Subject Index I13 Vertical asymptotes definition of, 239 rational functions and, 239, 240–241 multiplicity and, 240–241 Vertical components, of vectors, 641 Vertical compressions and stretches, graphing functions using, 115 Vertical lines, graphing polar equations and, 613, 614 Vertical-line test, 78 Vertical shifts, graphing functions using, 112–114 Viète, François, 573 Viewing angle, 485 Viewing window (rectangle), setting, 3 Villareal, Luis P., 273 Wallis, John, 634 Weight, 599 Wings of an airplane, 599 Work, computing, 656–657 x-axis, 2 projection of P on, 584 reflections about, graphing functions using, 117–118 testing equations for symmetry with respect to, 21, 22, 23 x-coordinates, 2 x-intercept, 9 of a quadratic function, 161–163 xy-plane, 2, 660 xz-plane, 660 Yang Hui, 903 y-axis, 2 projection of P on, 584 reflections about, graphing functions using, 117–118 testing equations for symmetry with respect to, 21, 22, 23 y-coordinates, 2 y-intercept, 9 of a line, finding, 38 yz-plane, 660 Zero, A4 Zero(s) (roots) complex, of a polynomial function, 230–234 Conjugate Pairs Theorem and, 231–232 definition of, 230 finding, 233–234 given, finding functions with, 232–233 of a polynomial function bounds on, 222–225 locating using Intermediate Value Theorem and a graphing utility, 225 Rational Zeros Theorem and, 218–219 specified, finding polynomial functions with, 232–233 real, of a polynomial function, 194–201, 214–229 definition of, 195 Factor Theorem and, 216–217 finding, 219–221 finding a polynomial function from, 195 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 Zero-level earthquakes, 325 Zero matrices, 798 Zero polynomials, 190, A22 Zero-Product Property, A4 Zero vector, 637, 638 Uniform motion formula, A70 Uniform motion problems, A70–A71 Uninhibited decay, law of, 357–358 Uninhibited growth, law of, 355–357 Uninhibited growth of cells, 355–357 Uninhibited radioactive decay, 357–358 Union of sets, A2, A3 Unit circles, 48–49, 397–398. See also Trigonometric functions, unit circle approach to Unit vectors definition of, 639 in the direction of v, 663 finding, 642–643, 663 Universal set, A2 Upper bound to zeros of a polynomial function, 222–225 Urschel, John, 139 Value of a function, 63, 65, 66–67 Variable(s), A6 domain of, A7 Variable costs, 46 Vector(s), 637–651 adding and subtracting algebraically, 641–642 adding geometrically, 638 algebraic, 640 angle between, finding, 652–653 column, product of a row vector and, 799 components of, 640, 641 cross product of. See Cross product decomposing into two orthogonal vectors, 654–656 direction cosines of, 665–666 direction of, 637, 643–644 dot product and. See Dot product equality of, 637, 641 finding from its direction and magnitude, 643–644 geometric, 637 graphs/graphing of, 639 historical feature on, 647 magnitude of, 637, 639, 642, 643–644 finding, 642 writing vectors in terms of direction cosines and, 666 modeling with, 644–646 multiplying by numbers geometrically, 638–639 operations on, 662–663 orthogonal, 654 finding, 672 parallel, 653–654 position, 640–641 in space, 661–662 product of. See Dot product row, product of a column vector and, 799 scalar multiples of, 642 in space, 659–668 angle between two vectors and, 664 direction angles of a vector and, 664–666 dot product and, 663–664 operations on, 662–663 position vectors and, 661–662 unit definition of, 639 in the direction of v, 663 finding, 642–643, 663 velocity, 643–644 zero, 637, 638 Vector product. See Cross product Vector projection of v onto w, 655 Velocity, instantaneous, 964–966 Velocity vectors, 643–644 Venn diagrams, A2 Vertex form, of a quadratic function, 160 Vertex/vertices of an ellipse, 693 of a graph, 837 of a hyperbola, 705 of a parabola, 159, 160–163, 682 of a right circular cone, 681 linear in sine and cosine, solving, Sum and Difference Formulas and, 518–520 quadratic in form, solving, 496–497 solutions of, 493 solving using a calculator, 496 using a graphing utility, 498 using fundamental identities, 497–498 using identities, 527–528 Trigonometric expressions, writing as an algebraic expression, 490 Trigonometric functions, 382–470 of acute angles, finding value using right triangles, 546–548 applications of, 545–598 damped motion and, 586–588 graphing the sum of two functions and, 588–589 Law of Cosines and, 572–573 Law of Sines and, 563–565 right triangle, 549–553 simple harmonic motion and, 583–586 definition of, 398, 399 domain of, 413–414 Double-angle Formulas for finding exact values of, 525 finding values of fundamental identities for, 417–419 given one of the functions and the quadrant of the angle, 419–422 odd-even properties for, 422–423 names of, 407 period of, 415–416 properties of, 412–426 range of, 413–414 right triangle trigonometry and, 546–559 Complementary Angle Theorem and, 548 finding value of functions of acute angles using right triangles and, 546–548 solving right triangles and, 548–549 signs of, in a given quadrant, 416–417 single, trigonometric equations involving, solving, 493–496 unit circle approach to, 397–412 calculators to approximate value of functions and, 405–406 circle of radius r to evaluate functions and, 406–407 exact values of functions for integer multiples of 6 30, 4 45 π π = ° = ° and 3 60 π = ° and, 404–405 exact values of functions of quadrantal angles and, 399–401 exact values of functions of 4 45 π − ° and, 401–402 exact values of functions of 6 30 π − ° and 3 60 π − ° and, 402–404 finding values of functions using a point on the unit circle and, 398–399 trigonometric functions of angles and, 399 unit circle and, 397–398 Trigonometric identities, 503–511 basic, 504 establishing, 505–508 double-angle formula and, 525–528 using Sum and Difference Formulas, 516–517 even-odd, 504 Pythagorean, 504 quotient, 504 reciprocal, 504 simplifying using algebra, 504–505 Trigonometry, historical feature on, 392 Trinomials, A22 Turning points of polynomial functions, 197–198 2 by 2 determinants, 669, 784 Unbounded graphs, 837 Unboundedness in the negative direction, polynomial functions and, 198
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