308 Index

discrete probability, 213 implicit function, 68 divergence test, 254 improper integral, 209 divergent sequence, 247 convergent, 209 divergent series, 252 diverges, 209 domain, 9 indefinite integral, 129 dot notation, 102 independent variable, 10 inflection point, 84 E integral improper, 209 ellipse of sec x, 193 Index 3 equation of, 304 of sec x, 193 ellipsoid, 228 properties of, 133 escape velocity, 212 integral sign, 125 expected value, 214 integral test, 258 exponential distribution, 217 integration Extreme Value Theorem, 91 by parts, 194 Intermediate Value Theorem, 41 F interval of convergence, 271 inverse sine, 180 Fermat’s Theorem, 78 involute, 240 A concave up, 84 frustum, 225 cone function, 8 K absolute extremum, 89 lateral area, 304 bounded, 40 algebraic precedence, 303 surface area, 304 differentiable, 41 kinetic energy, 211 alternating harmonic series, 261 volume, 304 implicit, 68 antiderivative, 125 continuous, 41 linear, 8 L arccosine, 181 convergent sequence, 247 rational, 197 arcsine, 180 convergent series, 252 L’Hˆopital’s Rule, 73 unbounded, 40 asymptote, 9, 85 coordinates lateral area, 100 function composition, 13 Cartesian, 229 law of cosines, 304 Fundamental Theorem of Algebra, 81 B converting rectangular to polar, 230 law of sines, 304 Fundamental Theorem of Calculus, 123 polar, 229 Leibniz notation, 35 bell curve, 216 rectangular, 229 limit, 27 bounded function, 40 cosines G limit at infinity, 73 law of, 304 limit of a sequence, 247 C critical value, 78 Gabriel’s horn, 213, 228 linear approximation, 113 cumulative distribution function, 216 geometric series, 252 linearity of the derivative, 46 cardioid, 230 cycloid, 239 global extremum, 89 local extremum, 77 Cartesian coordinates, 229 cylinder greatest integer, 80 local maximum, 77 Cauchy Principal Value, 210 lateral area, 304 local minimum, 77 center of mass, 205 surface area, 304 H long division of polynomials, 199 centroid, 206 volume, 304 chain rule, 53 harmonic series, 255 M chord, 18 D alternating, 261 circle Hooke’s Law, 161 Maclaurin series, 275 area, 303 definite integral, 129 hyperbolic cosine, 183 mean, 214, 217 circumference, 303 dependent variable, 10 hyperbolic sine, 183 Mean Value Theorem, 116 equation of, 7, 304 derivative, 20 hypercycloid, 240 moment, 205 unit, 7 dot notation, 102 hypocycloid, 240 complete the square, 200 Leibniz notation, 35 N completing the square, 303 difference quotient, 18 I composition of functions, 13, 31, 53 differentiable, 41 Napier’s constant, 171 concave down, 84 differential, 114 implicit differentiation, 68 Newton, 160

307

Index 309 310 Index

normal distribution, 221 of partial sums, 252 series, 246 W work, 158 O p-series, 258 absolute convergence, 266 witch of Agnesi, 52 one sided limit, 32 alternating harmonic, 261 optimization, 89 conditional convergence, 266 convergent, 252 P divergent, 252 geometric, 252 p-series, 258 harmonic, 255 parametric equations, 238 integral test, 258 partial fractions, 197 interval of convergence, 271 physicists, 102 Maclaurin, 275 point-slope formula, 304 radius of convergence, 271 polar coordinates, 229 Taylor, 278 power function, 43 sines power rule, 43 law of, 304 precedence sphere of algebraic operations, 303 surface area, 303 probability density function, 215 volume, 303 product rule, 48, 49 spiral of Archimedes, 231 generalized, 50 squeeze theorem, 63 properties of integrals, 133 standard deviation, 219 standard normal distribution, 217 Q standard normal probability density function, 216 quadratic formula, 303 subtend, 60 quotient rule, 50 sum rule, 47

R T radian measure, 59 tangent line, 18 radius of convergence, 271 Taylor series, 278 random variable, 214 Toricelli’s trumpet, 213, 228 rational function, 52, 197 torque, 203 rectangular coordinates, 229 torus, 228 related rates, 101 transcendental function, 59 Rolle’s Theorem, 116 transcendental number, 171 triangle inequality, 29 trigonometric identities, 304 S sequence, 246 U bounded, 250 bounded above, 250 unbounded function, 40 bounded below, 250 uniform distribution, 216 convergent, 247 uniform probability density function, 216 decreasing, 250 unit circle, 7 divergent, 247 increasing, 250 V monotonic, 250 non-decreasing, 250 variance, 219 non-increasing, 250 velocity, 23