Subject Index I3 3 by 3, 787–789 2 by 2, 784 Diagonal entries in square matrices, 803 Diastolic pressure, 441 Difference(s). See also Subtraction; Sum and Difference Formulas of complex numbers, A59 expressing as products, 536–537 limits of, 947–948 of logarithms, writing logarithmic expressions as, 39–330 of two cubes, formula for, A25 of two matrices, 796, 797 of two squares, formula for, A24 of vectors, 638 Difference function, 71, 72 Difference quotient of a function, 68–69 Differentiable functions, 963 Diophantus, 883 Direction angles of a vector, 644 finding, 664–666 Direction cosines, 665–666 Direction of a vector, 637, 643–644 Directrix of a conic, 729 of a parabola, 682 Discontinuous functions, determining, 956 Discriminant of a quadratic formula, A52 Disjoint sets, A2, A3 Distance Formula, 13–16 distance between two points in space and, 660–661 proof of, 14–15 Distance on real number line, A5–A6 Distributive Property, A3 of dot products, 652 of matrix multiplication, 802 of real numbers, A3, A4 Divergence of a sequence, 888–889 of a series, 891 geometric, 879–880 Dividend, 215, A25 Division. See also Quotient(s) of polynomials, using long division, A25–A27 of rational expressions, A36–A37 synthetic, A31–A35 Division algorithm for polynomials, 215 Divisor, 215, A25 Domain-restricted functions, inverse of, 289 Domains of a function, 63, 69–71 composite, 274–277 defined by an equation, 69–71 finding, 181 logarithmic, 314–315 rational, 236–239 trigonometric, 413–414 of a relation, 61 of a variable, A7 definition of, A7 Dot product, 651–659 angle between two vectors and, 652–653 computing work and, 656–657 decomposing vectors into two orthogonal vectors and, 654–656 definition of, 652, 663 determining whether vectors are orthogonal and, 654 determining whether vectors are parallel and, 653–654 finding, 652, 663–664 historical feature on, 657 properties of, 652, 664 of two vectors, 651–652 Double-angle Formulas establishing identities using, 525–528 finding exact values using, 525 Double roots, A48 Drag, 599 Counting formula, 912 Counting numbers, A3 Cramer’s Rule with inconsistent or dependent systems, 790 to solve a system of three equations containing three variables, 789–790 to solve a system of two equations containing two variables, 785–787 Cross product, 669–675 algebraic properties of, 670–671 definition of, 669 finding the area of a parallelogram and, 672–673 finding using determinants, 669–670 geometric properties of, 671–672 vectors orthogonal to two give vectors and, 672 Cube(s) of binomials, formula for, A25 difference of, formula for, A25 perfect, formula for, A25 sum of, formula for, A25 Cube root(s), A87 complex, 632, 633–634 Cube root functions, 101, 103 Cubic equations, solution of, 226 Cubic models, from data, 210–211 Curves bounding, 587 plane defined by rectangular equations, parametric equations for, 743–745 defined parametrically, rectangular equation for, 738–740 definition of, 736 graphs/graphing of, 736 rose, 620 Curvilinear motion, 740 Cycles of sinusoidal graphs, 432 Cycloids, 744–745 Damped motion, analyzing an object in, 586–588 Damping coefficient, 586 Damping factor, 586 Dantzig, George, 840n Decay, uninhibited, law of, 357–358 Decimals, A3 repeating, 880 Decomposition of a vector into two orthogonal vectors, 654–656 Decreasing functions, 90–91, 93 linear, 142 Degenerate conics, 681 Degrees of a monomial, A22 of a polynomial, 190, A22 Demand, Law of, 171 Demand Equation, 171, 172, 270 De Moivre, Abraham, 631 De Moivre’s Theorem, 631–632 Denominator, A36 rationalizing, A89 Dependence, relations and, 61 Dependent variable, 66 Depressed equations, 220 Derivative(s), 962–963 definition of, 962 difference quotient and, 68 Derivative function, 963 Descartes, René, 2n Descartes’ Rule of Signs, 217–218 Determinants Cramer’s Rule and to solve a system of three equations containing three variables, 789–790 to solve a system of two equations containing two variables, 785–787 expanding across a row or column and, 788 finding cross products using, 669–670 properties of, 791–792 equation of, 683, 684–687 graphs/graphing of, 683 with vertex at h k , ( ), 686–687 with vertex at the origin, 682–686 vertex of, 159, 160–163 polar equations of, 728–735 analyzing and graphing, 728–732 converting to a rectangular equation, 733 rotation of axes to transform equations of, 721–723 analyzing equations using, 724–725 identifying conics without, 726 Conjugate(s), 628 Conjugate axis of a hyperbola, 705 Conjugate hyperbolas, 719 Conjugate Pairs Theorem, 231–232 Constant(s), A6 limits of, 946 Constant functions, 90–91, 102 linear, 142 Constant rate job problems, A71–A73 Constant term of a polynomial function, 190 Constraints, 841 Consumer price index (CPI), 354 Continued fractions, A43 Continuous compounding, 348 Continuous function, 104 determining, 955–958 Continuous graphs, 191 Convergence of a sequence, 888 of a series, 891 geometric, 879–880 Cooling, Newton’s Law of, 359–360 Coordinate(s), 2 of P, 660, A4 Coordinate axes, 2 Copernicus, 392 Corner points, 837 Correlation coefficient, 153 Correspondence, relations and, 61 Cosecant function, 398 graphs/graphing of, 447–448 inverse definition of, 487 exact value of, 487–489 Cosine(s) direction, 665–666 Law of, 570–577 applied problems using, 572–573 historical feature on, 573 solving SAS triangles using, 571 solving SSS triangles using, 571–572 Cosine function, 398 graphs/graphing of, 429–430 inverse definition of, 475–476 exact value of, 476–477 name of, 407 properties of, 430 Sum and Difference Formulas for, 511–512 Costs fixed, 46 marginal, 169 variable, 46 Cotangent function, 398 graphs/graphing of, 445–447 inverse definition of, 487 exact value of, 487–489 Counting, 911–916 addition principle of, 912 finding all subsets of a set and, 911 general addition principle of, 913 multiplication principle of, 913–914 of the number of elements in a set, 911–913 problems involving n distinct objects and, 917–919 problems involving n nondistinct objects and, 922 problems using combinations and, 919–921
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