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Index

, 59, 116, 381 arctan, 303 of, 240 area acceleration, 310 as a , 14 due to gravity, 310 assumptions about, 84, 85 law, 193, 199 between graphs, 184 addition of points, 68 area function, 84, 89 addition rule for area, 87 as an , 183 addition theorem for area, 372 of box, 14 additivity of area, 85 of circle, 181 almost disjoint sets, 88 of circular sector, 349 Analyst 223 of ellipse, 181 and (logical operator), 51 of parabola, 19, 20 , 269, 328 of parabolic segment 30 theorem 270 of snowflake, 49 antidifferentiation of , 99 formulas for, 331 under power function, 154 Maple commands for, 330 Aristotle(384-322 B.C.), 14, 223, 235 Apollonius (c 260-170 B.C.), 220 Aryabhat¯ .a (circa 510), 199 approachable point, 224 associative law approximation, 120 for , 375 to n decimals, 121 for points, 69 strong approximation theorem, 122 assumptions about area, 84, 85 arc length, 191 average velocity, 235 arccos, 302 arccot, 304 Babylonians, 8, 24, 27, 35, 78 380 Banach, Stefan(1892–1945), 86 Archimedes (287–212 B.C.), 4, 8, 9, Berkeley, George (1685-1753), 223 27, 29, 30, 35, 190, 200, 202, Bernoulli, Daniel(1700-1782), 304 209 Bernoulli,Jacob (1654-1705) 26, 127, arcsin, 303 149 between, 288

397 398 INDEX bijective, 292 conic section, 220 Bolzano, Bernard(1781-1848), 236, 287 conservation of energy, 312 bound for a function, 84 constant sequence, 127 bounded constructivists, 54 function, 84 continuity, 256 set, 84 on a set 256 box, 13 contrapositive 54 circumscribed, 22 convergent sequence 122, 125 Brouncker, William(1620-1684), 108 convex downward (spills water), 314 convex upward (holds water), 313 cancellation laws 377 cosh 305 Cantor, 13 cosine Cartesian product, 57 defined, 191 Cauuchy, Augustin(1789-1857), 65, 236 integral of 206, 208 Cavalieri,Bonaventura (1598-1647), 39 cot, 247 chain rule, 250 counterclockwise, 191 change of scale for , 180 critical point, 263 chord of an arc, 199 critical point theorem I, 264 circle critical point theorem II, 265 area of, 181, 385 critical set, 263 circumference of, 385 csc, 247 defined, 80 unit, 80 decreasing function, 90 circular sector (area of), 349, 385 derivative 5, 6, 230 circumscribed fractional, 308 box, 22 higher order, 306 hexagon for snowflake, 49 of order 0, 306 clockwise, 191 of absolute value, 240 closed , 17 of arccos, 302 codomain, 58 of arcsin, 303 commutative law of arctan, 303 for numbers, 375 of arccot, 304 for points, 69 of ax, 301 composition of functions, 248, 259 of cos, 238 composition problem, 249 of cosh, 305 compound interest, 148 of exponential, 300 computer calculation of integrals, 113 of , 239 congruence problem, 86 of powers, 237 INDEX 399

of reciprocal, 246 Calculate ln(2), 106 of sin, 238 Composition problem, 249 1 of sinh, 305 lim{n n }, 148 of trigonometric functions, 247 Calculation of sines, 200 of xr 301 Discontinuous derivative problem, second 306 318 derived function 5 Falling bodies problem, 224 Descartes, Rene (1596-1660), 224 Bernoulli’s problem, 27 difference of sets, 16 Calculate e, 108 differential, 5, 242 Calculation of π, 8 differentiation, 5 Congruence problem, 86 logarithmic, 252 Pine Tree proble, 34 Dirichlet, P. G. Lejenue(1805-1859), 176 root problem, 9 discontinuous function, 258 Snowflake prioblem, 49 discreteness of Z, 380 end points of interval, 17 disjoint sets, 85 equal functions 60 almost disjoint sets, 88 equality 383 in Q, 381 between numbers, 117 of sets, 13 between points, 79 of propositions, 54 distributive law equation for line, 387 for numbers, 375 equivalent propositions, 52 for points, 69 error function, 330 Diver, Colin (1944-??), 399 etymology divergent sequence, 125 of corollary, 88 domain, 58 of sine, 199 double angle formulas, 195 Euclid (fl. c. 300 B.C.), 14, 35, 220, dummy 235 index, 15 Euler, Leonard (1707-1783), 62, 64, 127 variable 63, 167, 225 evalf, Maple command, 390 even function, 270 e, 108 exponential function, 293, 297 numerical calculation of, 147 derivative of 300 ellipse, area of, 181 exponents, laws of, 383 Emperor Yˇu(c.21st century B.C.), 78 exterme point, 261 entertainments local, 264 Archimedes sine integral, 211 extreme value property, 263 Area of a triangle, 99 400 INDEX

fluxion, 5, 242 induction, 65 Fourier, Joseph (1768–1830), 161, 309 inequalities, 380 fractional , 308 inequality rule function, 6, 58 for limits of functions, 229 bounded, 84 for sequences, 130 Euler’s definition of, 62 for integrals, 168 increasing, 90 inflection point, 316 functions injective, 290 equality of, 60 inner snowflake, 40 operations on, 166 instantaneous velocity, 235 fundamental theorem of calculus, 6, integrable function, 160 320, 324 integral, 160 Leibniz statement of, 325 and area, 183 change of scale for, 180 Galileo (1564-1642) 223, 224 Ei, 330 geometric , 141 indefinite, 213 finite, 33 of cos, 206, 208 G¯ougˇu,79 integral of sin, 208 graphs, 61 computer computation, 113 area between, 184 Si, 330 gravity, acceleration due to, 310 integration, 3 half angle formulas, 195 by parts, 335 Hausdorff, Felix (1868–1942), 86 by substitution, 340 height of box, 14 of rational functions, 353 Heine, Heinrich Eduard (1821-1881), interior point, 224 236 intermediate value property, 287, 288 Heron (between 250 BC and 150 AD), intersection, 14, 48 284 interval, 17 higher order derivatives, 306 inverse, horizontal, 387 additive, 376 hyperbolic functions, 305 multiplicative, 376 , 293 Ibn-al-Haitham(circa 1000 A.D.), 27 inverse function theorem, 298 image of f, 61 image of x under f, 58 Katyayana (c. 600 BC or 500BC??), implies, 53 79 increasing function, 90 kinetic energy, 312 indefinite integral, 213 Koch, Helga von(1870-1924) 49, 259 INDEX 401

Lagrange, Joseph Louis (1736–1813) maximum, 58 5, 242 local, 264 Leibniz, Gottfried(1646–1716) 5, 6, 62, of a function, 261 160, 167, 221, 241 284, 308, mean value theorem, 268 325 for integrals, 322 notation for sums, 65 Mercator, Nicolaus(1620-1687), 111 proof of product rule, 245 mesh of partition, 89 length of arc 191 minimum, 58 l’Hˆopital,GuillaumeFran¸cois(1661-1701), local, 264 124 of a function, 261 , 225 , 90 of a sequence, 125 piecewise, 173 infinite, 273 monotonicity one-sided, 274 of area, 88, 373 line 70, 387 of powers, 382 equation for, 387 Liouville, Joseph (1809–1882), 309 names for rational numbers, 380 ln(2) calculation, 113 Napier, John (1550-1632), 106 ln(a), 2 , 378 localization rule, 240 Newton, Isaac(1642–1727) 5, 6, 236, local maximum, 264 242 local minimum, 264 Newton’s law, 6 logarithm 104, 106, 159 nice function, 320 derivative of, 239 non-integrable function, 174 logarithmic differentiation 252 normalization property of area, 85 not (logical operator), 51 Maple 112, 329, 352 nowhere differentiable function, 259 approximate integration, 113 nth power theorem, 140 calculation of e, 147 rule for sequences, 132 integration, 216 null sequence rule, 128 mypi, 205 number as area, 14 routine, 205 short list of commands, 389 objects in a set, 12 sinsq, 205 odd function, 270 symbolic antidifferentiation, 330 open interval, 17 average, 113 operations on functions, 166 leftsum, 113 optical illusion, 190 rightsum, 113 optimization problems, 277 402 INDEX ordered pair, 57 for limits of functions, 229 order laws, 378 for sequences, 129 or (logical operator), 51 proofs without words, 24 outer snowflake, 40 propositions, 51 equal, 54 pair, ordered, 57 equivalent, 52 parabola, area of 19, 29 sets defined by, 56 parabolic segment, area of, 30 proposition form, 52 parallel lines, 388 Ptolemy, Claudius (fl 127-151), 199 parallelogram, 70 Pythagoras, (f. 530–510 B.C.), 78 area of, 384 Pythagoreans, 24 partition, 89 , 386 regular, 89 partition-sample sequence, 166 quadratic formula, 80 Pascal, Blaise (1623-1662), 24 quotient rule π (), 2, 8, 181, 113 for derivatives, 246 ancient values of, 9 for limits, 229 computer calculation of, 113 for limits of functions, 229 piecewise monotonic function, 173 for sequences ,130 non-piecewise monotonic function, 261 radian, 385 point Ramanujan, Srinivasa (1887–1920), 10 in , 12, 68 rate of change, 282 in a set, 12 rational functions, 353 of inflection, 316 rational numbers, 12 addition of points, 68 names for, 380 positive number, 378 , 6 potential energy, 312 reciprocal (derivative of), 246 power function, 297 reflection 30, 73 powers, 382 reflection law for sin and cos, 195 monotonicity of, 382 reflection theorem, 294 nth power theorem, 140 regular partition, 89 power-sums, (table of), 27 Rhind Papyrus, 8 prerequisites, 7, 375 Riemann, Bernhard (1826–1866) 160 product, Cartesian, 57 , 152 product rule right triangle, 95 for derivatives, 244 rituals for integration 341, 342, 346, for limits, 229 350, 351 INDEX 403

Rolle, Michel, (1652-1719), 267 , 386 Rolle’s theorem, 267 snowflake, 48, 259 rotation, 73 area of 49 ruler function, 176 spike function 171, 172, 226 problem, 9 Saint-Vincent, Gr´egoirede(1584-1667), squeezing rule 143 for limits of functions, 230 sample, 152 for sequences, 130 Sarasa, Alfons Anton de(1618-1667), stretch 179 106 Stringham, Irving, 106 schizophrenia, 54 of area, 87 sector, area of circluar 349, 385 subinterval of a partition, 89 sec, 247 subset, 13 segment, 70, 76 substitution in integrals, 350 sequences, 59 substitution, trigonometric, 345 constant, 127 summation formula 23, 24, 25, 26, 27 constant sequence rule, 128 summation notation, 63 convergent, 122, 125 sum rule divergent, 125 for derivatives, 242 inequality rule for, 130 for limits, 229 limit of, 125 for limits of functions, 229 nth root rule, 132 for sequences, 128 null sequence rule, 128 sum theorem product rule for, 129 for derivatives, 332 quotient rule for, 130 for indefinite integrals, 214 sum rule for, 128 for integrals, 166 translate of, 131 surjective, 292 translation rule for, 131 symmetric set, 270 set, 11 symmetry invariance, 85 bounded, 84 symmetry of square, 74 defined by a proposition, 56 laws, 378 , 219, 221, 231 sine tan, 247 defined, 191 transitivity of <, 379 etymology of word, 199 translate integral of, 208 of a sequence, 131 sine integral (Si), 330 of a set, 75 sinh, 305 translation invariance, 85 404 INDEX translation rule for sequences, 131 trapezoid, area of, 384 triangle area of, 99, 384 right, 95 , 117 trichotomy law, 378 trick, 135 trigonometric functions, 191 derivative of, 238, 247 trigonometric identities, 192, 193, 195, 199 trigonometric substitution, 345 union of sets 14, 48 uniqueness of inverses, 294 uniqueness theorem for convergence, 125 unit circle, 80 velocity 5, 219, 223 average, 235 instantaneous, 235 vertical, 387 Voltaire, Fran¸coisMarie Arouet de (1694- 1778), i

Weierstrass, Karl (1815-1897), 120, 259 width of box, 14 work, 4 zero-area set, 88, 183 zeroth order derivative, 306 ZˇuCh¯ongzh¯ı(429-500 A.D.), 9