTIONS TO GUIDE YOUR REVIEW216 (1)
PRACTICE EXERCISES217 (4)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,221 (4)
APPLICATIONS
APPLICATIONS OF DERIVATIVES225 (88)
EXTREME VALUES OF FUNCTIONS225 (12)
THE MEAN VALUE THEOREM AND DIFFERENTIAL237 (8)
EQUATIONS
THE SHAPE OF A GRAPH245 (12)
GRAPHICAL SOLUTIONS OF AUTONOMOUS257 (9)
DIFFERENTIAL EQUATIONS
MODELING AND OPTIMIZATION266 (17)
LINEARIZATION AND DIFFERENTIALS283 (14)
NEWTON'S METHOD297 (16)
QUESTIONS TO GUIDE YOUR REVIEW305 (1)
PRACTICE EXERCISES305 (4)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,309 (4)
APPLICATIONS
INTEGRATION313 (80)
INDEFINITE INTEGRALS, DIFFERENTIAL313 (9)
EQUATIONS, AND MODELING
INTEGRAL RULES; INTEGRATION BY SUBSTITUTION322 (7)
ESTIMATING WITH FINITE SUMS329 (11)
RIEMANN SUMS AND DEFINITE INTEGRALS340 (11)
THE MEAN VALUE AND FUNDAMENTAL THEOREMS351 (13)
SUBSTITUTION IN DEFINITE INTEGRALS364 (9)
NUMERICAL INTEGRATION373 (20)
QUESTIONS TO GUIDE YOUR REVIEW384 (1)
PRACTICE EXERCISES385 (4)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,389 (4)
APPLICATIONS
APPLICATIONS OF INTEGRALS393 (64)
VOLUMES BY SLICING AND ROTATION ABOUT AN393 (13)
AXIS
MODELING VOLUME USING CYLINDRICAL SHELLS406 (7)
LENGTHS OF PLANE CURVES413 (8)
SPRINGS, PUMPING, AND LIFTING421 (11)
FLUID FORCES432 (7)
MOMENTS AND CENTERS OF MASS439 (18)
QUESTIONS TO GUIDE YOUR REVIEW451 (1)
PRACTICE EXERCISES451 (3)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,454 (3)
APPLICATIONS
TRANSCENDENTAL FUNCTIONS AND DIFFERENTIAL457 (82)
EQUATIONS
LOGARITHMS457 (9)
EXPONENTIAL FUNCTIONS466 (11)
DERIVATIVES OF INVERSE TRIGONOMETRIC477 (8)
FUNCTIONS; INTEGRALS
FIRST-ORDER SEPARABLE DIFFERENTIAL EQUATIONS485 (14)
LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS499 (8)
EULER'S METHOD; POPULATION MODELS507 (13)
HYPERBOLIC FUNCTIONS520 (19)
QUESTIONS TO GUIDE YOUR REVIEW530 (1)
PRACTICE EXERCISES531 (4)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,535 (4)
APPLICATIONS
INTEGRATION TECHNIQUES, L'HOPITAL'S RULE, AND539 (68)
IMPROPER INTEGRALS
BASIC INTEGRATION FORMULAS539 (7)
INTEGRATION BY PARTS546 (9)
PARTIAL FRACTIONS555 (10)
TRIGONOMETRIC SUBSTITUTIONS565 (5)
INTEGRAL TABLES, COMPUTER ALGEBRA SYSTEMS,570 (8)
AND MONTE CARLO INTEGRATION
L'HOPITAL'S RULE578 (8)
IMPROPER INTEGRALS586 (21)
QUESTIONS TO GUIDE YOUR REVIEW600 (1)
PRACTICE EXERCISES601 (2)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,603 (4)
APPLICATIONS
INFINITE SERIES607 (110)
LIMITS OF SEQUENCES OF NUMBERS608 (11)
SUBSEQUENCES, BOUNDED SEQUENCES, AND619 (8)
PICARD'S METHOD
INFINITE SERIES627 (12)
SERIES OF NONNEGATIVE TERMS639 (12)
ALTERNATING SERIES, ABSOLUTE AND651 (9)
CONDITIONAL CONVERGENCE
POWER SERIES660 (9)
TAYLOR AND MACLAURIN SERIES669 (14)
APPLICATIONS OF POWER SERIES683 (8)
FOURIER SERIES691 (7)
FOURIER COSINE AND SINE SERIES698 (19)
QUESTIONS TO GUIDE YOUR REVIEW707 (1)
PRACTICE EXERCISES708 (3)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,711 (6)
APPLICATIONS
VECTORS IN THE PLANE AND POLAR FUNCTIONS717 (70)
VECTORS IN THE PLANE717 (11)
DOT PRODUCTS728 (10)
VECTOR-VALUED FUNCTIONS738 (11)
MODELING PROJECTILE MOTION749 (12)
POLAR COORDINATES AND GRAPHS761 (9)
CALCULUS OF POLAR CURVES770 (17)
QUESTIONS TO GUIDE YOUR REVIEW780 (1)
PRACTICE EXERCISES780 (4)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,784 (3)
APPLICATIONS
VECTORS AND MOTION IN SPACE787 (86)
CARTESIAN (RECTANGULAR) COORDINATES AND787 (9)
VECTORS IN SPACE
DOT AND CROSS PRODUCTS796 (11)
LINES AND PLANES IN SPACE807 (9)
CYLINDERS AND QUADRIC SURFACES816 (9)
VECTOR-VALUED FUNCTIONS AND SPACE CURVES825 (13)
ARC LENGTH AND THE UNIT TANGENT VECTOR T838 (9)
THE TNB FRAME; TANGENTIAL AND NORMAL847 (10)
COMPONENTS OF ACCELERATION
PLANETARY MOTION AND SATELLITES857 (16)
QUESTIONS TO GUIDE YOUR REVIEW866 (1)
PRACTICE EXERCISES867 (3)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,870 (3)
APPLICATIONS
MULTIVARIABLE FUNCTIONS AND THEIR DERIVATIVES873 (102)
FUNCTIONS OF SEVERAL VARIABLES873 (9)
LIMITS AND CONTINUITY IN HIGHER DIMENSIONS882 (8)
PARTIAL DERIVATIVES890 (12)
THE CHAIN RULE902 (9)
DIRECTIONAL DERIVATIVES, GRADIENT VECTORS,911 (14)
AND TANGENT PLANES
LINEARIZATION AND DIFFERENTIALS925 (11)
EXTREME VALUES AND SADDLE POINTS936 (11)
LAGRANGE MULTIPLIERS947 (11)
PARTIAL DERIVATIVES WITH CONSTRAINED958 (5)
VARIABLES
TAYLOR'S FORMULA FOR TWO VARIABLES963 (12)
QUESTIONS TO GUIDE YOUR REVIEW968 (1)
PRACTICE EXERCISES968 (4)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,972 (3)
APPLICATIONS
MULTIPLE INTEGRALS975 (78)
DOUBLE INTEGRALS975 (12)
AREAS, MOMENTS AND CENTERS OF MASS987 (13)
DOUBLE INTEGRALS IN POLAR FORM1000(7)
TRIPLE INTEGRALS IN RECTANGULAR COORDINATES1007(10)
MASSES AND MOMENTS IN THREE DIMENSIONS1017(7)
TRIPLE INTEGRALS IN CYLINDRICAL AND1024(13)
SPHERICAL COORDINATES
SUBSTITUTIONS IN MULTIPLE INTEGRALS1037(16)
QUESTIONS TO GUIDE YOUR REVIEW1046(1)
PRACTICE EXERCISES1047(2)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,1049(4)
APPLICATIONS
INTEGRATION IN VECTOR FIELDS1053(90)
LINE INTEGRALS1053(6)
VECTOR FIELDS, WORK, CIRCULATION, AND FLUX1059(11)
PATH INDEPENDENCE, POTENTIAL FUNCTIONS, AND1070(10)
CONSERVATIVE FIELDS
GREEN'S THEOREM IN THE PLANE1080(12)
SURFACE AREA AND SURFACE INTEGRALS1092(11)
PARAMETRIZED SURFACES1103(10)
STOKES' THEOREM1113(11)
DIVERGENCE THEOREM AND A UNIFIED THEORY1124(19)
QUESTIONS TO GUIDE YOUR REVIEW1136(1)
PRACTICE EXERCISES1136(3)
ADDITIONAL EXERCISES: THEORY, EXAMPLES,1139(4)
APPLICATIONS
APPENDICES1143(40)
A.1 MATHEMATICAL INDUCTION1143(3)
A.2 PROOFS OF LIMIT THEOREMS IN SECTION 1.21146(4)
A.3 PROOF OF THE CHAIN RULE1150(1)
A.4 COMPLEX NUMBERS1151(11)
A.5 SIMPSON'S ONE-THIRD RULE1162(1)
A.6 CAUCHY'S MEAN VALUE THEOREM AND THE1163(1)
STRONGER FORM OF L'HOPITAL'S RULE
A.7 LIMITS THAT ARISE FREQUENTLY1164(2)
A.8 PROOF OF TAYLOR'S THEOREM1166(1)
A.9 THE DISTRIBUTIVE LAW FOR VECTOR CROSS1167(2)
PRODUCTS
A.10 DETERMINANTS AND CRAMER'S RULE1169(7)
A.11 THE MIXED DERIVATIVE THEOREM AND THE1176(5)
INCREMENT THEOREM
A.12 THE AREA OF A PARALLELOGRAM'S1181(2)
PROJECTION ON A PLANE
ANSWERS1183
INDEXI-1
A BRIEF TABLE OF INTEGRALST-1