I10 Subject Index writing expressions containing fractional exponents as, A91 Radical equations, solving, A90 Radical sign, A9 Radioactive decay, uninhibited, 357–358 Radius of a circle, 48 of a sphere, 668 Range of a function, 63 of a relation, 61 of trigonometric functions, 413–414 Rate of interest, 345, A67 effective, 349 required to double a lump sum, 350–351 Rates of change average, 33 definition of, 94 of a function, 93–96 identifying linear functions using, 139–142 limits of, 951 instantaneous, 963–964 Rates of return, effective, 348–349 Rational equations, solving, A47–A48 Rational exponents factoring expressions containing, A91, A92 simplifying expressions with, A91–A92 Rational expressions, A35–A43. See also Partial fraction decomposition adding, A37–A39 complex, simplifying, A40–A42 dividing, A36–A37 least common multiple method and, A39–A40 multiplying, A36–A37 proper and improper, 813 reducing to lowest terms, A35–A36 subtracting, A37–A39 Rational functions asymptotes and, 238–244 horizontal, 239, 241–244 oblique, 239, 241–244 vertical, 239, 240–241 definition of, 236 domain of, 236–239 graphs/graphing of, 247–258 analyzing, 247–254 constructing a function from its graph and, 253–254 end behavior and horizontal asymptotes and, 239 with a hole, 252–253 solving applied problems involving, 254–255 with transformations, 238 in lowest terms, 237 multiplicity and, vertical asymptotes and, 240–241 proper and improper, asymptotes and, 240–241 standard form of, 246 Rationalizing the denominator, A89 Rationalizing the numerator, A89–A90, A89n Rational numbers, A3, A58 Rational Zeros Theorem, 218–219 Real axis in complex plane, 627 Real number(s), A3–A4 as coordinates of P, A4 positive and negative, A4 as scalars, 638–639 Real number line, A4 definition of, A4 distance on, A5–A6 Real zeros (roots) of a polynomial function, 194–201 definition of, 195 finding a polynomial function from, 195 multiplicity of, 196–197 Reciprocal functions, 103 of a complex number in standard form, A61 Reciprocal properties for inequalities, A80 solving inequalities using, A82 Rectangular coordinate(s) converting from polar coordinates to, 602–604 converting to polar coordinates from, 604–606 of equally likely outcomes, 928–929 of events, 927 exponential, 305 historical feature on, 931–932 probability models and, 925–927 tossing a fair coin and, 925 of the union of two events, 929–930 Probability models, 925–927 constructing, 927 definition of, 926 determining, 926 Product(s). See also Multiplication of complex numbers, 630–631, A60 cross. See Cross product dot. See Dot product expressing as sums, 535–536 expressing sums as, 536–537 limits of, 948 of matrices, 799–806 of a row vector and column vector, 799 scalar, finding, 662 of two vectors. See Dot product vector. See Cross product Product function, 71, 72 Product of inertia, 533 Product-to-Sum Formulas, 535–536 Projectile motion, 740–741 Projection of P on the x-axis, 584 Projection of P on the y-axis, 584 Prolate spheroids, 703 Proper fractions, 813 Proper rational functions, asymptotes and, 240–241 Proper subsets, 911 Ptolemy, 573 Pure imaginary numbers, A59 Pythagorean Identities, 419, 504 Pythagorean Theorem, A14–A15 converse of, A14 Pythagorean triples, A21 Quadrant(s), 2 Quadrantal angles, exact values of the trigonometric functions of, 399–401 Quadratic equations, A47–A48 definition of, A48 solutions of, character of, A64–A65 solving by factoring, A48–A49 in standard form, A48 Quadratic factors, irreducible, 222 nonrepeated, partial fraction decomposition and, 817–818 repeated, partial fraction decomposition and, 818–819 Quadratic formula, A64 discriminant of, A52 solving quadratic equations using, A51–A53 Quadratic functions, 157–170. See also Parabolas definition of, 157 finding, given vertex and one other point, 164 graphing with transformations, 158–160 using vertex, axis, and intercepts, 161–163 inequalities involving, 179–183 maximum or minimum value of, 164–165 vertex and axis of, 160–161 vertex form of, 160 x-intercepts of, 161–163 Quadratic models, 171–179 from data, 174–175 from verbal descriptions, 171–174 Quotient(s). See also Division of complex numbers, 630–631 limits of, 950–951 single, writing expressions as, A92 Quotient function, 72, 73, 215, A25 Quotient identities, 417, 504 Radical(s), A87 like, combining, A88 simplifying, A88 Polynomial(s), A22–A31 Chebyshëv, 526n coefficients of, 190, A22 completing the square and, A29 definition of, A23 degree of, 190, A22 dividing division algorithm for, 215 examples of, A23 using long division, A25–A27 using synthetic division, A31–A34 factoring, A27–A28 limits of, 949 prime, A27 recognizing, A23–A24 special products and, formulas for, A24–A25 standard form of, A22 zero, 190, A22 Polynomial equations, solving, 222 Polynomial functions, 190–235 complex, 230 complex zeros of, 230 Conjugate Pairs Theorem and, 231–232 definition of, 190 end behavior of graphs of, 198–201 graphs/graphing of analyzing, 206–210 end behavior and, 198–201 identifying graphs of polynomial functions and, 198 obtaining using bounds on zeros, 224 using transformations, 194 identifying, 190–191 power, 191–193 real zeros (roots) of, 194–201, 214–229 Factor Theorem and, 216–217 finding, 219–221 Intermediate Value Theorem and, 225 multiplicity of, 196–197 number of, 217 positive and negative, determining number of, 217–218 potential, listing using Rational Zeros Theorem, 218–219 Remainder Theorem and, 215–216 solving polynomial equations and, 222 theorem for bounds on, 222–225 with specified zeros, finding, 232–233 standard form of, 190 terms of, 190 turning points of, 197–198 Polynomial inequalities, solving graphically and algebraically, 259–261 multiplicity and, 260–261 Position vectors, 640–641 in space, 661–662 Positive real numbers, A4 Power(s). See also Exponent(s) of a complex number, in standard form, A62 of i, A62–A63 limits of, 949–950 Power functions, 191–193 AC generators and, 440 definition of, 191 of even degree, 192–193 of odd degree, 193 Present value, 346, 349 of a lump sum, 349–350 Present value formulas, 350 Prime polynomials, A27 Principal, 345, A67 Principal nth root, definition of, A87 Principal square root definition of, A9 of –N, definition of, A63 Principle of Mathematical Induction, 894 Probability, 925–935 compound, 928–929 computing using complements, 930 definition of, 925
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