I8 Subject Index scalar multiples of, 798–799 singular, 804, 806 square, 796 diagonal entries in, 803 identity matrix In, 803 inverse, 804 main diagonal in, 803 sum of, 796, 797–798 systems of linear equations and, 769–783 row operations on, 771–772 solving a system using, 773–779 writing the augmented matrix of a system, 770–771 writing the system from the augmented matrix, 771 zero, 798 Maxima of functions absolute definition of, 91 locating using graphs, 91–93 local definition of, 91 locating using graphs, 90–91, 93 Maximum value of a quadratic function, 164–165 May, Brian, 546 m by n matrices, 796 Mean, 938 arithmetic, A86 geometric, A86 Menelaus of Alexandria, 392 Method of elimination, solving a system of linear equations using, 758–760 Method of substitution, solving a system of linear equations using, 758 Metrica (Heron), 579 Midpoint formula, 16 Minima of functions absolute definition of, 91 locating using graphs, 91–93 local definition of, 91 locating using graphs, 90–91, 93 Minimum value of a quadratic function, 164–165 Minor axis of an ellipse, 693 Minors of the 3 by 3 determinant, 787–788 Mixture problems, A69 Modeling process, A66 Model(s)/modeling, A66 cubic, from data, 210–211 exponential, from data, 367–368 financial, 345–354 effective rates of return and, 348–349 future value of a lump sum and, 345–348 present value of a lump sum and, 349–350 rate of interest or time required to double a lump sum, 350–351 with linear functions from data, 149–157 from verbal descriptions, 143–145 for linearly related data, 151–152 logarithmic, from data, 368–369 logistic, 360–363 from data, 369–370 mathematical, A66 translating verbal descriptions into mathematical expressions and, A67 for object in simple harmonic motion, 583–585 probability, 925–927 constructing, 927 definition of, 926 determining, 926 quadratic, 171–179 from data, 174–175 from verbal descriptions, 171–174 sinusoidal, from data, 455–459 with vectors, 644–646 Modulus of a complex number, 628 Logarithmic functions, 313–326 changing between exponential statements and logarithmic statements and, 313–314 common, 318 definition of, 313 domain of, 314–315 evaluating, 314 fitting to data, 368–369 graphs/graphing of, 315–319 inverse of, 316–319 natural, 316 properties of, 316 Logarithmic models from data, 368–369 Logarithmic spirals, 622 Logarithmic statements, changing to exponential statements, 313–314 Logistic models, 360–363 from data, 369–370 Lovelace, Ada, 681 Lower bound to zeros of a polynomial function, 222–225 Lowest terms rational functions in, 237 reducing a rational expression to, A35–A36 Magnitude of a complex number, 628 of an earthquake, 325 of a vector, 637, 639, 643–644 finding, 642, 662 writing vectors in terms of direction cosines and, 666 Main diagonal in square matrices, 803 Major axis of an ellipse, 693, 729 Mapping inverse functions defined by, 283–284 of relations, 61, 63–64 Marginal cost, 169 Marginal propensity to consume, 885 Mathematical induction, 894–898 Extended Principle of, 897 Principle of, 894 proving statements using, 894–896 Mathematical modeling, A66 translating verbal descriptions into mathematical expressions and, A67 Matrix algebra, 795–812 Matrix/matrices, 795–812 adjacency, 811 arranging data in, 795–796 augmented, of a system of linear equations row operations on, 771–772 writing, 770–771 writing system of equations from, 771 coefficient, 771 column index of, 770, 795 computing revenue using, 800 definition of, 770 difference of, 796, 797 entries in, 795 diagonal, in square matrices, 803 equal, 796 examples of, 796 historical feature on, 808 identity property of, 803 inverse, 803–807 definition of, 803 finding, 805–806 lack of, showing, 806 multiplying a matrix by, 804–805 solving a system of linear equations using, 807 solving systems of linear equations using, 807 m by n, 796 multiplication of, 800–803 nonsingular, 804–805 product of, 799–806 in row echelon form, 773–779 row index of, 770, 795 Line(s), 32–48 coincident, 757 equations of finding given two points, 39 general form of, 39–40 of parallel lines, 40–41 of perpendicular lines, 42–43 point-slope form of, 36–37 slope-intercept form of, 37–38 of vertical lines, 36 family of, 47 graphing, of lines written in general form using intercepts, 39–40 horizontal, 37 length of, 15 midpoint of, finding, 16 normal, 820 parallel, equations of, 40–41 perpendicular, equations of, 42–43 point-slope form of, 36–37 secant, 95–96 slope-intercept form of, 37–38 slope of. See Slope tangent to a circle, 55 vertical, 33 equation of, 36 Linear algebra, 795 Linear equations dependent, 757 independent, 757 in one variable, definition of, A45 solving, A45–A47 using a graphing utility, 31 solving equations that lead to, A46–A47 systems of. See Systems of linear equations Linear functions definition of, 139 graphing, 139 identifying, 297–298 using average rate of change, 139–142 increasing, decreasing, or constant, 142 modeling using, 143–145 Linear programming problems, 840–847 definition of, 841 maximum, 844 minimum, 843 setting up, 840–841 solution to definition of, 842 location of, 842 solving, 841–844 simplex method for, 840n Linear relations, nonlinear relations vs., 150–152 Line of best fit, using graphing utility to find, 152–153 Line segments directed, 637 vectors and, 637 Local maxima and minima of functions, locating using graphs, 90–91, 93 Logarithm(s) change-of-base formula and, 332 historical feature on, 333 natural, 333 properties of, 327–335 establishing, 327–328 relating to exponents, 313 solving exponential equations using, 319 sum and difference of, writing logarithmic expressions as, 39–330 whose base is neither 10 nor e evaluating, 331–332 graphing, 332–333 Logarithmic equations, 319–320 solving, 336–338, 341 Logarithmic expressions exact value of, 314 writing as a single logarithm, 330–331 writing as a sum or difference of logarithms, 329–330
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