Index #, 2 \, 5, 4 [, 4 - definition, 38 ;, 2 indeterminate form, 2 22, 24 indeterminate form, 22 23, see Euler s Constant 2, 2, see infinity, 33 \, 6, 3, 2 3-6-9 triangle, 2 4-dimensional sphere, 85 45-45-9 triangle, 2 acceleration, 6 due to gravity, 7 8, 23 algebraic numbers, 9 alternating harmonic series, 239 alternating series, 238 24 Alternating Series Test, 239 24 angle-addition formulas, 26, 29 angle-subtraction formulas, 26 annulus, 73 antiderivative, 35, 4 linearity, 42 of a continuous function, 39 antidi erentiation, 4, see also integral AoPS, see Art of Problem Solving arccos, see inverse cosine Archimedes spiral, 277 arcsin, see inverse sine arctan, see inverse tangent area, 66 69 between curves, 66 ellipse, 67 69 in polar coordinates, 274 276 area under a curve, 26 35 as sum of rectangles, 3 3 Darboux sum, 3 definite integral, 3 lower area, 3 parabola, 28 3 upper area, 3 argument, 268 Art of Problem Solving, vi, 324 associativity, 8 astroid, 265 267 average value, 76 78 Binomial Theorem, 257 boundary point, 99 bounded, 9 Boundedness Theorem, 5, 55 proof of, 55 Boyle s Law, 22 C, 35 calculator, vii, 65 cardinality, 2 cardioid, 275 277 Cartesian plane, 6 Chain Rule, 73, 57 di erentiation, 72 74 integration, 47 53 notation, 74 36
proof, 84 86 characteristic polynomial, 292 circle as parametric curve, 259 26 in polar coordinates, 269 closed interval, codomain, 3 commutativity, 8 completeness, 9 complex numbers, 35 composition (of functions), 4 concave down, 92, 93 concave up, 93 concavity, 9 95 conditionally converge, 24 24 continuity, 48 52 algebraic properties, 49 antiderivative, 39 Boundedness Theorem, 5, 55 definite integral, 34 Extreme Value Theorem, 5 Intermediate Value Theorem, 5, 54 55 localness of, 49 continuous, 48 converge absolute, 238 24 conditionally, 24 24 improper integral, 27, 2 power series, 249 25 sequence, 22 225 geometric, 222 series, 227, 229 24 convex, 94 coordinates polar, see polar coordinates cos, see inverse cosine cosecant, 24 cosh, 34, 85 cosine, 2 24 angle-addition formula, 26 angle-subtraction formula, 26 derivative of, 7 domain and range of, 23 double-angle formula, 27 half-angle formula, 29 hyperbolic, 34 integral, 46 47 inverse, see inverse cosine period of, 24, 26 Taylor series, 25 252 cotangent, 24 counting numbers, 8 critical point, 96 99 cuboids, 69 curve in polar coordinates, 269 27 parametric, see parametric curve curve length, see length curve sketching, 9 95 cycloid, 26 265, 267 cylindrical shell, 72 damped oscillation, 4 Darboux sum, 3 properties, 32 decreasing, 88, 9, 223 definite integral, 26, 3, see also integral approximation techniques, 79 83 Simpson s Rule, 8 82, 88 9 Trapezoid Rule, 8 8 as continuous sum, 76 average value, 76 78 Chain Rule, 5 52 continuous function, 34 Fundamental Theorem of Calculus, 35 4 integrand, 32 limits of integration, 32 linearity, 42 Mean Value Theorem, 79 properties, 32 Simpson s Rule, 8 82, 88 9 Trapezoid Rule, 8 8 derivative, 57, 6, 6 as rate of change, 9, Chain Rule, 72 74 notation, 74 proof, 84 86 construction of, 59 6 definition, 6 exponential, 7 7 first derivative, 9 First Derivative Test, implicit, 8 82 inverse function, 75 76 inverse trig functions, 76 Leibniz Rule, 84 linearity, 65 logarithm, 7 7 Mean Value Theorem, 79 8 proof, 86 87 monomial, 65 67, 77 Newton s Method, 3 7 notation, 6 of a constant, 9 37
polynomial, 67 68 Product Rule, 68 69 Quotient Rule, 69, 7 related rates, 7 22 relationship to continuity, 62 Rolle s Theorem, 77 79 second derivative, 77 Second Derivative Test, tangent line approximation, 8 3 trig functions, 69 7 use in curve sketching, 9 95 where undefined, 64 zero, 9 di erentiable, 6 di erential equation, 279 295 characteristic polynomial, 292 Euler s Method, 297 299 exponential decay, 286 exponential growth, 285 286 first-order, 28 heating, 287 homogeneous, 29 initial condition, 279 integrating factor, 296 linear, 29 295 logistic, 287 289 radioactive decay, 286 relative rate (of growth), 285 second-order, 29 294 separable, 283 separation of variables, 282 285 slope field, 28 282 uniqueness of solution, 282 di erential notation, 6 discrete function, 22 disjoint (sets), 6 distributive, 8 diverge improper integral, 27, 2 sequence, 22 225 series, 227 Divergence Test, 23 domain, 3 dominate, 23 24 little-o notation, 24 double-angle formula, 27, 29 dummy variable, 3, 32 e, 3, 32 as a limit, 24 25 formula for, 249 element (of a set), ellipse, 67 69 empty set, 2, 4 endpoints, 259 epicycloid, 277 equilibrium, 287 errata, vii error function, 4, 85, 24 Euler s Constant, 257 Euler s Formula, 35, 252 Euler s Method, 282, 297 299 exp, see exponential exponential, 3, 85 88 derivative of, 7 7 integral, 46 47 Taylor series, 248 249 exponential function, 3, 32 exponential growth, 285 286 exponential indeterminate form, 24 26 Extended Mean Value Theorem, 25 extrema, 96 Extreme Value Theorem, 5 extreme values, 96 Fibonacci numbers Taylor series, 257 Fibonacci sequence, 22 field, 8 finite set, first derivative, 9 First Derivative Test, fixed point, 53 Fourier series, 32 frustum, 2 function, 3 6 antiderivative, 35 area under graph, see area under a curve asymptote horizontal, 92 vertical, 98 99 average value, 76 78 codomain, 3 composition, 4 concave down, 92, 93 concave up, 93 concavity, 9 95 continuous, 48 52 convex, 94 critical points, 96 98 decreasing, 88, 9 strictly, 89, 9 derivative of, 6, see also derivative di erentiable, 6 38
discrete, 22 domain, 3 dominate, 99, 23 24, 24 exponential, 29 32, 85 88 derivative of, 7 7 extrema, 96 extreme values, 96 First Derivative Test, global maximum, 99 global minimum, 99 graph, 6 9 image, 5 increasing, 88, 9 strictly, 89, 9 inflection point, 93 inverse, 4 length, 75 local maximum, local minimum, logarithm, 32 33, 85 88 derivative of, 7 7 maximum, 99 global, 99 local, relative, minimum, 99 global, 99 local, relative, monotonic, 88 strictly, 89 nondi erentiable, 64 odd, 4 optimization, 96 5 parametric, 259 periodic, 24 preimage, 5 range, 3 rational, 93 95, 236 real-valued, 3 relative maximum, relative minimum, Riemann zeta function, 235 scaling, 8 9 strictly decreasing, 89, 9 strictly increasing, 89, 9 strictly monotonic, 89 translation, 8 9 trigonometric, 9 29 derivative of, 69 7 Fundamental Theorem of Algebra, 52 Fundamental Theorem of Calculus, 26, 35 4, 78 gamma function, 24 geometric sequence, 22 22, 225 common ratio, 22 convergence, 222 geometric series, 226 229 global maximum, 99 global minimum, 99 Grand Integrator, 47 graph, 6 9 length, 75 gravity, 7 8, 23 greatest integer function, 48 49 greatest lower bound, Green s Theorem, 267 Hôpital, see l Hôpital s Rule half-angle formula, 27, 29 harmonic series, 23 23 alternating, 239 Harvard-MIT Mathematics Tournament, iv HMMT, see Harvard-MIT Mathematics Tournament homogeneous, 29 Hooke s Law, 294 horizontal asymptote, 92 Horizontal Line Test, 8, 9 hyperbolic cosine, 85 hyperbolic trig functions, 34 hypocycloid, 266, 267 identity element, 8 image, 5 imaginary number, 35 imaginary part, 35 implicit di erentiation, 8 82 related rates, 7 22 improper integral, 27 23 comparison test, 29 2 converge, 27, 2 diverge, 27, 2 improper at both ends, 2 23 increasing, 88, 9, 223 indefinite integral, 4 42, see also integral linearity, 42 indeterminate form, 2 22, 24, 22 23, 26 exponential, 24 26 index, 22 infimum, infinite set, infinity, 9 99 39
as a limit, 96 99 not a number, 97 inflection point, 93 initial condition, 279 integers, 7, 8 integral Chain Rule, 53 definite, see definite integral exponential, 46 47 improper, see improper integral indefinite, 4 42 integrand, 32 integration by parts, 53 57 limits of integration, 32 linearity, 42 partial fractions, 6 62 polynomials, 42 46 substitution, 57 6 trig substitution, 57 6 trigonometric, 46 47, 52 53, 65 integral sign, 32 Integral Test, 233 234, 237 integrand, 32 integrating factor, 296 integration, see integral Product Rule, 53 integration bee, see MIT Integration Bee integration by parts, 53 57 Intermediate Value Theorem, 5, 54 proof of, 54 55 intersection, 5 interval, 2 closed, half-open, open, partition of, 3 inverse, 8, 4 inverse cosine, 25 inverse function derivative of, 75 76 Inverse Function Rule, 75 76 inverse sine, 25 derivative of, 76 integral, 59, 66 inverse tangent, 25 derivative of, 76 integral, 59 Taylor series, 255 l Hôpital s Rule, 2 26 indeterminate forms, 2 22, 24 indeterminate forms, 22 23 exponential indeterminate forms, 24 26 proof, 25 29 least upper bound, left Riemann sum, 35 Leibniz Rule, 84 Leibniz, Gottfried, iii length, 73 76, 265 of parametric curve, 265, 278 limaçon, 273, 276 limit, 37 48 - definition of, 38 at infinity, 9 96 as finite limit, 95 96 comparison of, 43 infinite, 96 99 one-sided, 45 47 properties, 42 43 rational function, 93 95 sequence, 222 Squeeze Theorem, 44, 48 uniqueness of, 4 42 Limit Comparison Test, 235 237 limits of integration, 32 line as parametric curve, 26 26 in polar coordinates, 273, 277 linear, 65, 42 linear approximation, links, vii, 33 Lissajous curve, 267 little-o notation, 24 ln, 32, see also logarithm local, 49 local linearization, local maximum,, local minimum,, log, 32, see also logarithm logarithm, 32, 85 88 derivative of, 7 7 integral, 56 Taylor series, 252 254 logistic equation, 287 289 lower area, 3 lower bound, 9 lower Darboux integral, 3 lower Darboux sum, 3 Maclaurin polynomial, 245 Maclaurin series, 247 magnitude, 268 maximal element, maximum,, 99 32
Mean Value Theorem, 79 8, 86, 79 extended, 25 proof, 86 87 member (of a set), midpoint Riemann sum, 35 minimal element, minimum,, 99 MIT Integration Bee, 47, 84 monotonic, 88, 223 N, 8 natural logarithm, 32, 85 natural numbers, 8 Newton s Law of Heating, 287 Newton s Method, 3 7 Newton, Isaac, iii normal distribution, 3 numbers algebraic, 9 integers, 8 positive, 7 natural, 8 rational, 8 real, 9 completeness of, 9 construction of, 9 transcendental, 9 one-sided limits, 45 47 open interval, operator notation, 6 optimization, 96 5 p-series, 23, 232 235 parameterization, 259 parametric curve, 259 266 astroid, 265 267 circle, 259 26 cycloid, 26 265, 267 hypocycloid, 267 length, 265, 278 line, 26 26 polar curve, 27 speed, 264 265 tangent line, 262 264 parametric function, 259 partial fractions, 6 62 partial sum, 226, 227 partition, 3 period, 24 periodic, 24 polar coordinates, 267 276 Archimedes spiral, 277 area, 274 276 argument, 268 cardioid, 275 277 circle, 269 conversion to/from rectangular, 268 curves, 269 27 epicycloid, 277 limaçon, 273, 276 line, 273, 277 magnitude, 268 rose, 269 27, 278 area, 275 tangent line, 27 273 positive integers, 7 power series, 247 255 convergence, 249 25 radius of convergence, 249 25 preimage, 5 probability density function, 3 Product Rule, 69 di erentiation, 68 69 integration, 53 proof by contradiction, 9 proper subset, 3 Putnam, see William Lowell Putnam Mathematical Competition pyramid, 69 7 Q, 8 Q, 9 quadratic approximation, 24 243 Quotient Rule, 69, 7 R, 9 Racetrack Theorem, 23 radians, 2 radius of convergence, 249 25 range, 3 rate of change, 9, Ratio Test, 236 237 Taylor series, 25 25 rational function, 93 95, 98 99 series, 236 rational numbers, 8 real numbers, 9 construction of, 9 real part, 35 real-valued, 3 rectangular coordinates, 268 recursive, 22 recursive formula, 22 32
related rates, 7 22 relative maximum, relative minimum, resources, vi Riemann Hypothesis, 235 Riemann sum, 34 35 Riemann zeta function, 235 right Riemann sum, 35 Rolle s Theorem, 77 79 Root Test, 238 rose, 269 27, 278 area, 275 Rule of Seventy-two, 24 Russell s paradox, 34 secant, 24 secant line, 57, 58 second derivative, 77, 9 Second Derivative Test, second-order, 243 separable, 283 separation of variables, 282 285 sequence, 22 225 bounded, 222 225 converge, 22 225 decreasing, 222 224 diverge, 22 225 Fibonacci, 22 geometric, see geometric sequence increasing, 222 224 index, 22 limit, 222 monotonic, 222 224 recursive definition, 22 series, 226 255 absolute convergence, 238 24 alternating, 238 24 alternating harmonic, 239 Alternating Series Test, 239 24 conditionally converge, 24 24 converge, 227, 229 24 diverge, 227 Divergence Test, 23 geometric, 226 229 harmonic, 23 23 Integral Test, 233 234, 237 Limit Comparison Test, 235 237 Maclaurin, 247, see also Taylor series p-series, 23, 232 235 partial sum, 226, 227 power, see power series Ratio Test, 236 237 rational functions, 236 Root Test, 238 Series Comparison Test, 23 232 sum, 226 Taylor, see Taylor series telescoping, 229 Series Comparison Test, 23 232 set, 7 cardinality, 2 di erence, 6, 7 disjoint, 6 distributive law, 6 element, empty, 2, 4 finite, infinite, intersection, 5 member, Russell s paradox, 34 subset, 2 proper, 3 superset, 3 symmetric di erence, 33 union, 4 set di erence, 6, 7 sign graph, 9 Simpson s Rule, 8 82, 88 9 sin, see inverse sine sine, 2 24 angle-addition formula, 26 angle-subtraction formula, 26 derivative of, 69 7 domain and range of, 23 double-angle formula, 27 half-angle formula, 29 hyperbolic, 34 integral, 46 47 inverse, see inverse sine period of, 24, 26 Taylor series, 25 252 sinh, 34 slicing, 7 7 slope field, 28 282 Snell s Law, 24 solid of revolution, 7 73 sphere, 7 4-dimensional, 85 square wave, 32 Squeeze Theorem, 44, 48 strictly decreasing, 3, 89, 9, 223 strictly increasing, 89, 9, 223 strictly monotonic, 89 322
subset, 2 proper, 3 warning about notation, 3 substitution method, 57 6 sum, 226, 227 superset, 3 supremum,, 223 symmetric di erence, 33 tan, see inverse tangent tangent, 2 angle-addition formula, 29 derivative of, 7 domain and range of, 23 double-angle formula, 29 half-angle formula, 29 inverse, see inverse tangent period of, 26 tangent line, 57 6, 263 approximation using, 8 3 definition, 6 in polar coordinates, 27 273 to parabola, 57 58 to parametric curve, 262 264 tangent line approximation, 8 3, 24 error,, 3, 24 Newton s Method, 3 7 Taylor polynomial, 24 247 error, 245 247 Taylor series, 247 255 di erentiation, 254 255 exponential, 248 249 logarithm, 252 254 radius of convergence, 249 25 Ratio Test, 25 25 trigonometric, 25 252, 255 telescoping, 36, 229 torus, 78 transcendental number, 9 Trapezoid Rule, 8 8 triangle 3-6-9, 2 45-45-9, 2 trig substitution, 57 6 along parametric curve, 264 265 vertical asymptote, 98 99 Vertical Line Test, 7, 9 volume, 69 73 4-dimensional sphere, 85 by slicing, 7 7 cylindrical shell method, 72 pyramid, 69 7 solid of revolution, 7 73 sphere, 7 torus, 78 whole numbers, 8 William Lowell Putnam Mathematical Competition, iv Wolfram Alpha, vii Z, 8 union, 4 unit circle, 22 upper area, 3 upper bound, 9 upper Darboux integral, 3 upper Darboux sum, 3 velocity, 6 323