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ion Tower, Dallas, 582 Show Caves, Texas Hill Country, Sonora, 141 Southwestern University, Georgetown, 448 Texas Coins, 741 Titan, Arlington, 449 Problem-Solving Plan, xxviii Problem-Solving Strategies, S40–S49 Draw a Diagram, S40 Find a Pattern, S44 Guess and Test, S42 Make a Model, S41 Make a Table, S48 Make an Orga... |
through two points, 182, 183, 185, 186, 558 of parallel lines, 184–186, 188, 306 of perpendicular lines, 184–186, 189, 306, 617 point-slope form, 303, 305 of vertical lines, 182 Slope-intercept form, 188, 190, 191, 194 proof of, 196 Social Studies, 403 Solids, 654, see also Three-dimensional figures Platonic, 669 Solv... |
urns, see Rotations Two-column proofs, see Proofs, two-column Two-point perspective, 662 drawing figures in, 668 Two-Transversal Proportionality Corollary, 482 U Undefined terms, 6 Unit circle, 570 trigonometry and the, 570–571 University of Texas Longhorn Band, 833 Urban legends, 88 Use more than one method, 45 V Vani... |
(tr), Getty Images; 324 (cr), Imagebroker/Alamy Images; 326 (tl), Creatas/ Punchstock.com; 328 (tl), Creatas/Punchstock.com; 328 (c), Photodisc Red/RF/Getty Images; 331 (tr, br), Sam Dudgeon/HRW Photo; 332 (tr), Real Life Adventures by Gary Wise and Lance Aldrich; 337 (tl), Stefano Rellandini/Reuters/Corbis; 338 (tl), ... |
(br), ©Christer Fredriksson/Lonely Planet Images; 768 (tl) ©Photolibrary.com.pty.ltd./Index Stock Imagery, Inc.; 768 (b), ©Tony Freeman/PhotoEdit; 768 (cl), Scala/Art Resource, NY; 768 (bl), ©Photolibrary.com.pty.ltd./Index Stock Imagery, Inc.; 768 (br), Photo by Eisenmann, N.Y./Library of Congress; 772 (tr), Victoria ... |
two outputs, 1 and and 9, also have more than one output, 3 26 1. is not a function Two other inputs, 4 1, 1 2 1 , , 0, 0 1, 1 , 1 51 Although 1 appears as an output twice, each input has one and only one output. is a function. 4, 2 9, 3 26 , , , , 2 9, 3 , 0, 0 4, 2 51 because each input corresponds to one and only o... |
ns Definition of a Sequence Visual patterns exist all around us, and many inventions and discoveries began as ideas sparked by noticing patterns. Consider the following lists of numbers. 4, 1, 2, 5, 8, ? 1, 10, 3, 73, ? Analyzing the lists above, many people would say that the next number in the list on the left is 11 ... |
ay is added, three angles are formed. See the figure below. In Exercises 9–12, find the first five terms of the given sequence. 2 3 1 9. u1 4 and un 2un1 3 for n 2 10. u1 5 and un 1 3 un1 4 for n 2 Write a recursive formula for the number of angles formed with n rays if the same pattern continues. Graph the sequence. U... |
. Apply the formula with and that the common difference, u45 u1 1 45 1 2 44 4 2 21 1 181 ■ 5 u1 n 45 . d 5 Example 6 Finding Explicit and Recursive Formulas un6 If is an arithmetic sequence with 5 sive formula, and an explicit formula for u6 57 un. and u10 93, find u1 , a recur- Solution The sequence can be written as ... |
he first number for x and the second for y produces a true statement. The graph of an equation in two variables is the set of points in a plane whose coordinates are solutions of the equation. Thus, the graph is a geometric picture of the solutions. Recall that an arithmetic sequence is a sequence in which the differen... |
initial point and the remaining points can be found by using the equation’s slope. The slope determines how to find a second point from the initial point by moving vertically an amount equal to the numerator of the slope, which represents and then moving horizontally an amount equal to the denominator of the slope, whi... |
isease in year x, with corresponding to 1950. Round the slope of the line to one decimal place. b. Use the equation in part a to estimate the death rate in 1995 and in 2005. 57. According to the Center of Science in the Public Interest, the maximum healthy weight for a person who is 5 ft 5 in. tall is 150 pounds and fo... |
ression Lines It can be proved that for any set of data there is one and only one line for which the sum of the squares of the residuals is as small as possible. Such a line is called the least–squares regression line, and the computational process for finding it is called linear regression. Most graphing calculators h... |
ties. Exercises 1.5 1. a. In Example 2, find the equation of the line through the data points (1, 2) and (5, 5). b. Compute the sum of the squares of the errors for this line. Is it a better model than any of the models in the example? Why? 2. The linear model in Example 5 is the least squares regression line with coef... |
? d. Find the correlation coefficient for the model. Arizona Alaska Hawaii 8 34 51 52 561 525 510 515 58 Chapter 1 Number Patterns 1.6 Geometric Sequences Objectives • Recognize a geometric sequence • Find a common ratio • Graph a geometric sequence • Write a geometric sequence recursively and explicitly • Find partial... |
ur parents, how many ancestors do you have for the preceding ten generations? 41. A car that sold for $8000 depreciates in value 25% each year. What is it worth after five years? 42. A vacuum pump removes 60% of the air in a container at each stroke. What percentage of the original amount of air remains after six strok... |
un ; assume that the sequence is arith- 43. u1 3 44. u2 4 and the common difference is –6. and the common difference is 3. 45. u1 5 and u3 7. 46. u3 2 and u7 1. 70 Chapter Review 47. Find the 12th partial sum of the arithmetic sequence with u12 16. u1 3 and 48. Find numbers b, c, and d such that 8, b, c, d, 23 are the... |
ere 3 is said to be sum, or limit, of the infinite series, In the general case, an infinite series, or simply series, is defined to be an expression of the form a2 p an a5 a3 a4 p a1 in which each an is a real number. This series is also denoted by the sym- q bol a n1 an. NOTE a1, a2, a3, p If is a geometric sequence, ... |
aph y x4 2x2 3x 2 using a decimal window. (See Technology Tip.) Find the points where the graph crosses the x-axis. 2. Verify that the x-coordinates found in Step 1 are zeros of the function f. That is, the x-coordinates are solutions of x4 2x2 3x 2 0. y f x intersects the x-axis is of the form A point where the graph ... |
ion Techniques that can be used to solve all quadratic equations include • completing the square • using the quadratic formula Section 2.2 Solving Quadratic Equations Algebraically 89 Solving Quadratic Equations by Factoring The factoring method of solving quadratic equations is based on the Zero Product Property of re... |
x2 kx 49 0 71. kx2 8x 1 0 72. kx2 24x 16 0 In Exercises 61–68, find all exact real solutions of the equation. 61. y4 7y2 6 0 62. x4 2x2 1 0 63. x4 2x2 35 0 64. x4 2x2 24 0 65. 2y4 9y2 4 0 66. 6z4 7z2 2 0 67. 10x4 3x2 1 68. 6x4 7x2 3 73. Find a number k such that 4 and 1 are the solutions of x2 5x k 0. 74. Suppose a, b,... |
area of the walk is found by subtracting the area of the garden from the area of the outer rectangle. 40 2x, 24 2x. Section 2.3 Applications of Equations 103 Area of outer rectangle 1 1 2 Area of garden 2 Area of the walk 40 2x 24 2x > 2 1 > 660 960 128x 4x2 960 660 24 40 21 > 2 2 1 1 4x2 128x 660 0 x2 32x 165 0 165 2... |
t, what is the length of a side b of its base? Section 2.4 Other Types of Equations 107 29. Suppose that the open-top box being made from a sheet of cardboard in Example 7 is required to have at least one of its dimensions greater than 18 inches. What size square should be cut from each corner? 30. A homemade loaf of b... |
rip is to take 3 hours, how far from B should she land? Figure 2.4-9 Solution Refer to Figure 2.4-9. The basic formula for distance can be written in different ways. d rt or t d r 114 Chapter 2 Equations and Inequalities Let x represent the distance between C and B, t represent the time required to run from C to B, and... |
denotes the set of all real numbers x such that x b. x 7 b. • For the half-line to the left of b, denotes the set of all real numbers x such that denotes the set of all real numbers x such that 4 2 x b. x 6 b. Similar notation is used for the entire number line. q, q denotes the set of all real numbers. 2 b, q b, q 2 2... |
x 8 2x2 5x 3 6 1 Be alert for hidden behavior. 70. 1 x2 In Exercises 71–73, read the solution of the inequality from the given graph. 71. 3 2x 6 0.8x 7 y y = 3 − 2x (−1.43, 5.86) y = 0.8x + 7 8 6 4 2 x 2 4 −4 −2 0 −2 Section 2.5 Inequalities 125 72. 8 7 5x 7 − 5x| (0.4, 3) y = 3 (2.4, 3) x −8 −4 0 4 8 −4 −8 73. x2 3x 1... |
erwise. 1. 3. 5. 7. 9. 11. 13. 3x 2 3 2x 2x 3 5x 12 5 2x 2x . 4. 6. 8. 5x 1 4 5x 3x 1 2 3x 10. 12. 14. ` ` ` 5 6 3x 6 7 6 ` x 1 3x 15. 17. 19. 21. 23. 25. 27. 28. 29. 31. ■ 3x 1 1 2x ` ` 2 x2 4 3 0 1 x2 1 ` ` 2 x2 x 4 0 x2 3x 4 2 6 6 0 7 1 4x x3 0 1 4x 2 3x ` ` 6 1 x2 2 x2 2 6 1 7 4 0 0 x2 x 1 1 0 3x2 8x 2 x5 x3 1 0 6 ... |
e a scatter plot of the data. d. Estimate the length of the base that produces the maximum area, and state the approximate maximum area. Solution a. The base must be greater than 0 and less than 8.5 inches, and nt in the chart indicates that no triangle can be formed with a base length of 9 inches. b. The values shown ... |
h the rule of the function is defined for all values of t. Analogous comments apply to other applications. d t 1 2 A real-life situation may lead to a function whose domain does not include all the values for which the rule of the function is defined. 146 Chapter 3 Functions and Graphs Example 7 Finding the Domain of a... |
any two such points would lie on the same vertical line, this fact provides a useful test for determining whether a graph represents a function. Vertical Line Test A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once. Example 2 Determining Whether a Gr... |
−10 Figure 3.2-16 Technology Tip To change to parametric mode, choose PAR in the TI MODE menu or PARM in the TYPE submenu of the Casio GRAPH menu (on the main menu). Graphing or y f(x) in Parametric Mode x f(y) NOTE The graph in part a is the same as in Example 9. To obtain the equation in part a from the parametric e... |
3 x 6 For the function y-intercepts. Then sketch the graph. x 2 2x 3, x f 2 1 find the vertex and the x- and Solution In f x 1 2 x 2 2x 3, a 1, b 2, and c 3 c 3. Thus, the y-intercept is and the vertex is b 2a a b2 4ac 8, Since tions, so there are no x-intercepts. b 2abb , the quadratic equation 1 2 1, 2 2 1 1 2 2 x 2 ... |
opcorn per game when 20 vendors are working. For every additional vendor, each averages 1 fewer box sold per game. How many vendors should be hired to maximize sales? In Exercises 50–53, use the following equation for the height (in feet) of an object moving along a vertical line after t seconds: s 16t2 v0t s0 s0 is th... |
the y-axis, by a factor of 1 c . If c 66 1, the graph of g(x) f(c x) horizontally, away from the y-axis, by a factor of is the graph of f stretched 1 c . NOTE Some horizontal compressions can be expressed as vertical stretches, and vice versa. The graph of the function g below can be 1 3 obtained from a horizontal comp... |
the graph at a point Q such that the origin is the midpoint of 3.4.A-3. as shown in Figure PQ, y y = x3 4 P (x, y) O x Q (−x, −y) Figure 3.4.A-3 Using Figure 3.4.A-3, symmetry with respect to the origin can also be described in terms of coordinates and equations. Origin Symmetry Section 3.4.A Excursion: Symmetry 187 A ... |
21 5 194 Chapter 3 Functions and Graphs c. To find x 21 g f 1 g f 1 x 2 21 g 2 g f x 1 22 1 , replace x with the rule for f in g. x 1 2 x f 1 22 1 1 2 2 f x 1 1 4x 2 1 1 2 2 1 4x 2 3 , replace x with the rule for g(x) in f. d. To find 1 f g x f 2 21 f f g x 2 21 1 g x 1 1 22 g x 22 1 1 4 2 1 4 x 1 22 2 1 x 2b a 1 g 1 ... |
e nth iteration. 200 Chapter 3 Functions and Graphs NOTE Throughout this section all numerical results are displayed rounded to four decimal places, but computations are done using the full decimal expansion given by a calculator. Technology Tip If a function has been entered as Y1 in the equation memory, it can be ite... |
e 3 Finding an Inverse from an Equation Find g x 1 2 , the inverse of f 3x 2. x 1 2 Solution First, write the function in terms of x and y. y 3x 2 Exchange the x and the y. Thus, the inverse relation is that the relation can be represented in function notation. It is common to solve for y, so x 3y 2 x 3y 2 ■ Graphing E... |
0 1 16 2 64 3 3.5 4 4.5 5 144 196.5 256 324.5 400 Section 3.7 Rates of Change 215 To find the distance the rock falls from time the end of three seconds, the rock has fallen 16 had only fallen to 144 feet at the end of one second, note that at feet, whereas it 144 16 128 feet So during this time interval the rock trave... |
b. 20 to 60. d. 0 to 100. Section 3.7 Rates of Change 221 200 s e l a 100S 0 10 20 30 40 50 60 70 80 90 100 Pages 7. When blood flows through an artery (which can be thought of as a cylindrical tube) its velocity is greatest at the center of the artery. Because of friction along the walls of the tube, the blood’s veloc... |
tion f as x changes from a to b is the number The average rate of change of a function f as x changes from x to x h is given by the difference quotient of the function Review Exercises Section 3.1 1. Let f be the function given by the rule f 7 2x. Complete the table x 1 2 below. x f(x. If h 3. If f x 1 2 x 1 2 x 2 3x, ... |
l misses the rooftop on its way down and falls to the ground. Find the instantaneous velocity of the ball at seconds. t 2 Solution The height of the ball is given by the equation 2 48t 160. s t 2 1 16t t 2 The exact speed of the ball at can be approximated by finding the average speed over very small time intervals, sa... |
ion or a polynomial function. The context should clarify the meaning. Review addition, subtraction, and multiplication of polynomials in the Algebra Review Appendix, if needed. Polynomial functions of degree less than 5 are often referred to by special names. • First-degree polynomial functions are called linear functi... |
coefficient of the product of the factors. So, a 15. 15x3 x2 114x 72 15 x 3 1 2a x 2 3b a x 2 x 12 5 b x 12 5 b x 12 5 b 3 5 x 3 1 3 3b a 2a x 2 a 3b 5x 12 5 a 3x 2 21 x 3 x 3 1 1 2 21 2 ■ Example 8 A Polynomial with Specific Zeros Find three polynomials of different degrees that have 1, 2, 3, and zeros. 5 as Solution... |
ynomial is written as the product of irreducible factors with real coefficients, it is said to be completely factored over the set of real numbers. All linear polynomials are irreducible, and some quadratic polynomials are irreducible over the set of real numbers. NOTE Recall from 2 ± 212 2 algebra that can be simplifi... |
4 2x3 4x2 4x 1 34. x5 8x4 20x3 9x2 27x 27 35. x4 48x3 101x2 49x 50 36. 3x7 8x6 13x5 36x4 10x3 21x2 41x 10 22 37. a. Show that is an irrational number. Hint: x2 2. 22 Does this polynomial have any rational zeros? 23 b. Show that is irrational. is a zero of 38. Graph f 0.001x3 0.199x2 0.23x 6 in the x 1 2 x appear to hav... |
s of 1 Let c be a zero of multiplicity k of a polynomial f. • If k is odd, the graph of f crosses the x-axis at c. • If k is even, the graph of f touches, but does not cross, the x-axis at c. Example 1 Multiplicity of Zeros x 2 Find all zeros of State the multiplicity of each zero, and state whether the graph of f touc... |
2 54. g x 1 2 55. h 56. f x x 2 2 1 1 x3 3x2 4 48. g x 2 1 4x 4x3 3 0.25x4 2x3 4x2 0.25x4 2x3 3 3x3 18.5x2 4.5x 45 2x3 x2 4x 2 x5 3x3 x 1 0.25x4 x2 0.5 8x4 22.8x3 50.6x2 94.8x 138.6 32x6 48x4 18x2 1 g 57. Critical Thinking a. Graph x 2 the viewing window with 0 y 6 coincide with the horizontal line 1 0.01x3 0.06x2 0.12... |
Census Bureau] 9. a. Sketch a scatter plot of the data from 1985 to 1999, with x 0 corresponding to 1985. b. Decide whether a quadratic or quartic model seems more appropriate. c. Find an appropriate polynomial model. a. Sketch a scatter plot of the data, with x 0 corresponding to midnight. b. Find a quadratic polynom... |
a rational function on a calculator often depends on choosing an appropriate viewing window. For example, the following are graphs of f x 1 2 x 1 2x 4 in different viewing windows. 10 6 10 10 8 12 10 Figure 4.4-6a 6 Figure 4.4-6b x 2, but not at The vertical segment shown in Figure 4.4-6a is not a vertical asymptote. ... |
x 3 x2 x 6 In Exercises 13–22, find the horizontal or other asympis large, and tote of the graph of the function when find a viewing window in which the ends of the graph are within 0.1 of this asymptote. x 00 00 13. f x 1 2 3x 2 x 3 15. h x 2 1 5 x x 2 14. g x 1 2 3x2 x 2x2 2x 4 16. f x 1 2 4x2 5 2x3 3x2 x 17. g x 1 ... |
e enlarged again. There is a number system, called the complex number system, with the desired properties. and and x2 4 24 294 Chapter 4 Polynomial and Rational Functions Properties of the Complex Number System 1. The complex number system contains all real numbers. 2. Addition, subtraction, multiplication, and divisio... |
a bi number, then its conjugate is usually denoted Prove that for any complex that is, is a real number exactly number when z a bi. z a bi, z z z. z, 74. Critical Thinking The real part of the complex a bi number a bi The imaginary part of real number b (not bi). See Exercise 73 for notation. is defined to be the real... |
that c is not in the Mandelbrot set by finding a number in its orbit that is more than 2 units from the origin. How many iterations are needed to find the first such number? 7. c 0.4 9. c 0.7i 8. c 1.1 0.4i 10. c 0.2 0.8i 11. c 0.5 0.7i 12. c 0.4 0.6i In Exercises 13–18, determine whether or not c is in the Mandelbrot... |
the two zeros of x 2 ± 2 21 21 1 x2 2x 2 21 . 21 2 ⎧⎪⎪⎨⎪⎪⎩ x2 2x 2 x2 2x 2 ± 2i 2 2 2 2 2 ± 24 2 1 i x 1 and The complex factors are Section 4.6 The Fundamental Theorem of Algebra 313 The complete factorizations of the set of complex numbers are shown below. x f 1 2 over the set of real numbers and over x4 5x3 4x2 2x 8... |
multiplicity touch but do not cross the x-axis. Zeros of odd multiplicity cross the x-axis. The number of local extrema of the graph of a polynomial function is at most one less than the degree of a polynomial. Chapter Review 317 The number of points of inflection of the graph of a polynomial function is at most two le... |
ess than 11. (Why?) 2 660x, On the graph of 3 104x˛ y 4x˛ 4x˛ x f 1 2 • the x-coordinate of each point is the size of the square to be cut from each corner. • the y-coordinate of each point is the volume of the resulting box. The box with the largest volume corresponds to the point with the largest y-coordinate, that i... |
expressions are equivalent. 2 264 8 3 8 2 64 2 24 3 4˛ 1 4. 4 2 1 3 2 4˛ 3 2 4˛ This illustrates the definition of rational exponents. 330 Chapter 5 Exponential and Logarithmic Functions Definition of Rational Exponents Let c be a positive real number and let be a rational number t k with positive denominator. t c k is... |
ons on 1.5 y 1.5, the same screen, in order of increasing size and justify your In each of the following cases, arrange x answer by using the graphs. a. c. d, x x and 336 Chapter 5 Exponential and Logarithmic Functions 100. Graph f 2x in the standard viewing x 2 1 window. Then, without doing any more graphing, describe... |
graph of and graphs of y 2x y 3x, and less steeply than the graph of y 2x x f y 3x. has the same shape as the but it climbs more steeply than the graph of 2 1 ex Example 7 Population Growth If the population of the United States continues to grow as it has since 1980, then the approximate population, in millions, of t... |
of 1.08 every year, the balance in the account at the end of year x is given by B x 1 2 6000 1.08 1 x. 2 Therefore, the balance (to the nearest penny) in the account after 10 years is 10 B 1 2 6000 1.08 2 1 10 $12,953.55. ■ The pattern illustrated in Example 1 can be generalized as shown below. Compound Interest If P ... |
of its carbon-14, 2 64% 2 f x P 0.5 x 5730 0.36P P 1 0.36 0.5 0.5 x 5730 x 5730 x 5730 and 0.5 The point of intersection of the graphs of y1 is approximately (8445.6, 0.36) as shown in Figure 5.3-3. Therefore, the mastodon died about 8445.6 years ago. 0.36 y2 ■ 36% f x 1 2 of its carbon 0.36P. 2 1.5 0 0.5 15,000 Figur... |
ue of each year. 3% a dollar x years from today. b. How much will the dollar be worth in 5 years? in 10 years? c. How many years will it take before today’s b. How many dandelions will there be in 16 weeks? dollar is worth only a dime? 43. Average annual expenditure per pupil in 50. a. The half-life of radium is 1620 y... |
an investment of $2500 6.5% at b. If the investment doubles in 6 years, then 6 annual interest rate r, solve 6. To find the r 2 1 by graphing. The point of intersection of the graphs of and Y2 6 is approximately (0.1225, 6). Therefore, an annual interest rate of 12.25% 5.4-11. is needed for the investment to double in ... |
ntial and Logarithmic Functions Quotient Law of Logarithms For all v, w 77 0, log a ln a v wb v wb log v log w ln v ln w. The proof of the Quotient Law of Logarithms is similar to the proof of the Product Law of Logarithms. Example 3 Using the Quotient Law of Logarithms Use the Quotient Law of Logarithms to evaluate ea... |
b 3 log1 1 6 6x 6. a Therefore, Therefore, x x 125. x 0. 3. x 2 Therefore, x 1. d. If log6 6 x, then negative number, has no real solution. Because no real power of 1 6 is a ■ Basic Properties of Logarithms to Other Bases Logarithms are only defined for positive real numbers. That is, logb v is defined only when v 77 0... |
n bu bv for all real numbers b 77 0. If u v, then log b u log bv for all real numbers b 77 0. Because exponential and logarithmic functions are one-to-one functions, the converse is also true. If b u b v, then u v. If log b u log b v, then u v. Exponential Equations The easiest exponential equations to solve are those ... |
s in parts a and b are the same. In Exercises 35–44, solve the equation. (See Example 9.) 23. 9x 4 3x 3 0 u 3x. Hint: Note that 9x 2; let 3x 1 2 35. ln 1 3x 5 2 4x 1 ln 11 ln 2 log x 1 2 1 2 log 2 36. log 1 Section 5.6 Solving Exponential and Logarithmic Equations 387 2 59. Krypton-85 loses 6.44% of its mass each year.... |
es by approximately 1.029910 1.343, which is very close to the successive ratios of the data. ■ −5 0 Figure 5.7-1 100 NOTE Throughout this section, coefficients are rounded for convenient reading, but the full expansion is used for calculations and graphs. 50 −5 0 100 Figure 5.7-2 Section 5.7 Exponential, Logarithmic, ... |
r plot of the data, with 425 400 375 350 325 300 275 250 U.S. Population Projections: 2000–2050 403.687 377.350 351.070 324.927 299.862 281.422 2000 2010 2020 2030 2040 2050 Year a. How well do the projections in the graph compare with those given by the logistic model in Example 2? b. Find a logistic model of the U.S.... |
what is the interest rate? 25. Company sales are increasing at 6.5% per year. If sales this year are $56,000, write the rule of a function that gives the sales in year x (where x 0 corresponds to the present year). 26. The population of Potterville is decreasing at an annual rate of 1.5%. If the population is 38,500 n... |
. Solution y e x a. The slope of b. Using the point-slope form of a line with x 3 tion of the tangent line to is when y e e x x 3. 3 3 m e is and 3 3, e 2 1 , the equa- 5 y e 3 e y e at x 3 3 2 1 3x 2e 3 e x 2 3 1 2 c. The graphs of y e x and y e 3 x 2 2 1 are shown in Figure 5.C-5. ■ Exponential Functions with Bases O... |
should not be rounded until the end of the problem. Example 4 Evaluating Trigonometric Ratios on a Calculator Evaluate the six trigonometric ratios of 20°. Solution Your calculator should have buttons for sine, cosine, and tangent. To find the cosecant, secant, and cotangent, take the reciprocal of each answer. 418 Ch... |
y of applications of the trigonometric ratios. Example 5 Height Above Sea Level A straight road leads from an ocean beach at a constant upward angle of How high above sea level is the road at a point 1 mile from the beach? 3°. ocean 5 2 8 0 f t 3° r o a d Figure 6.2-6 h = height above sea level sea level 426 Chapter 6 ... |
ant 300 65° mph at a heading of a. How far east of the airport is the plane after . (See Exercise 55.) half an hour? b. How far north of the airport is the plane after 2 hours and 24 minutes? 57. A car on a straight road passes under a bridge. Two seconds later an observer on the bridge, 20 feet above the road, notes t... |
ting at a constant rate around its center, O, and P is a point on the outer edge of the wheel. There are two ways to measure the speed of point P, in terms of the distance traveled or in terms of the angle of rotation. The two measures of speed are called linear speed and angular speed. 440 Chapter 6 Trigonometry P Rec... |
rough the point 3, 2 . u, 1 2 Solution Using the values sin u x 3, y 2, 2 213 cos u and 3 213 r 2 3 2 2 1 1 tan u 2 2 2 3 2 213, 2 3 ■ Trigonometric Functions Trigonometric ratios have been defined for all angles. But modern applications of trigonometry deal with functions whose domains consist of real numbers. The bas... |
ind a coterIf an angle is less than 0 or greater than 2p . minal angle between 0 and Thus, the trigonometric functions of a real variable have the following property. by adding or subtracting multiples of 2p 2p Trigonometric Ratios of Coterminal Angles Any trigonometric function of a real number t is equal to the same ... |
tan 1 1 3 2p 4 2p 1 p 2 2 2 2 458 Chapter 6 Trigonometry Periodicity Identities The sine and cosine functions are periodic with period For every real number t, 2P. sin (t 2P) sin t and cos (t 2P) cos t P. The tangent function is periodic with period number t in the domain of the tangent function, For every tan (t P) ta... |
ew 465 11. A 40° b 10 12. C 35° a 12 13. A 56° a 11 14 15. From a point on level ground 145 feet from the base of a tower, the angle of elevation to the top of the tower is 57.3°. How high is the tower? 16. A pilot in a plane at an altitude of 22,000 feet observes that the angle of depression to a nearby airport is the... |
main of a basic trigonometric function that correspond to a given value of the range • Graph transformations of the sine, cosine, and tangent graphs Although a graphing calculator will quickly sketch the graphs of the sine, cosine, and tangent functions, it will not give you much insight into why these graphs have the ... |
ion and Vertical Stretch Graph g 1 2 t 2 1 sin t on the interval 2p, 2p . 4 3 Solution The graph of g is the graph of t f 1 2 sin t 1 2 compressed vertically by a factor of , as shown in Figure 7.1-6. reflected across the x-axis and y 1 −1 1 g(t) = − sin t 2 t 2π π f(t) = sin t −2π −π Figure 7.1-6 ■ Example 6 Vertical ... |
e cot t cost sint ing the quotient , the graph of cot t t f 1 2 can be obtained by graph- y cos t sin t . The cotangent function is not defined when whenever t is an integer multiple of f(t) the range of f(t) has vertical asymptotes at integer multiples of p. p. sin t 0 , and this occurs Therefore the domain of p, and ... |
g t 2 1 7 cos 3t b. h t 2 1 1 3 sin t 2 7 1 −π − 2π 3 − π 3 π 3 2π 3 t π Solution a. The function is the function by 7. Consequently, the graph of g is the graph of k (see Example 1a) stretched vertically by a factor of 7. multiplied g k t t 2 2 1 1 7 cos 3t cos 3t −7 Figure 7.3-8 y f(t) = sin t 2 t π 2π 1 h(t) = sin ... |
ing characteristics: amplitude a 00 00 phase shift c b period 2p b vertical shift d Example 4 Combined Transformations Describe the graph of g t 2 1 2 cos 3t 4 1 2 1. Solution Identify the amplitude, period, vertical shift and phase shift. 1 2 cos 3t 4 g t 1 2 1 amplitude a 0 0 3t 4 1 2 cos 1 2 2 period 2p b 3 2 4 2p 3... |
2 1 sin These results illustrate the following facts. Sinusoidal Graphs If b, d, k, r, and s are constants, then the graph of the function g(t) d sin(bt r) k cos(bt s) is a sinusoid and there are constants a and c such that d sin(bt r) k cos(bt s) a sin(bt c). Example 1 Sinusoidal Graphs Find a sine function whose grap... |
1 tan 2t 10. f t 2 1 cos t g cos t 2 1 1 2 a t b 1 In Exercises 11–13, sketch the graph of each function. 11. 13 cos t 2 sin t 3 12. h t 2 1 tan t 4 14. Which of the following functions has the graph shown below between p and p? a. b. c. d. e sin x, cos x, cos x 1 sin x, sin x e if x 0 if x 6 0 if x 0 if x 6 0 , −π 2 1... |
n points, so the equation has an infinite number of solutions. tan x 2. and Y1 tan x Y2 2 y 4 2 −2π −3π 2 −π − π 2 −2 π 2 π 2π 3π 2 5π 2 3π 7π 2 4π x One period Figure 8.1-1 5 π 2 π 2 5 Figure 8.1-2 NOTE Solutions in this chapter are often rounded, but the full decimal expansion given by the calculator is used in all c... |
inverse sine function without using a calculator. Technology Tip If you attempt to use a calculator to evaluate the inverse sine function at a number not in its domain, such as sin you will get an error message. 1 2 , 2 1 Example 1 Special Values Evaluate: a. sin 1 1 2 Solution b. 1 sin 22 2 b a a. 1 2 sin 1 is the num... |
two solutions (intersection points) on the interval one full period of the cosine function. in Figure 8.3-1 show that there are which is p, p , 3 4 Y2 0.6 The definition of the inverse cosine function states that cos 10.6 Using the inverse cosine function, of by using the identity 0, p is the number in the interval 3 1... |
r are there exactly 11 hours of daylight? b. What day has the maximum amount of daylight? 55. A weight hanging from a spring is set into motion moving up and down. Its distance d (in centimeters) above or below the equilibrium point at time t seconds is given by d 5 1 sin 6 t 4 cos 6 t . 2 At what times during the firs... |
valently, a 8, b 2p 5 , and c 0. 1 2 3 4 5 t Therefore, the motion of the moving weight can be described by this function: 8 sin h t 2 1 2p 5 t Figure 8.4-8 b. The graph of h 8 sin t 2 1 2p 5 t is shown in Figure 8.4-8. c. The value of h 3 1 2 gives the height of the weight after 3 seconds. 8 sin h 3 1 2 2p 5 a 3 b 8 s... |
.1 39.3 27.4 [Source: National Climatic Data Center] 13. The table shows the average monthly precipitation, in inches, in San Francisco, CA, based on data from 1971 to 2000. Month Precipitation Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. 4.45 4.01 3.26 1.17 0.38 0.11 0.03 0.07 0.20 1.04 2.49 2.89 [Source... |
n tan real number b a 0 Let teristics. and b 7 0. The following functions have the given charac- f t 2 1 a sin bt c 2 1 d and g t a cos bt c 0 2 1 a amplitude fmax 1 2 1 1 fmin2 0 , phase shift c b fmin2 fmax period 2p b vertical shift d 1 2 1 d 2 563 564 Chapter Review Review Exercises In Exercises 1–6, solve each equ... |
ling with identities, but some suggestions follow. The phrases “prove the identity” and “verify the identity” mean “prove that the given equation is an identity.” 571 572 Chapter 9 Trigonometric Identities and Proof 4 Graphical Testing 2π 2π 4 Figure 9.1-1 When presented with a trigonometric equation that might be an i... |
x tan x sec x cos x tan x˛1 2 tan x˛1 tan x sec x cos x 2 2 ■ Look carefully at how identity b was proved in Example 7. First prove AD BC B tan x, identity a, which is of the form C tan x, D sec x cos x Then divide both sides by BD, that . A sec x, (with and sec x cos x 2 is, by tan x˛1 to conclude that , 2 A B C D . ... |
7 for the signs of the functions in each quadrant. y ( 7, 3) x x 4 3 x 42 − 32 = 7 sin x = 3 4 Figure 9.2-1a y 1 y 3 32 − 12 = 8 = 2 2 (−1, −2 2) cos y = − 1 3 Figure 9.2-1b Section 9.2 Addition and Subtraction Identities 585 mine in which of the following intervals x y lies: p 2 b , 0, a 3p 2 b , or 3p 2 a , 2p . b So... |
and M is 2 negative angle identity, and the addition identity 1 and by periodicity, the 592 Chapter 9 Trigonometric Identities and Proof p tan tan 1 3 b a tan 1 tan b tan a 1 tan b tan a m k 1 mk . This completes the proof when similar. b a. The proof in the case a b is ■ Example 3 The Angle Between Two Lines If the sl... |
2, write each expression as a product. 19. sin 3x sin 5x 20. cos 2x cos 6x 21. sin 9x sin 5x 22. cos 5x cos 7x 23. sin x 5 13 , for 0 6 x 6 p 2 24. sin x 4 5 , for p 6 x 6 3p 2 25. cos x 3 5 , for p 6 x 6 3p 2 26. cos x 1 3 , for p 2 6 x 6 p 27. tan x 3 4 , for p 6 x 6 3p 2 28. tan x 3 2 , for p 2 6 x 6 p 29. csc x 4, ... |
tion in the form sin a NOTE The value of not needed to find the maximum or minimum of the function. is x f 1 2 5 3 5 sin x 4 5 a cos x b 5 cos a sin x sin a cos x 2 1 Because the sine function varies between 5 sin is 5 and the minimum is x a 5. 1 2 x a 1 2 and 1, the maximum of ■ 1 5 sin 608 Chapter 9 Trigonometric Ide... |
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