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In what year was the rate the highest? x x x 48 x x Section 4.2 Real Zeros 259 40. During the first 150 hours of an experiment, the growth rate of a bacteria population at time t 0.0003t3 0.04t2 0.3t 0.2 g hours is bacteria per hour. a. What is the growth rate at 50 hours? at 100 t 2 1 hours? b. What is the growth rat... |
1 2x6 3x5 4x4 5x3 4x2 3x 2 x 2 g 1 c. Let a12x12 a11x11 p a2x2 a1x a0. f x 1 2 If c is a zero of f, what conditions must the coefficients ai satisfy so that 1 c is also a zero? 260 Chapter 4 Polynomial and Rational Functions 4.3 Graphs of Polynomial Functions Objectives β’ Recognize the shape of basic polynomial functi... |
7 Section 4.3 Graphs of Polynomial Functions 261 Polynomial Functions of Even Degree When the degree of a polynomial function in the form is even, its graph has the form shown in Figures 4.3-2a or 4.3-2b. Again, the graph x when a is negative, is the reflection of Figure 4.3-2a across of 2 1 the x-axis. axn, x f f 1 2 ... |
look the same. β’ In the viewing window 20 x 20 and 10,000 y 10,000, graph f and g. Do the graphs look almost the same? β’ In the viewing window 100 x 100 and 1,000,000 y 1,000,000, graph f and g. Do the graphs look virtually identical? The reason that the answer to the last question is βyesβ can be understood x by obse... |
2 f x 1 2 3x6 2 β4 β2 4 2 0 β2 2 4 x 2 4 β4 β2 Figure 4.3-4b x x 264 Chapter 4 Polynomial and Rational Functions 0.4x3 x2 2x 3 f x 1 2 y 4 2 0.4x3 g x 1 2 y 4 2 β4 β2 0 2 4 β4 β2 0 2 4 x β2 β4 Figure 4.3-4c 0.6x5 4x2 2 β4 0.6x5 g x 1 2 y 4 2 β4 β2 0 2 4 β4 β2 0 2 4 x β2 β4 β2 β4 x x Figure 4.3-4d Intercepts Consider a... |
the graph of f crosses the x-axis at c. β’ 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 touches or crosses the x-axis at each corresponding x-intercept. x 3 x 1 3. 21... |
function changes. The number of inflection points on the graph of a polynomial is governed by the degree of the function. Number of Points of Inflection β’ The graph of a polynomial function of degree n, with n 2, has at most n 2 points of inflection. β’ The graph of a polynomial function of odd degree, with n 7 2, has ... |
-axis, you should see that the y-coordinates increase, then decrease, and then increase again. y x3, Zoom in on the portion of the graph between 0 and 1, as shown in Figure 4.3-8b. Observe that the graph actually has two local extrema, one peak and one valley, which is the maximum possible number of local extrema for a... |
b 50 β The graphs shown in Examples 2β4 were known to be complete because they included the maximum possible number of local extrema. Many graphs, however, may have fewer than the maximum number of possible peaks and valleys. In such cases, use any available information and try several viewing windows to obtain the mos... |
x2 5 Section 4.3 Graphs of Polynomial Functions 271 28. f x 1 2 0.001x5 0.01x4 0.2x3 x2 x 5 In Exercises 29β34, find a single viewing window that shows a complete graph of the function. 29. 30. 31. 32. 33. 34 x3 8x2 5x 14 x3 3x2 4x 5 x4 3x3 24x2 80x 15 x4 10x3 35x2 50x 24 2x5 3.5x4 10x3 5x2 12x 6 x5 8x4 20x3 9x2 27x 7... |
a partial view of the graph of a cubic polynomial whose leading coefficient is negative. Which of the patterns shown in Exercise 41 does this graph have? β10 20 β10 10 43. A complete graph of a polynomial function g is 46. The figure below is a partial view of the graph of a fourth-degree polynomial. Sketch the genera... |
trace feature to verify that the graph is x 1 Find a viewing window that shows actually rising from left to right between and this. x 3. x 1 c. Show that it is not possible for the graph of a 1 2 f x to contain a horizontal polynomial segment. Hint: A horizontal line segment is part of the horizontal line for some con... |
quartic polynomial functions to model the data. Example 1 A Polynomial Model The following data, which is based on statistics from the Department of Health and Human Services, gives the cumulative number of reported cases of AIDS in the United States from 1982 through 2000. Find a quadratic, a cubic, and a quartic reg... |
or h provides a reasonable model for the given data, knowledge of polynomial graphs suggests that the cubic and quartic models, should not be used for predicting future results. As x gets y 219.2x3 larger, the graphs of g and h will resemble respectively those of and, which turn downward. However, the cumulative numbe... |
1 2 745,843.98 and h 803,155.18 54 2 1 That is, the estimated population of San Francisco in 1995 was approximately 745,844 and the estimated population of San Francisco in 2004 was approximately 803,155. β In Example 2, a model may not be accurate when applied outside the 5,817,115, range of points used to construct ... |
nomial Models 277 What does the model predict? d. Is this model a reasonable one? 6. The table, which is based on the U.S. National Center for Educational Statistics, shows actual and projected enrollment (in millions) in public high schools in selected years. Year 1975 1980 1985 1990 Enrollment 14.3 13.2 12.4 11.3 Yea... |
, with x 0 corresponding to midnight. b. Find a quadratic polynomial model for the data. c. What is the predicted temperature for noon? for 9 a.m.? for 2 p.m.? 278 Chapter 4 Polynomial and Rational Functions d. Use the model to predict the median income in a. Sketch a scatter plot of the data with x 0 2002. e. Does thi... |
, such as β’ Identify holes β’ Describe end behavior β’ Sketch complete graphs 1 x f x 2 1 4x 3 2x 1 x t 1 2 2x3 5x 2 x2 7x 6 k x 1 2 Section 4.4 Rational Functions 279 Although a polynomial function is defined for every real number x, a rational function is defined only when its denominator is nonzero. Domain of Rational... |
) 4 (2, 0) β8 β4 0 β8 β16 Figure 4.4-1 x 8 Solution The y-intercept is f 0 1 2 02 0 2 0 1 2. 2 1 x2 x 2 0 The x-intercepts are solutions of x 1 0. Solutions of x2 x 2 0 that are not solutions of can be found by factoring. x 1 x2 x 2 0 x 2 0 1 x 1 or x 2. x 1 0, 21 2 1 Neither graph of f, as shown in Figure 4.4-1. nor 2... |
a y 10 5 0 β5 β10 x 1 2 3 4 Figure 4.4-2b By the Big-Little Concept, x 1 2x 4 3 little f x 2 1 3 1 little 3 big 1 2 very big when This fact can be confirmed by a table of values for x 2.01, 2.001, 2.0001, etc., as shown in Figure 4.4-2a. In graphical terms, the points with x-coordinates slightly greater than 2 have ver... |
function might have a vertical asymptote at or it might behave differently. You have often cancelled factors to reduce fractions. x2 4 x 2 1 But the functions x 2 x 2 21 x 2 2 x 2 x2 4 x 2 x p 1 2 and q x 2 x 1 2 are not the same, because when x 2 22 4 0 2 2 0 2 2 4. 2 x p q 1 1 2 2, which is not defined, but For any ... |
urate Rational Function Graphs Getting an accurate graph of 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 s... |
5 3x 2x 3 2 f x 1 2 Thus, when 0 horizontal line x 0 is large, the graph of f gets very close to the y 3 2 which is called a horizontal asymptote of, the graph. The dashed lines in Figure 4.4-7 indicate the vertical and horizontal asymptotes of the graph. b. The zeros of the denominator of g x 2 1 x x2 4 are Β± 2 and n... |
rational function, the x-axis is the horizontal asymptote of the graph. When the numerator and denominator have the same degree, as in Examples 4a and 4c, the horizontal asymptote is determined by the leading coefficients of the numerator and denominator: Function 3x 6 2x 5 2x3 x x3 1 f x 1 2 h x 1 2 Horizontal asympt... |
f x2 4x 5 close to 0. Therefore, of f approaches the parabola The curve x2 4x 5 y x2 4x 5 is the quotient in the division. x 2 1 and by the Big-Little Concept 6 x 1 x. The graph 0 y x2 4x 5, as shown in Figure 4.4-11. is called a parabolic asymptote. Note that for large values of is very 0 β Section 4.4 Rational Funct... |
function in factored form. Then read off the relevant information. x 1 x2 21 2 1 x 3 and Vertical Asymptotes: x 2 Intercepts: y-intercept: 0 1 02 0 6 1 6 f 0 1 2 x-intercept: x 1 Horizontal Asymptote: y 0 zeros of the denominator but not the numerator zero of numerator but not of denominator degree of numerator is les... |
shown in Figure 4.4-13. y 16 8 0 β8 β8 β4 x 4 8 Figure 4.4-13 β 290 Chapter 4 Polynomial and Rational Functions Exercises 4.4 In Exercises 1β6, find the domain of the function. 1. f x 1 2 3x 2x 5 2. g x 2 1 x3 x 1 2x2 5x 3 In Exercises 23β50, analyze the function algebraically: list its vertical asymptotes, holes, and... |
2 x3 3x2 4x 1 x2 x 23. f x 1 2 1 x 5 25. k x 1 2 3 2x 5 27. f x 1 2 3x x 1 29. f x 1 2 2 x x 3 24. q x 1 2 26 28. p x 1 2 x 2 x 30. g x 1 2 3x 2 x 3 1 x 1 2 32. g x 1 2 x 2x2 5x 3 x 31. f x 1 2 33. f x 1 2 35. h x 1 2 36. f x 1 2 2 1 x 3 x2 x 2 1 3 x2 6x 5 x 5 2 1 x2 1 x3 2x2 x 1 38. k x 1 2 x2 1 x2 1 40. F x 2 1 x2 x... |
3 57. h x 1 2 3x2 x 4 2x2 5x 58. f x 1 2 2x2 1 3x3 2x 1 59. g x 1 2 x 4 2x3 5x2 4x 12 60. h x 1 2 x2 9 x3 2x2 23x 60 In Exercises 61β66, find a viewing window or windows that show(s) a complete graph of the functionβif possible, with no erroneous vertical line segments. Be alert for hidden behavior. 61. f x 1 2 2x2 5x ... |
2 c. Use the different quotient in part a to 1 x determine the average rate of change of 1 x changes from 3 to 3.1, from 3 to 3.01, and from 3 to 3.001. Estimate the instantaneous rate x 3. of change of at as x f f 2 1 2 change of d. How are the estimated instantaneous rates of x 2 1 x2 related to the x 3? and x 2 val... |
the cost of the box be $20? 74. A truck traveling at a constant speed on a reasonably straight, level road burns fuel at the rate of of the truck in miles per hour and gallons per mile, where x is the speed is given by x g x g 1 2 2 1 800 x2 200x. g x 1 2 a. If fuel costs $1.40 per gallon, find the rule of x the cost ... |
ptote of the average cost function. Explain what the asymptote means in this situation, that is, how low can the average cost possibly be? 78. Radioactive waste is stored in a cylindrical tank, whose exterior has radius r and height h as shown in the figure. The sides, top, and bottom of the tank are one foot thick and... |
negative integers makes it possible to find the solution to this equation. By enlarging the number system to include rational numbers, it is possible to solve equations that have no integer solution, such as Similarly, the equation x2 2 x 12 x 12 are real numhas no rational solution, but ber solutions. The idea of enl... |
i 3 4b i β NOTE The mathematicians who invented the complex numbers in the seventeenth century were very uneasy about a number i such that i2 1. Consequently, they called numbers of the form bi, where b is a real number i 21, and numbers. imaginary The existence of imaginary numbers is as real as any of the familiar nu... |
i4 i 1 i i 1 21 2 1 1 Definition of i 296 Chapter 4 Polynomial and Rational Functions The powers of i form a cycle. Any power of i must be one of four values: i, 1, i, in, divide n by 4 and match the remainder to one of the powers listed above. or 1. To find higher powers of i, such as Example 4 Powers of i Find i54. ... |
1 1 2 2 3 6i 4i 8i2 1 4i 2 11 2i 5 11 5 2 5 i β Section 4.5 Complex Numbers 297 Square Roots of Negative Numbers Because i2 1, 21 is defined to be i. Similarly, because 1 2 52i2 25 25, 5i 225 1 is defined to be 5i. In general, 2 1 2 Square Roots of Negative Numbers Let b be a positive real number. 2b is defined to be ... |
, the equation has solutions in the complex number system. is not a real number, this equation has no real number 223 i223. is an imaginary number, namely, 223 x 1 Β± 223 4 Note that the two solutions, 1 4 conjugates. 1 Β± i223 4 223 4 i and Β± 223 1 4 4 223 4 1 4 i, i are complex β NOTE See the Algebra Review Appendix to... |
4. Use the Do this problem in matrix form by computing x key for the exponent. 1 3 e. Do each of the following calculations and interpret the answer in terms of complex numbers. A B 1, C 3 4 3 4 3 1. 3. 5. 7. 8. 9. 11. 13. 15. 29. 31. 33. 35. 300 Chapter 4 Polynomial and Rational Functions Exercises 4.5 In Exercises 1... |
x 66. 3x2 4 5x 68. x3 125 0 x4 81 0 70. i i2 i3 p i15 i i2 i3 i4 i5 p i15 71. Simplify: 72. Simplify: 73. Critical Thinking If is a complex z 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 re... |
2i β6 β4 0 β2 β2i 4 6 2 β i 5 Real β3 β 4i β4i β4i β6i Figure 4.5.A-1 Orbits of Complex Numbers The concepts and processes described in Section 3.5.A apply equally well to functions with complex number inputs. For instance, the process of finding the orbit of a complex number under a complex-valued function is the sam... |
0.25 0.125i 0.203125 0.187500i 0.243896 0.173828i 0.220731 0.165208i.2439 0.1738i 0.2207 0.1652i 0.2286 0.1771i 2 2 2 0.25 0.125i 1 0.203125 0.187500i 2 Use iteration on your calculator to find orbit of 0 is approaching a number near c 0.25 0.25i, can be shown that for. 0 It suggests that the f 15 0.2277 0.1718i. In f... |
) z 2 c Describe the orbit of 0 under ber c. f z 1 2 z2 c for the given complex num- a. c 1 i b. c 0.25 0.625i Solution a. The first seven iterations of f z 2 1 z2 1 i 1 2 are shown in Figure 4.5.A-4. If each of these numbers is plotted in the complex plane, successive iterations get farther and farther from the origin... |
i oscillates. Neither orbit approaches infinity, so orbit of both numbers are in the Mandelbrot set. Example 2 shows that the orbits of approach infinity, so these numbers are not in the Mandelbrot set. c 0.25 0.625i c 1 i and c 0.25 0.25i Diagram of the Mandelbrot Set Although the Mandelbrot set is defined analyticall... |
that ation produces a point more than 3 units from the origin and that 0.25 0.625i is in the yellow region. 1 i The border of the Mandelbrot set is very jagged and chaotic. The varying rates at which the orbits of these border points approach infinity produce some interesting patterns of great complexity. When specifi... |
0.4 0.6i In Exercises 13β18, determine whether or not c is in the Mandelbrot set. 13. c i 15. c 1 14. c i 16. c 0.1 0.3i 17. c 0.2 0.6i 18. 0.1 0.8i Section 4.6 The Fundamental Theorem of Algebra 307 4.6 The Fundamental Theorem of Algebra Objectives β’ Use the Fundamental Theorem of Algebra β’ Find complex conjugate zer... |
omials with complex coefficients. In the rest of this section, βpolynomialβ means βpolynomial with complex, possibly real, coefficientsβ unless specified otherwise. Fundamental Theorem of Algebra Every nonconstant polynomial has a zero in the complex number system. 308 Chapter 4 Polynomial and Rational Functions Althou... |
of x c c1, c2, p, cn polynomial of degree n has exactly n complex zeros. Example 1 Finding a Polynomial Given Its Zeros Find a polynomial a zero of multiplicity 3, and f x f 1 2 24. 2 1 2 of degree 5 such that 1, 2, and 5 are zeros, 1 is Section 4.6 The Fundamental Theorem of Algebra 309 Solution must be a factor of. ... |
) Let number z is a zero of f, then its conjugate be a polynomial with real coefficients. If the complex is also a zero of f. z 310 Chapter 4 Polynomial and Rational Functions Example 3 A Polynomial with Specific Zeros Find a polynomial with real coefficients whose zeros include the numbers 2 and 3 i. Solution 3 i Beca... |
ugate must also be a zero. Thus, some ck is the conjugate of with a and b real numbers. Thus, x cn2 x c22 a bi, a bi f x, 2 1 and say, cj, p ck cj 1 Section 4.6 The Fundamental Theorem of Algebra 311 x cj21 x ck2 1 1 1 x 2 4 x a bi 1 a bi a bi a bi x 2 4 3 3 x2 x 2 x2 ax bix ax bix a2 x2 2ax a2 b2 1 x ck2 x cj21 1 2 1 ... |
Because the degree of the polynomial is 4, there are exactly 4 complex zeros. Find all rational zeros: The possible rational zeros are factors of 8. Β± 1, Β± 2, Β± 4, Β± 8 312 Chapter 4 Polynomial and Rational Functions Graph sible zeros are the rational zeros. x4 5x3 4x2 2x 8 x f 2 1 and determine which of the pos- y 20 ... |
estimating one zero, which yields the real linear factor, using synthetic division to determine the quadratic factor, and then using the quadratic formula to estimate the remaining two zeros. Exercises 4.6 In Exercises 1β6, determine if without using synthetic or long division. g(x) is a factor of 1. 2. 3. 4. 5. 6 x10... |
write linear factors. f(x) 28. degree 3; zeros 2 29. zeros include 2 i and 2 i, 2; f 0 1 1, 1 2 2 314 Chapter 4 Polynomial and Rational Functions 30. zeros include 1 3i and 1 3i 54. x4 6x3 29x2 76x 68; zero 2 of multiplicity 2 31. zeros include 2 and 2 i 32. zeros include 3 and 4i 1 33. zeros include 3, 1 i, 1 2i 34. ... |
be complex z a bi w c di numbers (a, b, c, d are real numbers). Prove the given equality by computing each side and comparing the results. z w z w a. (The left side says: βFirst find z w and then take the conjugate.β The right side says: βFirst take the conjugates of z and w and then add.β) z w z w b. 58. Let 1 2 h x ... |
................. 239 Constant polynomial...................... 240 Zero polynomial.......................... 240 Degree of a polynomial.................... 240 Leading coefficient........................ 240 Polynomial function....................... 240 Polynomial division....................... 240 Synthetic divisio... |
253 Complete factorization..................... 253 Bounds Test............................. 256 Real zeros of polynomials.................. 257 axn 2 1 f x Graph of....................... 260 Continuity.............................. 261 End behavior............................ 263 Intercepts............................. |
............... 280 Big-Little Concept......................... 280 Holes.................................. 283 End behavior............................ 284 Horizontal asymptotes..................... 284 Other asymptotes......................... 285 Complex numbers........................ 294 Imaginary numbers............ |
polynomial and r is a real number that satisfies any of the following statements, then r satisfies all the statements. β’ r is a zero of the polynomial function β’ r is an x-intercept of the graph of f β’ r is a solution, or root, of the equation β’ 0 f is a factor of the polynomial expression β’ There is a one-to-one corr... |
x3 1 22 22 2x2 0 2. What is the remainder when h. 3. What is the remainder when x 1? x 0 x4 3x3 1 is divided by x2 1? x112 2x8 9x5 4x4 x 5 is divided by 4. Is x 1 a factor of f x 1 2 14x87 65x56 51? Justify your answer. 5. Use synthetic division to show that x6 5x5 8x4 x3 17x2 16x 4, x 2 is a factor of and find the ot... |
is a lower bound for the real zeros of x4 4x3 15. In Exercises 22 and 23, find the real zeros of the polynomial. 22. x6 2x5 x4 3x3 x2 x 1 23. x5 3x4 2x3 x2 23x 20 Section 4.3 24. List the zeros of the polynomial and the multiplicity of each zero. 3 3 5 x 4 x 2 x 17 x2 4 1 2 1 21 25. List the zeros of the polynomial an... |
1 x3 2x2 3 x4 3x 2 Section 4.3.A 38. HomeArt makes plastic replicas of famous statues. Their total cost to produce copies of a particular statue is shown in the table on the next page. a. Sketch a scatter plot of the data. b. Use cubic regression to find a function C(x) that models the dataβthat 320 Chapter Review is,... |
46. f x 1 2 x 3 x2 x 2 48. h x 1 2 x4 4 x4 99x2 100 47. g x 1 2 x2 x 6 x3 3x2 3x 1 49. k x 1 2 x3 2x2 4x 8 x 10 50. Which of these statements is true about the graph of x 1 x2 1 x 3 2 x2 1? 2 1 2 f x 1 1 21 21 a. The graph has two vertical asymptotes. x 3. b. The graph touches the x-axis at c. The graph lies above the... |
-world situations require you to find the largest or smallest quantity satisfying certain conditions. For instance, automotive engineers want to design engines with maximum fuel efficiency. Similarly, a cereal manufacturer who needs a box of volume 300 cubic inches might want to know the dimensions of the box that requ... |
each corner. Then, Volume of box Length Width Height > > 1300 0 0 11 Figure 4.C-3 β§βͺβ¨βͺβ© β§βͺβ¨βͺβ© 30 2x > 22 2x x 2 2 660x 1 4xΛ 3 104xΛ x x 2 1 3 104xΛ 2 660x gives the volume y of the box Thus, the equation square from each corner. Because the that results from cutting an shortest side of the cardboard is 22 inches, the... |
Since the can pr 2 h 58, or equivalently, h 58 pr 2. Therefore, surface area 2prh 2pr 2 2pr 58 pr 2b a 2pr 2 116 r 2pr 2. Note that r must be 1 or greater because the diameter 2r must be at least 2. Furthermore, r cannot be more than 5 because if then would be at least the volume which is greater than 58. h 1, r 7 5 a... |
be used to construct it b. the smallest possible amount of material will be used to construct it (how much material is needed?) 6. If x c 1 2 is the cost of producing x units, then c x 1 x 2 is x the average cost per unit. Suppose the cost of producing x units is given by 2 10,000x c 1 than 300 units can be produced p... |
6. 326 Chapter Outline 5.1 Radicals and Rational Exponents 5.2 Exponential Functions 5.3 Applications of Exponential Functions 5.4 Common and Natural Logarithmic Functions 5.5 Properties and Laws of Logarithms 5.5.A Excursion: Logarithmic Functions to Other Bases 5.6 5.7 Solving Exponential and Logarithmic Equations Ex... |
y = xn x y = 0 c 66 0 No solution y y = xn c x y = c Figure 5.1-1 Figure 5.1-2 Figure 5.1-3 Figure 5.1-4 The figures illustrate the following definition of nth roots. nth Roots Let c be a real number and n a positive integer. The nth root of c is denoted by either of the symbols 1 n2c or c n and is defined to be β’ the... |
0913 1 40Λ b. The fraction 1 11 repeating decimal 1 11 fraction is equivalent to the 0.090909 p. The Figure 5.1-5 is not equivalent to this decimal if it is rounded off, as Figure 5.1-6 shown at left. Therefore, it is better to leave the exponent in fractional form. 225 1 11 Λ 1.6362 β Rational Exponents Rational expon... |
Exponents Let c and d be nonnegative real numbers and let r and s be rational numbers. Then 1. 2. crcs crs cr cs crs (c 0) 3. (cr)s c rs 4. 5. 6. (cd)r cr dr r cr d r c db a (d 0) cr 1 c r (c 0) and d 1, If c 1 cr cs cr dr β’ β’ r s. if and only if if and only if c d. Example 3 Simplifying Expressions with Rational Expo... |
the Denominator Rationalize the denominator of each fraction. a. 7 25 Solution b. 2 3 26 a. Multiply the fraction by 1 using a suitable radical fraction. 7 25 7 25 1 7 25 25 25 a b 725 5 a2 b2 to determine b. Use the multiplication pattern a suitable radical fraction equivalent to 1. a b 1 21 2 2 3 26 3 26 3 26 3 26 B... |
real exponents. 334 Chapter 5 Exponential and Logarithmic Functions 5. 20.0081 6. 20.000169 47. 2x7 x 5 2 x 3 2 48. x A 1 2 y3 2 x0 y7 A B Exercises 5.1 Note: Unless directed otherwise, assume all letters represent positive real numbers. In Exercises 1β15, evaluate each expression without using a calculator. 2. 23 64 ... |
b a ab 1 bx 2 x b 1 49. 51. 53. 55. A 1 1 1 1 A 7a 5a 2a 4a 2 2 2 2 In Exercises 57β66, write each expression without radicals, using only positive exponents. 57. 23 a 2 b2 59. 34 24 a3 61. 25 t 216t5 63. 65. 23 xy 2 3 5 B c c51 1 2 2 3 42 5 6 c 2 A 1 58. 24 a 3 b3 60. 323 a 3b4 62. 2x 23 x 2 24 x 3 64. Q 34 r 14 s 3 ... |
of the equation xn c Verify that rs is a xn d. and s is a solution of solution of xn cd. b. Explain why part a shows that c 2n cd 2n 2n d. 91. Write laws 3, 4, and 5 of exponents in radical notation in the case when r 1 m and s 1 n. 92. a. Graph f x5 and explain why this x 2 1 function has an inverse function. b. Show... |
feet long. gives an estimate of the s 230fd 98. Using a viewing window with 0 x 4 and graph the following functions on the 0 y 2, same screen, x, 4 in order of increasing size and justify In each of the following cases, arrange and your answer by using the graphs. a. x 99. Using a viewing window with 3 x 3 and graph t... |
even number as its denominator. Because within any interval there are infinitely many rational numbers that have an even denominator, a 6 0 f. Therefore, the function is not well-behaved for, so it is not defined for those values. has an infinite number of holes in every interval when a 6 0 ax x 2 1 The following two ... |
by graphing. x 2 1 2x 6 a. g x 1 2 2x3 b. h x 1 2 2x3 4 Solution a. If f x 2x, then g x 2x3 f x 3. So the graph of g is the 1 2 2 1 1 graph of f shifted horizontally 3 units to the left, as shown in Figure 5.2-2. x f 2x, x h 2 1 h(x) is the graph of f 1 and vertically 4 units downward, as shown in Figure 5.2-3. shifte... |
from the graph of f to the graph of each function below. Verify by graphing. x 1 2 3x g x 2 1 30.2x h x 1 2 30.8x k x 1 2 3 x p 0.4x 3 x 1 2 Section 5.2 Exponential Functions 339 30.8x Solution The graphs of g x 1 2 izontally by a factor of 30.2x 1 0.2 h and 5 x 2 1 and 30.8x 1 0.8 are the graph of f stretched hor- 1.... |
x, x 1 2 1 2 a. Assuming that the value of your stock continues growing at this rate, how much will your investment be worth in 4 years? b. When will your investment be worth $8000? Solution a. Letting x 4, 5000 In 4 years your stock is worth about $5627.54. 4 5627.54. 1.03 4 f 2 1 2 1 b. Find the value of x for which... |
Pu 1 x 2 1 that remains after x Estimate the 0.99997x. 2 Solution Because M is an exponential function with a base smaller than 1 but very close to 1, its graph falls very slowly from left to right. The fact that the graph falls so slowly as x gets large means that even after an extremely long time, a substantial amoun... |
P 85 corresponds to the year 2065. Therefore, the population will reach half a billion approximately by the year 2065. and y 500 227e0.0093t t t 1 2 2 1 500. β Other Exponential Functions In most real-world applications, populations cannot grow infinitely large. The population growth models shown previously do not tak... |
viewing window with graph each function below on the same screen, and observe their behavior. 10 5 x 5 e0.4x e e2x e e3x e 10 10 and Y3 Y2 0.4x Y1 2x 3x 1 2 1 2 2 1 How does the coefficient of x affect the shape of the graph? Predict the shape of the graph of answer by graphing. y Y1 80. Confirm your Figure 5.2-14 Sec... |
into the graph of the given function. (Section 3.4 may be helpful.) 7. x f 1 2 3x 4 9. x k 1 2 1 4 1 3x 2 11. f 13. g x x 1 1 2 2 32x 4 1 0.15x 3 2 8. x g 1 2 x 3 10. g x 1 2 30.4x 12. x f 1 2 8 5 3x 1 2 In Exercises 14β19, sketch a complete graph of the function. 14. 16. 18. x 4 23x 25x 15. 17. 19. x f x 1 2 5 2b a 3... |
x is 53. According to data from the National Center for Health Statistics, the life expectancy at birth for a person born in year x is approximated by the function below. D x 1 2 79.257 1 9.7135 1024 e 1900 x 2050 1 2 0.0304x a. What is the life expectancy of someone born in 1980? in 2000? b. In what year was life exp... |
other ax g ax x x 2 1 1 1 2 2 59. Critical Thinking For each positive integer n, let fn be the polynomial function below. 1 x x2 2! x3 3! x4 4! fn1 x 2 p xn n! 5 y 55, a. Using the viewing window with ex 4 x 4 f41 x and and graph the same screen. Do the graphs appear to coincide? g x 1 2 2 on 2 2 x f41, then by that o... |
10 years is 10 B 1 2 6000 1.08 2 1 10 $12,953.55. β The pattern illustrated in Example 1 can be generalized as shown below. Compound Interest If P dollars is invested at interest rate r (expressed as a decimal) per time period t, then A is the amount after t periods. A P(1 r)t Notice that in Example 1, or years, is t ... |
. Because the interest is compounded every 1 365 A 6800 and of a year, the interest rate per period is. r 0.07 365 A P 1 6800 5000 t 1 r 2 1 0.07 365 b a t 8,000 The point of intersection of the graphs of 5000 y1 t 1 0.07 365 b a and 6800 y2 be worth is approximately (1603.5, 6800). Therefore, the investment will $6800... |
dollars is invested at an annual interest rate of r, compounded continuously, then A is the amount after t years. A Pert Example 5 Continuous Compounding 5% If you invest much is in the account at the end of 3 years? $4000 at annual interest compounded continuously, how Section 5.3 Applications of Exponential Function... |
, 1000a5 7600. lation grows. and Solve for a to find the factor by which the bacteria popu- 5 1 2 f 7600 1000a5 7600 a5 7.6 a 25 7.6 a 7.6 1 5 7.60.2 7.60.2. 1000 7.60.2x f x Therefore, the functionβs growth factor is Find f 24, 2 1 1 2 the bacteria population after 24 hours. 1000 7.60.2 24 2 16,900,721 24 f 1 1 2 Afte... |
of 1 0.3 0.7. ing in the water was changing by a factor of This pattern is true in general for exponential decay. 30% 0.3 Exponential decay can be described by a function of the form f(x) Pax, x 0 where f(x) is the quantity at time x, P is the initial quantity when changes when x increases by 1. If the quantity decayi... |
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 β 36% f x 1 2 of its carbon 0.36P. 2 1.5 0 0.5 15,000 Figure 5.3-3 Section 5.3 Applic... |
for 4 at 7.2% compounded quarterly for 5 at 5.9% compounded quarterly for 8 In Exercises 19β26, use the compound interest formula. Given three of the quantities, A, P, r, and t, find the remaining one. 19. A typical credit card company charges 18% annual interest, compounded monthly, on the unpaid balance. If your cur... |
or you can receive 6% 28. If money can be invested at 7% interest, compounded quarterly, which is worth more: $9000 in 5 years? now or $12,500 29. If an investment of $1000 grows to $1407.10 in seven years with interest compounded annually, what is the interest rate? 30. If an investment of $2000 $2700 years, with an ... |
0.05 n b c a n 3 d a. Complete the following table: n 1 0.05 n b a n 1000 10,000 500,000 1,000,000 5,000,000 10,000,000 b. Compare the entries in the second column of e0.05, and the table in part a to the number complete the following sentence: As n gets larger and larger, the value of 1 0.05 n b a number gets closer ... |
of Mexico was 100.4 million in 2000 and is expected to grow by approximately 1.4% each year. a. If x g is the population, in millions, of Mexico corresponds to the year x 0 in year x, where 2000, find the rule of the function g. (See Example 6.) 2 1 b. Estimate the population of Mexico in the year 2010. 42. The number... |
-life of radium is 1620 years. Find the $5550 elementary and secondary schools was 1989β1990 and has been increasing at about each year. a. Write the rule of a function that gives the in 3.68% expenditure per pupil in year x, where x 0 rule of the function that gives the amount remaining from an initial quantity of 100... |
, as explained in Section 3.6. Recall that the graphs of inverse functions are reflections of one another across the line The exponential function f and its inverse function are graphed in Figure 5.4-2. y x. 10x x x f 1 2 10x 1 2 y 1 f(x) = 10x x y y = x f(x) 1 x 1 g(x) Figure 5.4-1 Figure 5.4-2 The inverse function of... |
a. If log b. If x 2, 10x 29, Figure 5.4-3. then 102 x. Therefore, x 100. then log 29 x. Therefore, x 1.4624, as shown in β NOTE Logarithms are rounded to four decimal places and an equal sign is used rather than the βapproximately equalβ sign. The word βcommonβ will be omitted except when it is necessary to distinguis... |
186. is undefined for real numbers because there is no exponent of β Section 5.4 Common and Natural Logarithmic Functions 359 In a few cases you can evaluate ln x without a calculator. ln e 1 because e1 e ln 1 0 because e0 1 Example 4 Solving by Using an Equivalent Statement Solve each equation by using an equivalent ... |
numbers greater than 3. is all positive real numbers. g x 2 1 Range of h: log x The range of vertical stretch has no effect on the range. is all real numbers, so the x g 2 1 The graphs of g and h are shown in Figure 5.4-9. The points a 9 1, 0 1, and 2 1, and 2 2 log 10, 1 2 13, 2 2 1 x 3 2 4, 0 1 h x continues to appr... |
needed in order for the investment in part a to double in 6 years? Solution a. The annual interest rate r is 0.065. Find D(0.065). 0.065 D 1 2 ln 2 1 0.065 2 ln 1 11.0067 Therefore, it will take approximately 11 years to double an investment of $2500 6.5% at b. If the investment doubles in 6 years, then 6 annual inter... |
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