318 Index dot notation, 130 alternating, 269 Leibniz notation, 47 Hooke’s Law, 209 second, 111, 112 hyperbolic cosine, 100 difference quotient, 30 hyperbolic sine, 100 differentiable, 53 hypercycloid, 248 differential, 142 hypocycloid, 248 discrete probability, 220 divergence test, 262 I divergent sequence, 255 Index divergent series, 260 implicit differentiation, 87 domain, 21 implicit function, 88 dot notation, 130 improper integral, 217 convergent, 217 E diverges, 217 indefinite integral, 157 ellipse independent variable, 22 area, 313 inflection point, 113 equation of, 314 integral ellipsoid, 236 improper, 217 error estimate, 183 indefinite, 157 A completing the square, 313 escape velocity, 219 of sec x, 173 composition of functions, 25, 43, 65 of sec3 x, 173 absolute extremum, 117 exp function, 83 concave down, 112 expected value, 221 properties of, 161 algebraic precedence, 313 concave up, 112 exponential distribution, 225 integral sign, 153 alternating harmonic series, 269 cone exponential function, 80 integral test, 266 antiderivative, 153 lateral area, 314 Extreme Value Theorem, 119 integration arc length, 230 surface area, 314 by parts, 175 arccosine, 94 volume, 314 Intermediate Value Theorem, 53 arcsine, 92 continuous, 53 F interval of convergence, 280 area convergent sequence, 255 inverse function, 80 Fermat’s Theorem, 106 between curves, 189 convergent series, 260 inverse sine, 92 frustum, 232 under curve, 149 coordinates involute, 248 asymptote, 21, 114 Cartesian, 237 function, 20 bounded, 51 converting rectangular to polar, 238 K B polar, 237 differentiable, 53 rectangular, 237 implicit, 88 kinetic energy, 219 bell curve, 224 cosines linear, 20 rational, 178 bounded function, 51 law of, 314 L critical point, 106 unbounded, 52 C critical value, 106 function composition, 25 L’Hˆopital’s Rule, 96 cumulative distribution function, 223 Fundamental Theorem of Algebra, 109 lateral area of a cone, 128 cardioid, 238 curve sketching, 105 Fundamental Theorem of Calculus, 151 law of cosines, 314 Cartesian coordinates, 237 cycloid, 247 law of sines, 314 Cauchy Principal Value, 218 cylinder G Leibniz notation, 47 center of mass, 213 lateral area, 314 limit, 39 centroid, 214 surface area, 314 Gabriel’s horn, 220, 236 limit at infinity, 96 chain rule, 65 volume, 314 geometric series, 260 limit of a sequence, 255 chord, 30 global extremum, 117 linear approximation, 141 circle D greatest integer, 109 linearity of the derivative, 58 area, 313 local extremum, 105 circumference, 313 definite integral, 157 H local maximum, 105 equation of, 19, 314 dependent variable, 22 local minimum, 105 unit, 19 derivative, 32, 46 harmonic series, 263 logarithm, 80 317 Index 319 320 Index logarithmic function, 80 S long division of polynomials, 180 Trapezoid Rule, 183 second derivative, 111, 112 triangle inequality, 41 M sequence, 254 trigonometric identities, 314 bounded, 258 Maclaurin series, 284 bounded above, 258 U mean, 222, 225 bounded below, 258 Mean Value Theorem, 144 convergent, 255 unbounded function, 52 moment, 213 decreasing, 258 uniform distribution, 223 divergent, 255 uniform probability density function, 223 increasing, 258 N unit circle, 19 monotonic, 258 non-decreasing, 258 Newton, 208 V Newton’s method, 137–139 non-increasing, 258 normal distribution, 229 of partial sums, 260 variance, 227 series, 254 velocity, 35 p-series, 266 O absolute convergence, 274 W one sided limit, 44 alternating harmonic, 269 optimization, 117 conditional convergence, 274 witch of Agnesi, 64 convergent, 260 work, 207 divergent, 260 P geometric, 260 harmonic, 263 p-series, 266 integral test, 266 parametric equations, 246 interval of convergence, 280 partial fractions, 178 Maclaurin, 284 physicists, 130 radius of convergence, 280 point-slope formula, 314 Taylor, 286 polar coordinates, 237 Simpson’s Rule, 185 power function, 55 sines power rule, 55, 87, 90 law of, 314 precedence sphere of algebraic operations, 313 surface area, 314 probability density function, 222 volume, 314 product rule, 60, 61 spiral of Archimedes, 239 generalized, 62 squeeze theorem, 75 properties of integrals, 161 standard deviation, 227 standard normal distribution, 224 Q standard normal probability density function, 224 quadratic formula, 313 subtend, 72 quotient rule, 62 sum rule, 59 R T radian measure, 71 tangent line, 30 radius of convergence, 280 Taylor series, 286 random variable, 222 Toricelli’s trumpet, 220, 236 rational function, 64, 178 torque, 211 rectangular coordinates, 237 torus, 236 related rates, 129 transcendental function, 71 Rolle’s Theorem, 144.