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inomial Theorem a formula that can be used to expand any binomial combination a selection of objects in which order does not matter common difference the difference between any two consecutive terms in an arithmetic sequence common ratio the ratio between any two consecutive terms in a geometric sequence complement of ...
nd the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See Example 4. • The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See Example 5. • In some probability problems, we need to use permutatio...
ings to choose from. How many ways are there to top a frozen yogurt? 21. How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E? 3 1 y  22. Use the Binomial Theorem to expand  _ _ x − 2 2 5 . 1  23. Find the seventh term of  x2 − __ 2 expanding the binomial. 13 without...
ues of x sufficiently close to a but greater than a. Both a and L are real numbers. Understanding Two-Sided Limits In the previous example, the left-hand limit and right-hand limit as x approaches a are equal. If the left- and right-hand limits are equal, we say that the function f (x) has a two-sided limit as x approa...
limit as x approaches a if, as x approaches a, the left-hand limit is not equal to the right-hand limit. GRAPHICAl For the following exercises, estimate the functional values and the limits from the graph of the function f provided in Figure 14. y 6 5 4 3 2 1 −5 −4 −3 −1 −2 −1 −2 −3 −4 −5 −6 f(x) 21 3 4 5 x Figure 14 ...
d another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots. Example 4 Eval...
t 4.25% interest is given by the formula A = A0 e 0.0425t, where A0 is the initial amount invested. Find the average rate of change of the balance of the account from t = 1 year to t = 2 years if the initial amount invested is $1,000.00. 59. The height of a projectile is given by s(t) = −64t 2 + 192t Find the average r...
termine if the function is continuous at x = a. 1. Check Condition 1: f (a) exists. 2. Check Condition 2: lim x → a 3. Check Condition 3: lim x → a f (x) = f (a). f (x) exists at x = a. 4. If all three conditions are satisfied, the function is continuous at x = a. If any one of the conditions is not satisfied, the func...
where it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous. 23. f (x) = x3 − 2x − 15 27. f (x) = \| x − 2 \| ______ x2 − 2x 24. f (x) = x2 − 2x − 15 ___________ x − 5 28. f (x) = tan(x) + 2 25. f (x) = 2 ⋅ 3x + 4 26. f (x) = −sin(3x) 5 _ 29. f (x) = 2x + x 30. f (x...
dx df ___ dx d ___ dx The expression f ′(x) is now a function of x ; this function gives the slope of the curve y = f (x) at any value of x. The derivative of a function f (x) at a point x = a is denoted f ′(a). How To… Given a function f, find the derivative by applying the definition of the derivative. 1. Calculate f...
x) ______________________________ h = x2 + 2xh + h2 − 100x − 100h − x2 + 100x __________________________________ h = 2xh + h2 − 100h ______________ h = h(2x + h − 100) ______________ h = 2x + h − 100 = 2x − 100 f ′(x) = 2x − 100 f ′(200) = 2(200) − 100 = 300 Formula for a derivative Substitute f (a + h) and f (a). Mult...
a certain velocity at a given instant. That means that if the object traveled at that exact velocity for a unit of time, it would travel the specified distance. instantaneous velocity Let the function s(t) represent the position of an object at time t. The instantaneous velocity or velocity of the object at time t = a ...
ly if x → a− f (x) = lim lim x → a+ f (x). Key equations average rate of change AROC = f (a + h) − f (a) _____________ h derivative of a function f ′(a) = lim h → 0 f (a + h) − f (a) _____________ h CHAPTER 12 review 1071 Key Concepts 12.1 Finding Limits: Numerical and Graphical Approaches • A function has a limit if t...
f (x) = 3 ______ 5 + 2x 15. f (x) = 3 _ — √ x 16. f (x) = 2x 2 + 9x 17. For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable. y 5 4 3 2 1 x 21 3 4 5 f(x) −1−2−3−4−5 –1 –2 –3 –4 –5 Figure 2 1076 CHAPTER 12 introduction to calculus For the following ex...
k ≥ 7; in interval notation, this would be (−∞, 1] ∪ [7, ∞). Section 1.7 4. The domain of function 3. Yes 2. Yes 1. h(2) = 6 f −1 is ( −∞, −2) and the range of function f −1 is (1, ∞). 5. a. f (60) = 50. In 60 minutes, 50 miles are traveled. b. f −1(60) = 70. To travel 60 miles, it will take 70 minutes. 8. f −1(x) = (2...
_____ ≈ ln(5) = 2.861 x3(x + 5)  _______ . (2x + 3) 13. 14. — 4 Section 4.6 1. x = −2 2. x = −1 2 4. The equation has no solution. 11 11 __ __  or ln  6. t = 2ln   3 3 7. t = ln  1 1 __ _  = − ln(2) — 2 2 √ 10. x = e5 − 1 11. x ≈ 9.97 ln(0.8) ______ ln(0.5) 1 __ 3. x = 2 5. x = ln(3) _ 2 _  ln  3 8. x = ln(2)...
+ 4(1) −1(−3) + −3(13(1) + −4(−1)  =  1(1) + 1(−1) =  1 0  1 0 3. A−1 =  1 2 4 −3 −5 3  1 2 6 1(−4) + 4(1)  −1(−4) + −3(1) −3(4) + −4(−3)  1(4) + 1(−3) 4. X =  4 38 58  BA =  −3 1 −4 1   1 4 −3 −1 3 __ 5 2. A−1 =  − 2 __ 5 1 __ 5  1 __ 5 Section 9.8 1. (3, −7) 2. −10 3 3.  −2, _ , 5 12 _  5 Chapter 10...
x 43. domain: (−∞, ∞) 45. domain: (−∞, ∞) y 5 4 3 2 1 –1 –2 –3 –4 –5 –2 –1 1 2 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x 47. f (−3) = 1; f (−2) = 0; f (−1) = 0; f (0) = 0 49. f (−1) = −4; f (0) = 6; f (2) = 20; f (4) = 34 51. f (−1) = −5; f (0) = 3; f (2) = 3; f (4) = 16 53. (−∞, 1)∪(1, ∞) 55. y y 104 96 8...
in of f (x): [−7,∞); f −1 (x) = √ 5. y = f −1(x) 11. f −1(x) = − 2x _ x − 1 x − 7 — 15. Domain of f (x): [0, ∞); f −1 (x) = √ 17. f ( g(x)) = x and g( f (x)) = x 21. One-to-one 23. Not one-to-one — x + 5 19. One-to-one 25. 3 27. 2 33. 6 37. 0 39. 1 31. [2, 10] 35. −4 41. x f −1(x) 1 3 4 6 7 9 12 13 16 14 5 _ 45. f −1 (...
f a = 0 then the function becomes a linear function. 5. If possible, we can use factoring. Otherwise, we can use the 7. g(x) = (x + 1)2 − 4; vertex: (−1, −4) 33  _ 4 quadratic formula. 2 5 − 33 5 ; vertex:  −  9. f (x)=  11. k(x) = 3(x − 1)2 − 12; vertex: (1, −12) 2 13. f (x) = 3  x − 5 5 ; vertex:   _ _ , − 6 6...
_ _ , f (x) → −∞ 2 2 1 _ End behavior: x → ±∞, f (x) → 3 33. y = 2x 31. y = 2x + 4 35. Vertical asymptote at x = 0, horizontal asymptote at y = 2 y 10 8 6 4 2 y = 2 −10 −4−6−8 −2−2 −4 −6 −8 −10 42 6 8 10 x x = 0 37. Vertical asymptote at x = 2, horizontal asymptote at y = 0 y 10 8 6 4 2 −10 −4−6−8 y = 0 −2−2 −4 −6 −8 ...
2 21 3 4 5 x f (x) = −4(2)x + 2 27. Horizontal asymptote: h(x) = 3; domain: all real numbers; range: all real numbers strictly greater than 3. h(x1−2−3−4−5 21 3 4 5 x 29. As x → ∞, f (x) → −∞; as x → −∞, f (x) → −1 31. As x → ∞, f (x) → 2; as x → −∞, f (x) → ∞ 33. f (x) = 4x − 3 35. f (x) = 4x − 5 37. f (x) = 4−x 39. ...
populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic 3. Regression analysis is model best describes populations. the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the...
2 1 2 3 1 3 −π 0 π− 3 1− 2 2− 3 −1 π 2 π 3π 2 2π x −2π 3π− 2 −π 4 3 2 1 0 π− 2 −1 −2 −3 −4 π 2 π 3π 2 2π x −2π − 3π 2 11. Amplitude: 1; period: π; midline: y = 0; maximum: y = 1 occurs at x = π; minimum: y = −1 occurs at π _ x = ; for one period, the 2 graph starts at 0 and ends at π. f(t) 1 13. Amplitude: 4; period: ...
−0.6 −0.8 −1 −1 −0.6 0.2 0.6 1 x −1 −0.6 y 1 0.8 0.6 0.4 0.2 0 −0.2 −0.2 −0.4 −0.6 −0.8 −1 0.2 0.6 1 x 9π− 4 −2π 7π− 4 3π− 2 5π− 4 −π 3π− 4 4 3 2 1 7π− −3π 2 5π− 2 −2π 3π− 2 −π π − 2 −1 −2 −3 −4 π 2 π 3π 2 2π 5π 2 3π 7π 2 4π 9π 2 5π 2 11π 6π 2 13π 7π 2 15π 8π 2 17π x 9. Amplitude: none; period: π; midline: y = 0; asym...
_ ____ ____ ___ ___ ____ ___ __ __ 21. , , , , , , , , 12 12 12 12 6 12 6 6 6 π π 5π 5π 5π 3π π __ __ ___ ___ ___ ___ __ 23. 0, 25. 27. , π, , π 11π 7π 1 1 ___ ____ __ __ 31. π − sin−1  −  ,  , 2π + sin− 2π π 1 1 1   ,  sin−1    ,  sin−1    ,  sin− − + 33. 3 10 3 3 10 3 10 3 9 9 9 5π 4π 1 1 1   ,   , ...
0 10 20 30 21. Cardioid 23. One-loop/dimpled limaçon 45. (θ from 0 to 8) 47. (θ from −π to π) (θ from 0 to 2π) (θ from 0 to 2π 11 0.5 1 1.5 2 0.5 1 1.5 2 25. One-loop/dimpled limaçon (θ from 0 to 2π) 27. Inner loop/two-loop limaçon 49. (θ from 0 to 2π) 51. (θ from 0 to 3π) 1 2 3 2 4 6 8 10 1 3 5 7 9 1 2 3 4 53. (θ from...
the second account tops: 45, Low-tops: 15 more information. 73. $12,500 in 75. High- 77. Infinitely many solutions. We need — 3  Section 9.2 1. No, there can be only one, zero, or infinitely many solutions. 3. Not necessarily. There could be zero, one, or infinitely many solutions. For example, (0, 0, 0) is not a solu...
. x − 3z = 7 y + 2z = − 5 with infinite solutions C-37 53.  −2 2 1 7 2 −8 5 0 19 −10 22 3 \|  55.  1 0 3 −1 4 0 0 1 2 \| 12 0 −7  59. No solutions exist 57. No solutions exist  63. No inverse exists 1 2 7  __ 61. 6 1 8 67. (−1, 0.2, 0.3) 1 __  77. ( x, 5x + 3) 71. 0 73. 6 75.  6, 2 69. 17% oranges, 34% bananas, 3...
s (2, 0) 2 4 6 8 x 24 16 8 Focus (0, 9) −32 −24 −16 −8 8 16 24 32 y = −9 −8 −16 −24 37. −10 −7.5 −5 −2.5 y 2.5 x = 25 6 2.5 x 5 35. y 7.5 5 2.5 x − 5 3 7 3 −10 −7.5 −5 −2.5 x 2.5 39. −2.5 y 5 y = 0 −5 −20 −10 −15 Focus (−4, −2) −5 −10 −15 −20 41. x 5 x = −4 −10 −5 −2.5 −5 −7.5 −10 5 10 15 Focus (0, −5) 25 6 y 10 5 −5 −...
arithmetic sequences and linear functions difference. have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers. 7. The common 21. a 1 = 5 9. Th...
values of x get infinitely close to a but x → a never equal a. As the values of x approach a from the left and right, the limit is the value that the function is approaching. 3. −4 5. −4 11. 4 13. Does not exist 17. Answers 19. Answers will vary 21. Answers will vary will vary 23. 7.38906 25. 54.59815 27. e 6 ≈ 403.42...
heorem 703, 704, 747 58 domain of a rational function 283 dot product 739, 747 double-angle formulas 584, 585, 634 doubling time 401, 405, 429 Dürer 688 E eccentricity 923, 931 electrostatic force 41 elimination 788 ellipse 721, 788, 865, 866, 867, 869, 872, 896, 923, 927, 931 ellipsis 938 end behavior 226, 287, 317 en...
titution method 761, 854 sum and difference formulas for cosine 572 sum and difference formulas for sine 573 sum and difference formulas for tangent 575 summation notation 969, 970, 1009 sum-to-product formula 599, 634 surface area 299 symmetry test 682 synthetic division 261, 270, 318 system of equations 817, 818, 820...
ponential functions to model and solve real-life problems (p. 223). Section 3.2 Recognize and evaluate logarithmic functions with base a (p. 229). Graph logarithmic functions (p. 231). Recognize, evaluate, and graph natural logarithmic functions (p. 233). Use logarithmic functions to model and solve real-life problems ...
rcises further explores and expands upon concepts learned in this chapter. 1. As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. (a) Write a linear equation for your current monthly wage W1 in terms o...
, page 37 • Cost, Revenue, and Profit, • Fuel Use, Exercise 97, page 52 Exercise 67, page 82 • Digital Music Sales, Exercise 89, page 64 • Fluid Flow, Exercise 70, page 68 • Consumer Awareness, Exercise 68, page 92 • Diesel Mechanics, Exercise 83, page 102 1 333202_0101.qxd 12/7/05 8:29 AM Page 2 2 Chapter 1 Functions ...
, 22.65 Simplify. So, you would estimate the 2003 revenue to have been about $22.65 billion, as shown in Figure 1.11. (The actual 2003 revenue was $22.5 billion.) FIGURE 1.11 Now try Exercise 49. 333202_0101.qxd 12/7/05 8:30 AM Page Much of computer graphics, including this computer-generated goldfish tessellation, con...
ent increase in the cost of a 30-second spot (a) from Super Bowl XXIII in 1989 to Super Bowl XXVII in 1993 and (b) from Super Bowl XXVII in 1993 to Super Bowl XXXVII in 2003. 59. Music The graph shows the numbers of recording artists who were elected to the Rock and Roll Hall of Fame from 1986 to 2004. 16 14 12 10 1987...
(0, 7) (1, 4) (2, 1) x 2 6 4 (3, − 2) 8 10 (4, − 5) −4 −2 −2 −4 −6 FIGURE 1.15 Now try Exercise 5. 333202_0102.qxd 12/7/05 8:31 AM Page 16 16 Chapter 1 Function and Their Graphs One of your goals in this course is to learn to classify the basic shape of a graph from its equation. For instance, you will learn that the l...
onfirm algebraically the estimate you found in part (b). Solution a. You can use a calculator to complete the table, as shown at the left. b. The table of values can be used to sketch the graph of the equation, as shown in Figure 1.27. From the graph, you can estimate that a height of 71 inches corresponds to a weight ...
tercept form. The Slope-Intercept Form of the Equation of a Line The graph of the equation y mx b is a line whose slope is m and whose y- intercept is 0, b. Exploration Use a graphing utility to compare the slopes of the lines m 0.5, 1, 2, 1, 2, and obtain a true geometric perspective. What can you conclude about the s...
horizontal change 22 in. 288 in. 0.076. Because 1 12 0.083, the slope of the ramp is not steeper than recommended. y 22 in. 24 ft x FIGURE 1.41 Now try Exercise 97. Example 6 Using Slope as a Rate of Change A kitchen appliance manufacturing company determines that the total cost in x dollars of producing units of a bl...
rofits (in millions) for Applebee’s International, Inc. for the years 1994 through 2003. (Source: Applebee’s International, Inc.) In Exercises 89–92, find a relationship between such that two points. y is equidistant (the same distance) from the x, y and x 89. 90. 91. 92. 4, 1, 2, 3 6, 5, 1, 8 , 7, 1 3, 5 2 2, 4, 7 1 2...
00, (2001, (2002, 5.3) 7.6) 11.0) 16.0) 24.1) 33.8) 44.0) 55.3) 69.2) 86.0) 109.5) 128.4) 140.8) (a) Sketch a scatter plot of the data. Let x 0 corre- spond to 1990. (b) Use a straightedge to sketch the line that you think best fits the data. (c) Find the equation of the line from part (b). Explain the procedure you us...
to general, gu v gu gv. gx g2. Let a. gx x2 4x 1. g2 gt b. c. Find each function value. gx 2 Solution a. Replacing with 2 in x gx x2 4x 1 yields the following. g2 22 42 1 4 8 1 5 b. Replacing with yields the following. t x gt t2 4t 1 t 2 4t 1 c. Replacing with x yields the following. gx 2 x 22 4x 2 1 x 2 x 2 4x 4 4x 8 ...
b (c) (d) c, 0, b, 0, a, 3 (b) 333202_0104.qxd 12/7/05 8:35 AM Page 49 Circulation of Newspapers In Exercises 11 and 12, use the graph, which shows the circulation (in millions) of daily newspapers in the United States. (Source: Editor & Publisher Company) 50 40 30 20 10 Morning Evening ) 1992 1994 1996 1998 2000 2002 ...
part (c) with the actual data. (e) Use a graphing utility to find a linear model for the data. Let correspond to 1996. How does the model you found in part (b) compare with the model given by the graphing utility? x 6 True or False? In Exercises 103 and 104, determine whether the statement is true or false. Justify yo...
e minimum occurs is 2 3, 10 Now try Exercise 49. 3 You can also use the table feature of a graphing utility to approximate numerically the relative minimum of the function in Example 5. Using a table that begins at 0.6 and increments the value of by 0.01, you can approximate that the minimum of occurs at the point 0.67...
ns of dollars) from sales of digital music from 2002 to 2007 can be approximated by the model r r 15.639t3 104.75t2 303.5t 301, 2 ≤ t ≤ 7 t where 2002. represents the year, with (Source: Fortune) t 2 corresponding to (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 20...
as x f x x the greatest integer less than or equal to x. Some values of the greatest integer function are as follows. 1 greatest integer ≤ 1 1 1 2 greatest integer ≤ 1 1 10 greatest integer ≤ 1 1 1.5 greatest integer ≤ 1.5 1 0 10 2 The graph of the greatest integer function f x x y 3 2 1 −4 −3 −2 −1 1 2 3 4 x ( ) = [[ ...
: 1, 3, 3, 9 76. L1: 1, 7, 4, 3 L2: 1, 5, 2, 7 333202_0107.qxd 12/7/05 8:41 AM Page 74 74 Chapter 1 Functions and Their Graphs 1.7 Transformations of Functions What you should learn • Use vertical and horizontal shifts to sketch graphs of functions. • Use reflections to sketch graphs of functions. • Use nonrigid transf...
and 3. FIGURE FOR 5 FIGURE FOR 6 (a) (b) f x x2 c, x2 c, f x x c2, x c2 In Exercises 5–8, use the graph of to sketch each graph.To print an enlarged copy of the graph go to the website www.mathgraphs.com. f 7. (a) (b) (c) (d) (e) (f) (g 2x 8. (a) (b) (c 10 (f) 3x (g) y f 1 (d) (e) 333202_0107.qxd 12/7/05 8:41 AM Page 8...
of and f f gx f x gx is g 2x 1 x 2 2x 1 x 2 2. When x 2, the value of this difference is f g2 22 2 2. Now try Exercise 5(b). In Examples 1 and 2, both numbers. So, the domains of numbers. Remember that any restrictions on the domains of considered when forming the sum, difference, product, or quotient of and have domai...
hat represents the total number (a) Find the function t, (b) Interpret the value of p5. (c) Let represent the population of the United States in corresponds to 2000. Find and t 0 nt t, year where interpret ht pt nt. 59. Military Personnel The total numbers of Army personnel from (in thousands) and Navy personnel (in th...
5 AM Page 95 y y = x The Graph of an Inverse Function Section 1.9 Inverse Functions 95 y = f (x) (a, b) y = f −1(x) (b, a) x x FIGURE 1.93 f −1( ) = ( + 3− y 6 (1, 2) (3, 3) (2, 1) (1, 1)− 6 (0, 3)− − ( 1, 1) − ( 3, 0) −6 − − ( 5, 1) y x= − − ( 1, 5) FIGURE 1.94 y (3, 9) f (x) = x2 (2, 4) (4, 2) y = x (9, 3) (1, 1) −1 ...
2x 3 f x 3x 1 f x x5 2 f x x3 1 f x x f x x 2, f x 4 x2, f x x2 2 44 50. 48. f x 3x 1 f x 6x 4 4x 5 52. 54. f x x35 f x 8x 4 2x 6 39. 41. 43. 45. 46. 47. 49. 51. 53. In Exercises 55–68, determine whether the function has an inverse function. If it does, find the inverse function. 55. f x x4 57. gx x 8 59. px 4 56. f x...
are shown in the table. Construct a scatter plot that represents the data and find the least squares regression line for the data. (Source: indy500.com) Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Prize money, p 8.06 8.11 8.61 8.72 9.05 9.48 9.61 10.03 10.15 10.25 Solution t 5 represent 1995. The scatter plo...
del on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: The World Almanac and Book of Facts) In Exercises 3– 6, sketch the line that you think best approximates the data in the scatter plot.Then find an equat...
S In Exercises 49–54, write a sentence using the variation terminology of this section to describe the formula. 49. Area of a triangle: A 1 50. Surface area of a sphere: 2bh 51. Volume of a sphere: S 4r 2 r3 V 4 3 52. Volume of a right circular cylinder: V r 2h 53. Average speed: r d t 54. Free vibrations: kg W 333202...
mally and verify that two functions are inverse functions of each other (p. 93). Use graphs of functions to determine whether functions have inverse functions (p. 95). Use the Horizontal Line Test to determine if functions are one-to-one (p. 96). Find inverse functions algebraically (p. 97). Section 1.10 Use mathematic...
many correct answers.) f g such f gx hx. hx 6x 53 135. 136. hx 3x 2 333202_010R.qxd 12/7/05 8:49 AM Page 121 137. Electronics Sales The factory sales (in millions of from 1997 to vt dt dollars) for VCRs 2003 can be approximated by the functions vt 31.86t2 233.6t 2594 and DVD players and dt 4.18t2 571.0t 3706 t where 19...
o graph both equations in the same viewing window. Find the point of intersection. What does it signify? (d) You think you can sell $20,000 per month. Should you change jobs? Explain. 2. For the numbers 2 through 9 on a telephone keypad (see figure), create two relations: one mapping numbers onto letters, and the other...
y f x, For instance, in Figure 2.5, notice how the graph of to produce the graphs of f x x 2 1 and y f x ± c, y f x ± c, y f x. can be transformed y x 2 gx x 22 3. are rigid transformations of the graph of y 2 (0, 1) y x= 2 −2 f(x) = − x2 + 1 x 2 −1 −2 Reflection in x-axis followed by an upward shift of one unit FIGUR...
2, 5; 4, 1; 3, 4; 2, 3; 5, 12; 2, 2; 1 ; 4, 3 2 point: 2, 0 point: 1, 0 In Exercises 53–56, determine -intercept(s) of the graph visually. Then find the Graphical Reasoning the x x -intercepts algebraically to confirm your results. 53. y x 2 16 y −8 −4 x 8 55. y x 2 4x 5 y 54. y x 2 6x 56. y 2x 2 5x 3 x −6 −4 8 −4 −4 ...
ions as sketching aids. • Use the Intermediate Value Theorem to help locate zeros of polynomial functions. Why you should learn it You can use polynomial functions to analyze business situations such as how revenue is related to advertising expenses, as discussed in Exercise 98 on page 151. Graphs of Polynomial Functio...
ant aspect of using a graphing utility is to find a viewing window that shows all significant features of the graph. For instance, the viewing window in part (a) illustrates all of the significant features of the function in Example 4. a. 3 − 4 b. −2 − 3 0.5 −0.5 5 2 If you are unsure of the shape of a portion of the g...
8x 6 f x 1 f x x 6 4 fx 1 4 x6 2 (c) (e) (b) (d) (f) f x x 26 4 f x 1 4x 6 1 fx 2x6 1 In Exercises 13–22, describe the right-hand and left-hand behavior of the graph of the polynomial function. 13. 15. 17. 18. 19. 20. 21. 22. 14. 16. f x 2x 2 3x 1 hx 1 x 6 2x 3x 2 3x 3 5x f x 1 gx 5 7 f x 2.1x 5 4x 3 2 f x 2x 5 5x 7.5 ...
g or decreask? ing? If so, is this behavior determined by Explain. a, h, or g (c) Use a graphing utility to graph the function given by Hx x 5 3x 3 2x 1. Use the graph and the result of part (b) to determine whether form can be written Hx ax h5 k. Explain. the in H Skills Review In Exercises 105–108, factor the express...
and is called ________ if the degree of the numerator is less than that of the denominator. 3. An alternative method to long division of polynomials is called ________ ________, in which the divisor must be of the form x k. 4. The ________ Theorem states that a polynomial f x has a factor x k if and only if f k 0. 5. ...
btraction of Complex Numbers If their sum and difference are defined as follows. are two complex numbers written in standard form, a bi c di and a bi c di a c b di Sum: Difference: a bi c di a c b di The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the add...
replaced by 100% concentrate to bring the mixture up to 60% concentration? 333202_0205.qxd 12/7/05 9:36 AM Page 169 2.5 Zeros of Polynomial Functions Section 2.5 Zeros of Polynomial Functions 169 What you should learn • Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. • F...
fficients can be written as Every polynomial of degree the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. n > 0 333202_0205.qxd 12/7/05 9:36 AM Page 174 174 Chapter 2 Polynomial and Rational Functions A quadratic factor with no real zeros is said to be pr...
ates that if n linear factors f x an x c1 x c2 f x . . . x cn is a polynomial of degree where c1, c2, . . . , cn n n > 0, then has precisely f are complex numbers. 3. The test that gives a list of the possible rational zeros of a polynomial function is called the ________ ________ Test. 4. If a bi is a complex zero of ...
ed and is is the total cost (in dollars) is the number of units produced. The total profit p 140 0.0001x, x of the product and sold. The cost equation C 80x 150,000, C and obtained by producing and selling units is the product where for x x P R C xp C. You are working in the marketing department of the company that pro...
ominator as follows. f x x2 x 2 x2 x 6 x 1x 2 x 2x By setting the denominator you can determine that the graph has the line (of the simplified function) equal to zero, as a vertical asymptote. x 3 Now try Exercise 9. y 4 3 2 1 f(x) = 2x 2 x 2 − 1 Horizontal asymptote: y = 2 −4 −3 −2 −1 1 2 3 4 x Vertical asymptote: x =...
ches wide. What should the dimensions of the page be so that the least are amount of paper is used? 11 2 y 11 in. 2 1 in. x 1 in. 11 in. 2 FIGURE 2.47 Graphical Solution Let be the area to be minimized. From Figure 2.47, you can write A Numerical Solution A Let be the area to be minimized. From Figure 2.47, you can wri...
y vertical asymptotes. 82. The graph of a rational function can never cross one of its asymptotes. Think About It In Exercises 83 and 84, write a rational function that has the specified characteristics. (There are many correct answers.) f 83. Vertical asymptote: None 84. Vertical asymptote: Horizontal asymptote: y 2 x...
ng a Rational Inequality Solve 2x 7 x 5 ≤ 3. Solution 2x 7 x 5 ≤ 3 3 ≤ 0 2x 7 x 5 2x 7 3x 15 x 5 Write original inequality. Write in general form. ≤ 0 ≤ 0 Find the LCD and add fractions. Simplify. x 8 x 5 Critical numbers: Test intervals: Test: x 5, x 8 , 5, 5, 8, 8, Zeros and undefined values of rational expression Is...
ble. 333202_0207.qxd 12/7/05 9:40 AM Page 206 206 Chapter 2 Polynomial and Rational Functions 74. Safe Load The maximum safe load uniformly distributed over a one-foot section of a two-inch-wide wooden beam Load 168.5d 2 472.1, is approximated by the model where is the depth of the beam. d (a) Evaluate the model for d ...
i2 3i i 2 2 1 6i5 2i 72. 74. i6 i3 2i 333202_020R.qxd 12/7/05 9:43 AM Page 210 210 Chapter 2 Polynomial and Rational Functions In Exercises 75 and 76, write the quotient in standard form. 75. 6 i 4 i 76. 3 2i 5 i In Exercises 77 and 78, perform the operation and write the result in standard form. 77. 4 2 3i 2 1 i 78. 1...
niz (1702), Jean d’Alembert (1746), Leonhard Euler (1749), JosephLouis Lagrange (1772), and Pierre Simon Laplace (1795). The mathematician usually credited with the first correct proof of the Fundamental Theorem of Algebra is Carl Friedrich Gauss, who published the proof in his doctoral thesis in 1799. f x Linear Facto...
.6x Solution Value x 3.1 x x 3 2 Function Value f 3.1 23.1 f 2 f 3 0.632 2 a. b. c. Graphing Calculator Keystrokes Display 2 2 .6 > > 3.1 ENTER ENTER > 3 2 ENTER 0.1166291 0.1133147 0.4647580 The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chap...
For quarterly compounding, you have balance is n 4. So, in 5 years at 9%, the nt A P1 r n 12,0001 0.09 4 4(5) Formula for compound interest Substitute for r,P, n, and t. $18,726.11. Use a calculator. b. For monthly compounding, you have n 12. So, in 5 years at 9%, the balance is nt A P1 r n 12,0001 0.09 12 12(5) Formu...
sure kilometers). The data are shown in the table. h True or False? the statement is true or false. Justify your answer. In Exercises 71 and 72, determine whether 71. The line y 2 is an asymptote for the graph of f x 10 x 2. e 271,801 . 99,990 72. Think About It nents to determine which functions (if any) are the same....
graph of the natural exponential function introduced in Section 3.1 on page 388, you will see that is one-to-one and so has an inverse function. This inverse function is called the natural logarithmic x function and is denoted by the special symbol ln read as “the natural log of ” or “el en of ” Note that the natural l...
to an amount times is given by the original principal after years, where t ln K0.095. (a) Complete the table and interpret your results. invested at P, t t 1 2 4 6 8 10 12 K t (b) Sketch a graph of the function. 89. Human Memory Model Students in a mathematics class were given an exam and then retested monthly with an ...
r to lie in a line (see Figure 3.24). Choose any two points to 0, 0, determine the slope of the line. Using the two points you can determine that the slope of the line is 0.421, 0.632 and m 0.632 0 0.421 0 1.5 3 2 . By the point-slope form, the equation of the line is ln y 3 X ln x. You can therefore conclude that 2 ln...
for solving exponential or logarithmic equations. The first is based on the One-to-One Properties and was used to solve simple exponential and logarithmic equations in Sections 3.1 and 3.2. The the following second is based on the Inverse Properties. For properties are true for all and a > 0 and are defined. for which...
4 eln t e476164 t e476164 t 18 Write original equation. Substitute 357 for y. Add 119 to each side. Divide each side by 164. Exponentiate each side. Inverse Property Use a calculator. The solution is number of endangered animals reached 357 in 1998. Because t 18. t 10 represents 1990, it follows that the W RITING ABOUT...
following would result in the highest value of the investment? Explain your reasoning. r, P t (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years. 125. Think About It Are the times required for the investments in Exercises 107 and 108 to quadruple twice as long as the times f...
iors roughly followed the normal distribution given by y 0.0035ex518 225,992, 200 ≤ x ≤ 800 x where From the graph, estimate the average SAT score. is the SAT score for mathematics. Sketch the graph of this function. (Source: College Board) Solution The graph of the function is shown in Figure 3.34. On this bell-shaped...
ons of dollars) of a famous painting can be modeled by V 10ekt t represents the year, with corresponding to where 1990. In 2004, the same painting was sold for $65 million. Find the value of and use this result to predict the value of the painting in 2010. k, t 0 39. Bacteria Growth The number N of bacteria in a cultur...
23s 2 t1 t2 (a) Use the regression feature of a graphing utility to find a for the data. linear model and an exponential model t4 t3 (b) Use a graphing utility to graph the data and each model in the same viewing window. (c) Create a table comparing the data with estimates obtained from each model. (d) Use the results ...
of the function in the context of the problem. (b) Use a graphing utility to graph the function and identify any asymptotes. (c) As the plane approaches its absolute ceiling, what can be said about the time required to increase its altitude? (d) Find the time for the plane to climb to an altitude of 4000 feet. 96. Hum...
x 12 x2 x 2 26. f x x2 4x 3 x2 2x 3 In Exercises 28 and 29, solve the inequality. Sketch the solution set on the real number line. 28. 3x3 12x ≤ 0 29. 1 x 1 ≥ 1 x 5 In Exercises 30 and 31, use the graph of to describe the transformation that yields the graph of f x 2 gx 2.2x 4 gx 2 f x 2.2x, x3 g. x , 30. 31. f 5 5 In ...
xercise 108, page 293 • Respiratory Cycle, Exercise 73, page 330 • Security Patrol, Exercise 97, page 351 • Machine Shop Calculations, • Data Analysis: Meteorology, • Navigation, Exercise 69, page 310 Exercise 75, page 330 Exercise 29, page 360 • Sales, Exercise 88, page 320 • Predator-Prey Model, Exercise 77, page 341...
Now try Exercise 87. The formula for the length of a circular arc can be used to analyze the motion of a particle moving at a constant speed along a circular path. Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. the particle is is the length of the arc traveled...
he linear speed (in feet per minute) of one of the 24 cutting teeth as they contact the wood being cut. 104. Linear and Angular Speeds A carousel with a 50-foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed of the platform rim of the...
d the unit circle, as shown in Figure 4.25. The values of sint 2 Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result correspond to those of cost 2 cos t. sin t and and 0, 2 sint 2n sin t and cost 2n cos , ... , ... t. for any integer and r...

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