Index. Excerpt from "Calculus" 2013 AoPS Inc. Copyrighted Material INDEX

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1 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, triangle, 2 4-dimensional sphere, triangle, 2 acceleration, 6 due to gravity, 7 8, 23 algebraic numbers, 9 alternating harmonic series, 239 alternating series, Alternating Series Test, 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, between curves, 66 ellipse, in polar coordinates, area under a curve, 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, average value, 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, Cartesian plane, 6 Chain Rule, 73, 57 di erentiation, integration, notation, 74 36

2 proof, characteristic polynomial, 292 circle as parametric curve, 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, continuity, algebraic properties, 49 antiderivative, 39 Boundedness Theorem, 5, 55 definite integral, 34 Extreme Value Theorem, 5 Intermediate Value Theorem, 5, localness of, 49 continuous, 48 converge absolute, conditionally, improper integral, 27, 2 power series, sequence, geometric, 222 series, 227, 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, inverse, see inverse cosine period of, 24, 26 Taylor series, cotangent, 24 counting numbers, 8 critical point, cuboids, 69 curve in polar coordinates, parametric, see parametric curve curve length, see length curve sketching, 9 95 cycloid, , 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, Simpson s Rule, 8 82, 88 9 Trapezoid Rule, 8 8 as continuous sum, 76 average value, 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, notation, 74 proof, construction of, 59 6 definition, 6 exponential, 7 7 first derivative, 9 First Derivative Test, implicit, 8 82 inverse function, inverse trig functions, 76 Leibniz Rule, 84 linearity, 65 logarithm, 7 7 Mean Value Theorem, 79 8 proof, monomial, 65 67, 77 Newton s Method, 3 7 notation, 6 of a constant, 9 37

3 polynomial, Product Rule, Quotient Rule, 69, 7 related rates, 7 22 relationship to continuity, 62 Rolle s Theorem, 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, characteristic polynomial, 292 Euler s Method, exponential decay, 286 exponential growth, first-order, 28 heating, 287 homogeneous, 29 initial condition, 279 integrating factor, 296 linear, logistic, radioactive decay, 286 relative rate (of growth), 285 second-order, separable, 283 separation of variables, slope field, uniqueness of solution, 282 di erential notation, 6 discrete function, 22 disjoint (sets), 6 distributive, 8 diverge improper integral, 27, 2 sequence, series, 227 Divergence Test, 23 domain, 3 dominate, little-o notation, 24 double-angle formula, 27, 29 dummy variable, 3, 32 e, 3, 32 as a limit, formula for, 249 element (of a set), ellipse, 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, exp, see exponential exponential, 3, derivative of, 7 7 integral, Taylor series, exponential function, 3, 32 exponential growth, exponential indeterminate form, 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, average value, codomain, 3 composition, 4 concave down, 92, 93 concave up, 93 concavity, 9 95 continuous, convex, 94 critical points, decreasing, 88, 9 strictly, 89, 9 derivative of, 6, see also derivative di erentiable, 6 38

4 discrete, 22 domain, 3 dominate, 99, 23 24, 24 exponential, 29 32, 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, 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, global maximum, 99 global minimum, 99 Grand Integrator, 47 graph, 6 9 length, 75 gravity, 7 8, 23 greatest integer function, greatest lower bound, Green s Theorem, 267 Hôpital, see l Hôpital s Rule half-angle formula, 27, 29 harmonic series, 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, 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, index, 22 infimum, infinite set, infinity,

5 as a limit, not a number, 97 inflection point, 93 initial condition, 279 integers, 7, 8 integral Chain Rule, 53 definite, see definite integral exponential, improper, see improper integral indefinite, 4 42 integrand, 32 integration by parts, limits of integration, 32 linearity, 42 partial fractions, 6 62 polynomials, substitution, 57 6 trig substitution, 57 6 trigonometric, 46 47, 52 53, 65 integral sign, 32 Integral Test, , 237 integrand, 32 integrating factor, 296 integration, see integral Product Rule, 53 integration bee, see MIT Integration Bee integration by parts, Intermediate Value Theorem, 5, 54 proof of, intersection, 5 interval, 2 closed, half-open, open, partition of, 3 inverse, 8, 4 inverse cosine, 25 inverse function derivative of, Inverse Function Rule, 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, exponential indeterminate forms, proof, 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, definition of, 38 at infinity, 9 96 as finite limit, comparison of, 43 infinite, one-sided, properties, rational function, sequence, 222 Squeeze Theorem, 44, 48 uniqueness of, 4 42 Limit Comparison Test, limits of integration, 32 line as parametric curve, 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, derivative of, 7 7 integral, 56 Taylor series, logistic equation, 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

6 Mean Value Theorem, 79 8, 86, 79 extended, 25 proof, 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, open interval, operator notation, 6 optimization, 96 5 p-series, 23, parameterization, 259 parametric curve, astroid, circle, cycloid, , 267 hypocycloid, 267 length, 265, 278 line, polar curve, 27 speed, tangent line, parametric function, 259 partial fractions, 6 62 partial sum, 226, 227 partition, 3 period, 24 periodic, 24 polar coordinates, Archimedes spiral, 277 area, argument, 268 cardioid, circle, 269 conversion to/from rectangular, 268 curves, epicycloid, 277 limaçon, 273, 276 line, 273, 277 magnitude, 268 rose, , 278 area, 275 tangent line, positive integers, 7 power series, convergence, radius of convergence, preimage, 5 probability density function, 3 Product Rule, 69 di erentiation, 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, Quotient Rule, 69, 7 R, 9 Racetrack Theorem, 23 radians, 2 radius of convergence, range, 3 rate of change, 9, Ratio Test, Taylor series, rational function, 93 95, 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

7 related rates, 7 22 relative maximum, relative minimum, resources, vi Riemann Hypothesis, 235 Riemann sum, Riemann zeta function, 235 right Riemann sum, 35 Rolle s Theorem, Root Test, 238 rose, , 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, sequence, bounded, converge, decreasing, diverge, Fibonacci, 22 geometric, see geometric sequence increasing, index, 22 limit, 222 monotonic, recursive definition, 22 series, absolute convergence, alternating, alternating harmonic, 239 Alternating Series Test, conditionally converge, converge, 227, diverge, 227 Divergence Test, 23 geometric, harmonic, Integral Test, , 237 Limit Comparison Test, Maclaurin, 247, see also Taylor series p-series, 23, partial sum, 226, 227 power, see power series Ratio Test, rational functions, 236 Root Test, 238 Series Comparison Test, sum, 226 Taylor, see Taylor series telescoping, 229 Series Comparison Test, 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, inverse, see inverse sine period of, 24, 26 Taylor series, sinh, 34 slicing, 7 7 slope field, 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,

8 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, to parabola, to parametric curve, tangent line approximation, 8 3, 24 error,, 3, 24 Newton s Method, 3 7 Taylor polynomial, error, Taylor series, di erentiation, exponential, logarithm, radius of convergence, Ratio Test, trigonometric, , 255 telescoping, 36, 229 torus, 78 transcendental number, 9 Trapezoid Rule, 8 8 triangle 3-6-9, , 2 trig substitution, 57 6 along parametric curve, vertical asymptote, Vertical Line Test, 7, 9 volume, 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