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ons as ratios of the sides of a right triangle. 45° 1 2 45° 1 FIGURE 4.28 In Example 1, you were given the lengths of two sides of the right triangle, but not the angle Often, you will be asked to find the trigonometric functions of a given acute angle To do this, construct a right triangle having as one of its angles.... |
an 26 (a) sin tan (c) sec 5, cos (a) cot90 (c) cos 1 3 sec (a) cot (c) tan 5 cot (a) tan90 (c) (b) (d) cos sec90 (b) (d) cot sin (b) (d) sin sin90 (b) (d) cos csc In Exercises 33–42, use trigonometric identities to transform the left side of the equation into the right side 0 < < /2. 33. tan cot 1 cos sec 1 tan cos sin... |
d tangent functions at the four quadrant angles 0, 3 . 2 2 , , and Solution To begin, choose a point on the terminal side of each angle, as shown in Figure 4.38. For each of the four points, and you have the following. cos 0 x r 1 1 1 0 cos x r 2 cos x r 0 1 1 1 1 r 1, tan 0 y x tan y x 2 tan y x cos 3 2 x r 0 1 0 tan ... |
R 92 Synthesis d θ 6 mi Not drawn to scale S 23.1 0.442t 4.3 cos t 6 S is measured in thousands of units and where the time in months, with Predict sales for each of the following months. is representing January 2006. t 1 t (a) February 2006 (b) February 2007 (c) June 2006 (d) June 2007 89. Harmonic Motion The displace... |
ntal translation (shift) of the basic sine and cosine curves. y a sinbx c, Comparing you find that the graph of y a sinbx c By solving for bx c 0 you can find the interval for one cycle to be completes one cycle from bx c 2. y a sin bx with to x, Left endpoint Right endpoint Period This implies that the period of y a s... |
d sets up a wave y 0.001 sin 880t, motion that can be approximated by where is the time (in seconds). t (a) What is the period of the function? (b) The frequency f is given by f 1p. What is the frequency of the note? Month, t Tallahassee, T Chicago, C Model It 1 2 3 4 5 6 7 8 9 10 11 12 63.8 67.4 74.0 80.0 86.5 90.9 92... |
graphs of the two remaining trigonometric functions can be obtained from the graphs of the sine and cosine functions using the reciprocal identities sec x 1 cos x csc x 1 sin x and . y For instance, at a given value of x. the -coordinate of cos Of course, when x, exist. Near such values of of the tangent function. In ... |
r of the function in the same viewing window. Describe the behavior of the function as increases without bound. x gx ex 22 sin x f x 2x4 cos x 65. 67. 66. 68. f x ex cos x hx 2x24 sin x Exploration In Exercises 69–74, use a graphing utility to graph the function. Describe the behavior of the function as approaches zero... |
sed interval y arcsin x 2, 2. is the closed interval 1, 1 y x Now try Exercise 17. 333202_0407.qxd 12/7/05 11:10 AM Page 345 Section 4.7 Inverse Trigonometric Functions 345 Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval shown in Figure 4.73. 0 ≤ x ≤ , as y y = cos... |
750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let be the height of the shuttle. s Section 4.7 Inverse Trigonometric Functions 351 94. Granular Angle of Repose Different types of granular substances naturally settle at different angles when stored is called the angle of in... |
tes, rotates, or is moved by wave motion. For example, consider a ball that is bobbing up and down on the end of a spring, as shown in Figure 4.84. Suppose that 10 centimeters is the maximum distance the ball moves vertically upward or downward from its equilibrium (at rest) position. Suppose further that the time it t... |
ake the trip? (b) How far east and south is the yacht after 12 hours? (c) If a plane leaves Myrtle Beach to fly to Freeport, what bearing should be taken? 33. Surveying A surveyor wants to find the distance across a 32 is N W. C to swamp (see figure). The bearing from A, The surveyor walks 50 meters from the bearing to... |
e other inverse trigonometric functions (p. 345). Evaluate compositions of trigonometric functions (p. 347). 109–114, 123, 126 115–122, 124, 125 127–132 Section 4.8 Solve real-life problems involving right triangles (p. 353). Solve real-life problems involving directional bearings (p. 355). Solve real-life problems inv... |
changed from is changed from 1 5 1 10 (b) (a) (c) to A b k y 0.2 0.1 −0.1 −0.2 t 5π 149. Graphical Reasoning The formulas for the area of a A 1 s r, and is the angle measured circular sector and arc length are r respectively. ( in radians.) is the radius and 2 r2 (a) For 0.8, r. write the area and arc length as functi... |
in radians. (a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Study the pattern in the polynomial approximation of the arctangent function and guess the next term. Then repeat part (a). How does the accuracy of the ap... |
ssion with one of the following. (b) tan x sec2 x (a) csc x (d) sin x x tan (e) 21. 23. 25. sin x sec x sec4 x tan4 x sec2 x 1 sin2 x (c) (f) sin2 x sec2 x tan2 x 22. 24. 26. cos2 xsec2 x 1 cot x sec x cos22 x cos x In Exercises 27–44, use the fundamental identities to simplify the expression. There is more than one co... |
identities sin2 x cos 2 x sin x cos x tan2 x 2 Reciprocal identity Rule of exponents Quotient identity Numerical Solution Use the table feature of a graphing utility set in radian mode to create a table that shows the tan2 x 1cos2 x 1 y1 and values of tan2 x y2 for different values of as shown in Figure 5.2. From the t... |
the interval has solutions x 6 2, has a period of x 6 2n and x 5 6 2n General solution where n is an integer, as shown in Figure 5.3 = sin x FIGURE 5.3 Another way to show that the equation sin x 1 2 solutions is indicated in Figure 5.4. Any angles that are coterminal with 56 will also be solutions of the equation. has... |
Analytic Trigonometry 5.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The equation 2 sin 1 0 has the solutions 7 6 2n and 11 6 2n, which are called ________ solutions. 2. The equation 2 tan2 x 3 tan x 1 0 is a trigonometric equation that is of ________ type. 3. A solution to an equation that does not satisfy the... |
400 Chapter 5 Analytic Trigonometry 5.4 Sum and Difference Formulas What you should learn • Use sum and difference formulas to evaluate trigonometric functions, verify identities, and solve trigonometric equations. Why you should learn it You can use identities to rewrite trigonometric expressions. For instance, in Exe... |
two lines whose equations are y1 m1x b1 and y2 m2 x b2. Assume that both lines have positive slopes. Derive a formula for the angle between the two lines.Then use your formula to find the angle between the given pair of lines. y 6 4 θ y1 = m1x + b1 −2 2 4 x y2 = m2x + b2 93. y x and 94. y x and y 3 x y 1 3 x 95. Conjec... |
uct formula Simplify. 2 sin 4x equal to zero, you can find that the solutions in the By setting the factor 0, 2 interval , x 0, are cos x 0 The equation the solutions are of the form yields no additional solutions, and you can conclude that x n 4 is an integer. You can confirm this graphically by sketching the graph of... |
function. Use a graphing utility to rule out incorrectly rewritten functions. (c) Add a trigonometric term to the function so that it becomes a perfect square trinomial. Rewrite the function as a perfect square trinomial minus the term that you added. Use a graphing utility to rule out incorrectly rewritten functions. ... |
π y1 x 4 −4 In Exercises 117 and 118, use the zero or root feature of a graphing utility to approximate the solutions of the equation. y x 3 4 cos x 117. 118. y 2 1 2 x2 3 sin x 2 333202_050R.qxd 12/5/05 9:07 AM Page 423 5 Chapter Test Chapter Test 423 Take this test as you would take a test in class. When you are fini... |
e same amount of daylight. What are these days called? (c) Which city has the greater variation in the number of daylight hours? Which constant in each model would you use to determine the difference between the greatest and least numbers of hours of daylight? (d) Determine the period of each model. 10. The tide, or de... |
Law of Sines also be used to solve a right triangle? If so, write a short paragraph explaining how to use the Law of Sines to solve each triangle. Is there an easier way to solve these triangles? a. AAS B 50° c = 20 b. ASA B 50° a = 10 C A C A 333202_0601.qxd 12/5/05 10:40 AM Page 436 436 Chapter 6 Additional Topics in... |
uded Angle—SAS Find the remaining angles and side of the triangle in Figure 6.12. C b = 15 cm FIGURE 6.12 a 115° A c = 10 cm B Solution Use the Law of Cosines to find the unknown side a in the figure. a2 b2 c2 2bc cos A a2 152 102 21510 cos 115 a2 451.79 a 21.26 a 21.26 Because the reciprocal form of the Law of Sines t... |
ip between and x (b) Write x . as a function of x. ) yields positive values of . (Select the sign that 47. Geometry The lengths of the sides of a triangular parcel of land are approximately 200 feet, 500 feet, and 600 feet. Approximate the area of the parcel. 48. Geometry A parking lot has the shape of a parallelogram ... |
451 Vector addition and scalar multiplication share many of the properties of ordinary arithmetic. c d w 1. vu , , and Properties of Vector Addition and Scalar Multiplication Let be vectors and let and be scalars. Then the following properties are true. u v v u u 0 u cdu cdu cu v cu cv du cu du 1u u, 0u 0 8. 6. 2. 4. ... |
iven. Write a linear combination of the standard unit vectors i and j. Initial Point 3, 1 0, 2 1, 5 6, 4 43. 44. 45. 46. Terminal Point 4, 5 3, 6 2, 3 0, 1 In Exercises 47–52, find the component form of and sketch the specified vector operations geometrically, w i 2j. where u 2i j and v 47. 48. 49. 50. 51. 52. v 3 2u v... |
dditional Topics in Trigonometry Example 4 Finding the Angle Between Two Vectors Find the angle between u 4, 3 and v 3, 5. v = 〈3, 5〉 Solution cos u v u v 4, 3 3, 5 4, 3 3, 5 u = 〈4, 3〉 27 534 θ This implies that the angle between the two vectors is FIGURE 6.34 arccos 27 534 22.2 x as shown in Figure 6.34. Now try Exer... |
pret the result in the context of the problem. (b) Identify the vector operation used to increase the prices by 5%. 66. Revenue The vector u 3240, 2450 gives the numbers of hamburgers and hot dogs, respectively, sold at a fast-food stand in one month. The vector gives the prices (in dollars) of the food items. v 1.75, ... |
try Powers of Complex Numbers The trigonometric form of a complex number is used to raise a complex number to a power. To accomplish this, consider repeated use of the multiplication rule. z rcos i sin z 2 rcos i sin rcos i sin r 2cos 2 i sin 2 z3 r 2cos 2 i sin 2rcos i sin r 3cos 3 i sin 3 z4 r 4cos 4 i sin 4 z5 r5cos... |
umber, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. 89. Square roots of 90. Square roots of 91. Cube roots of 5cos 120 i sin 120 16cos 60 i sin 60 8cos i sin 2 3 5 6 2 3 5 6 92. Fifth roots of 93. Square roots of 94. Fourth roots of 95. Cube roots of 96. Cube roots of i... |
the vectors. u cos i sin v cos i sin cos 45i sin 45j v cos 300i sin 300j u 22, 4, u 3, 3, v 4, 33 v 2, 1 81. 82. 83. 84. In Exercises 85–88, determine whether orthogonal, parallel, or neither. u and v are 85. 87. u 3, 8 v 8, 3 u i v i 2j 86. 88. u 1 4, 1 2 v 2, 4 u 2i j v 3i 6j In Exercises 89–92, find the projection ... |
per hour, with a The wind at the altitude of the plane has a velocity of 50 kilometers E. What is the true direction of the plane, and what bearing of 60 per hour with a bearing of N is its speed relative to the ground? 30. 45. A force of 85 pounds exerted at an angle of above the horizontal is required to slide an obj... |
700.qxd 12/5/05 9:38 AM Page 495 Systems of Equations and Inequalities 7.1 7.2 Linear and Nonlinear Systems of Equations Two-Variable Linear Systems 7.3 Multivariable Linear Systems 7.4 7.5 7.6 Partial Fractions Systems of Inequalities Linear Programming 77 Systems of equations can be used to determine the combinations... |
he intersect feature. Use this feature to find the points of intersection of the graphs in Figures 7.1 to 7.3. Be sure to adjust your viewing window so that you see all the points of intersection. Graphical Approach to Finding Solutions From Examples 2, 3, and 4, you can see that a system of two equations in two unknow... |
Each item can be produced for $2.16. (a) How many items must be sold to break even? (b) How many items must be sold to make a profit of $8500? 65. DVD Rentals The weekly rentals for a newly released DVD of an animated film at a local video store decreased each week. At the same time, the weekly rentals for a newly rel... |
imination to solve a system of two linear equations x in and perform the following steps. y, 1. Obtain coefficients for y terms of one or both equations by suitably chosen constants. (or ) that differ only in sign by multiplying all x 2. Add the equations to eliminate one variable, and solve the resulting equation. 3. ... |
. Does the problem involve more than one unknown quantity? 2. Are there two (or more) equations or conditions to be satisfied? If one or both of these situations occur, the appropriate mathematical model for the problem may be a system of linear equations. Example 8 An Application of a Linear System An airplane flying ... |
ckets was further reduced to $18.95. After the last jacket was sold, total receipts for the clearance sale were $5108.30. How many jackets were sold before noon and how many were sold after noon? the price of Fitting a Line to Data squares regression line In Exercises 57–62, find the least y ax b for the points x1, y1 ... |
the other -terms from the first column. x x x 3y x 2y 3z 9 4 y 3z 5 x 2y 3z 9 y 3z 5 2x 5y 5z 17 2x 4y 6z 18 2x 5y 5z 17 y z 1 x 2y 3z 9 y 3z 5 y z 1 Write Equation 1. Write Equation 2. Add Equation 1 to Equation 2. Adding the first equation to the second equation produces a new second equation. Multiply Equation 1 by ... |
nt invested in municipal bonds. The total interest earned during the first year was $1120. How much was invested in each type of fund? Solution x, y, Let represent the amounts invested in the money-market fund, municipal bonds, and mutual funds, respectively. From the given information, you can write the following equa... |
sed to play 32 songs within two hours. You are to choose the songs from the latest rock, dance, and pop albums. You want to play twice as many rock songs as pop songs and four more pop songs than dance songs. How many of each type of song will you play? 63. Acid Mixture A chemist needs 10 liters of a 25% acid solution.... |
te that the techniques vary slightly, depending on the type of factors of the denominator: linear or quadratic, distinct or repeated. Example 1 Distinct Linear Factors Write the partial fraction decomposition of x 7 x 2 x 6 . Solution The expression is proper, so be sure to factor the denominator. Because x 2 x 6 x 3x ... |
4x 2 1 2xx 12 x 3 x 22x 22 x 3 x 3 x 2 x 2 Graphical Analysis In Exercises 53–56, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the... |
to be 2, 3. the shaded region in Figure 7.25. 1, 0 Now try Exercise 37. 333202_0705.qxd 12/5/05 9:45 AM Page 545 Section 7.5 Systems of Inequalities 545 When solving a system of inequalities, you should be aware that the system might have no solution or it might be represented by an unbounded region in the plane. Thes... |
er account. Find and graph a system of inequalities to describe the various amounts that can be deposited in each account. 333202_0705.qxd 12/5/05 9:45 AM Page 550 550 Chapter 7 Systems of Equations and Inequalities 72. Ticket Sales For a concert event, there are $30 reserved seat tickets and $20 general admission tick... |
equation represents a where family of lines, each of slope Of these infinitely many lines, you want the one that has the largest -value while still intersecting the region determined by the constraints. In other words, of all the lines whose slope is you want the one that has the largest -intercept and intersects the ... |
ion: z 2x y Constraints: (See Exercise 6.) 10. Objective function: z 25x 35y Constraints: x ≥ y ≥ 0 8x 9y ≤ 7200 8x 9y ≥ 3600 0 y 60 40 20 (0, 45) (30, 45) (60, 20) (0, 0) (60, 0) 20 40 60 x y 800 400 (0, 800) (0, 400) (900, 0) x 400 (450, 0) 11. Objective function: z 25x 30y Constraints: (See Exercise 9.) 12. Objectiv... |
.2 Use the method of elimination to solve systems of linear equations in two variables (p.507). Interpret graphically the numbers of solutions of systems of linear equations in two variables (p. 510). Use systems of linear equations in two variables to model and solve real-life problems (p. 513). Section 7.3 Use back-s... |
processes, for which the required times per unit are shown in the table. Process Process Process II III I Hours for walking shoe Hours for running shoe Hours available per day 4 2 24 1 2 9 1 1 8 What is the optimal production level for each type of shoe? What is the optimal profit? 85. Optimal Cost A pet supply compan... |
o a, 1, 2, 3 c and so that the lin- b, as its only solution? ear system shown has x 2y 3z a x y z b 2x 3y 2z c Equation 1 Equation 2 Equation 3 13. The following system has one solution: x 1, y 1, and z 2. 4x 2y 5z 16 x y 0 x 3y 2z 6 Solve the system given by (a) Equation 1 and Equation 2, (b) Equation 1 and Equation 3... |
w operation, the new row-equivalent matrix that is displayed on your graphing utility is stored in the answer variable. You should use the answer variable and not the original matrix for subsequent row operations. 333202_0801.qxd 12/5/05 10:59 AM Page 575 Section 8.1 Matrices and Systems of Equations 575 In Example 3 i... |
y substituting values for to obtain a few solutions. Then check each solution in the original equation. Now try Exercise 65. It is worth noting that the row-echelon form of a matrix is not unique. That is, two different sequences of elementary row operations may yield different row-echelon forms. This is demonstrated i... |
le shows the numbers of people (in millions) in the United States who participated in snowboarding for selected years from 1997 to 2001. (Source: National Sporting Goods Association) y Year Number, y 1997 1999 2001 2.8 3.3 5.3 (a) Use a system of equations to find the equation of y at2 bt c that passes through the para... |
t is, A similar property holds. That is, if is an matrix consisting entirely of zeros, then O matrices. For example, the following matrices are the additive identities for the set of all 2 3 is the additive identity for the set of all In other words, c. is the matrix and m n m n 2 2 and O 0 0 matrices. 0 0 0 0 and O 0 ... |
6. 2X 2A B 2A 4B 2X In Exercises 27–34, if possible, find the result. AB and state the order of 27. 28. 29. 30. 31. 32 17 , 13 10 12 , 33. 34 11 16 0 4 4 0 B 6 2 1 6 In Exercises 35– 40, use the matrix capabilities of a graphing utility to find if possible. AB, 35. 36. 37. 38. 39. 40. 2 10 A 5 A 11 14 6 6 5 5 12 10 2 3... |
er the real x, To solve this equation for multiply each side of the number equation equation by ax b. (provided that a 0 ). a1 ax b a1ax a1b 1x a1b x a1b a1 The number definition of the multiplicative inverse of a matrix is similar. is called the multiplicative inverse of a because a1a 1. The A matrix and let be the De... |
). 2 5 7 8 3 2 6 4 12 3 2 5 7 1 12 5 44. 40. 43. 42. 41 In Exercises 45– 48, use the inverse matrix found in Exercise 13 to solve the system of linear equations. 45. 47. x 2y 5 2x 3y 10 x 2y 4 2x 3y 2 46. 48. x 2y 0 2x 3y 3 x 2y 2x 3y 1 2 In Exercises 49 and 50, use the inverse matrix found in Exercise 21 to solve the ... |
There you found the cofactors of the entries in the first row to be 4. 1, 5, and C13 C11 C12 So, by the definition of a determinant, you have a12C12 A a11C11 a13C13 First-row expansion 01 25 14 14. Now try Exercise 37. In Example 3, the determinant was found by expanding by the cofactors in the first row. You could ha... |
2 Now try Exercise 1. 28 14 2 14 14 1 So, the solution is Check this in the original system. 333202_0805.qxd 12/5/05 11:05 AM Page 621 Section 8.5 Applications of Matrices and Determinants 621 Example 2 Using Cramer’s Rule for a 3 3 System Use Cramer’s Rule to solve the system of linear equations. x 2x 3x 2y 4y 3z 1 z ... |
2x 2y 2z x 3y 4z x 2y z 2x 3y 5z 4 3x 5y 9z 7 5x 9y 17z 13 In Exercises 15–24, use a determinant and the given vertices of a triangle to find the area of the triangle. 15. y 16. y 5 4 3 2 1 (1, 5) (0, 0) (3, 14, 5) (0, 0) x −1 −2 4 1 (5, −2) −4 −2 2 4 x (−2, −3) (2, −3) (−2, 1) (3, −1) 2 −2 x 4 19. y 20. y (6, 10) 8 4 ... |
the matrix capabilities of a graphing utility to find the product. 4 11 12 2 4 61. 62 10 2 1 5 3 1 2 2 63. Manufacturing A tire corporation has three factories, each of which manufactures two products. The number of units of product in one day is aij represented by A 80 40 i in the matrix produced at factory 140 80 120... |
(The other 45,000 households do not subscribe.) The percent changes in cable subscriptions each year are shown in the matrix below. Percent Changes From Gold 0.70 0.20 0.10 From From NonGalaxy subscriber 0.15 0.80 0.05 0.15 0.15 0.70 Percent Changes To Gold To Galaxy To Nonsubscriber (a) Find the number of subscribers ... |
n. 0! 0! 1 1! 1 2! 1 2 2 3! 1 2 3 6 4! 1 2 3 4 24 5! 1 2 3 4 5 120 The value of does not have to be very large before the value of extremely large. For instance, 10! 3,628,800. n n! becomes 333202_0901.qxd 12/5/05 11:28 AM Page 645 Section 9.1 Sequences and Series 645 Factorials follow the same conventions for order of... |
the sum. 86. 10 j1 3 j 1 85. 87. 88. 6 j1 24 3j 4 k0 4 k0 1k k 1 1k k! In Exercises 89–98, use sigma notation to write the sum. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. . . . 1 39 . . . 22 1 32 5 1 33 5 1 31 27 81 243 729 1 1 2 1 1 1 12 32 22 1 2 4 . . . 1 128 1 8 1 4 6 . . . 1 202 . . . 1 2 . . . 1 6 10 12 1 15 3 2 8 6... |
ces and Partial Sums 655 Example 3 Writing the Terms of an Arithmetic Sequence The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first 11 terms of this sequence. Solution You know that d 13th terms of the sequence are related by 65. 20 and a13 a4 So, you must add the common difference ... |
0. 0.5, 0.9, 1.3, 1.7, . . . , n 12 n 10 61. 40, 37, 34, 31, . . . , 63. 62. 75, 70, 65, 60, 100, a25 15, a100 a1 a1 64. . . . , 220, 307, n 10 n 25 n 25 n 100 65. Find the sum of the first 100 positive odd integers. 66. Find the sum of the integers from 10 to 50. In Exercises 67–74, find the partial sum. 67. 69. 50 n1... |
this sequence, the Begin with n 1. 36 12 3 c. The sequence whose n th term is n 1 is geometric. For this sequence, the 3 1 3. common ratio of consecutive terms is 1 81 , . . . , 1 3 , 1 27 1 3 Begin with , . . . n 1. n 1 9 , , 19 13 1 3 Now try Exercise 1. The sequence 1, 4, 9, 16, . . . , whose th term is n n2, is not... |
10 2, n 10 4, r 1 6, r 1 3, n 12 100, r ex, n 9 1, r 3, n 8 500, r 1.02, n 40 1000, r 1.005, n 60 a1 a1 a1 a1 a1 a1 29. 31. 32. 33. 34. 35. 9th term: 7, 21, 63, . . . 36. 7th term: 3, 36, 432, . . . 37. 10th term: 5, 30, 180, . . . 40. 1st term: 39. 3rd term: 38. 22nd term: 4, 8, 16, . . . 27 a1 4 a2 a4 a3 16, 3, a5 18... |
you will use finite differences to find a model that represents the number of individual income tax returns filed in the United States from 1998 to 2003. Introduction In this section, you will study a form of mathematical proof called mathematical induction. It is important that you see clearly the logical need for it,... |
12/5/05 11:35 AM Page 679 Section 9.4 Mathematical Induction 679 Sums of Powers of Integers The formula in Example 3 is one of a collection of useful summation formulas. n This and other formulas dealing with the sums of various powers of the first positive integers are as follows. Sums of Powers of Integers 1. 2. 3. 4... |
ial terms that divide out from the expression. 2 factors 3 factors 8C2 8 7 2 1 and 10 3 10 9 8 3 2 1 2 factors 3 factors Example 2 Finding Binomial Coefficients Find each binomial coefficient. b. 7 4 c. 12C1 d. 12 11 a. 7C3 Solution a. b. c. d. 7 6 5 35 7C3 12 12 1 12! 1! 11! 12C1 12 11 35 12 11! 1! 11! 12 1 12 Now try... |
ng utility to find a cubic model for the data. Let represent the year, with corresponding to 1990. t 0 t (b) Use a graphing utility to plot the data and the model in the same viewing window. (c) You want to adjust the model so that corresponds to 2000 rather than 1990. To do this, you 10 units to the left to obtain shi... |
collection of n Te c h n o l o g y Most graphing calculators are programmed to evaluate Consult the user’s guide for your calculator and then evaluate 8P5. You should get an answer of 6720. nPr. Permutations of n Elements Taken r at a Time The number of permutations of elements taken at a time is n r nPr n! n r! nn 1n... |
no seating restrictions? (b) the two members of each couple wish to sit together? 22. Single File In how many orders can four girls and four boys walk through a doorway single file if (a) there are no restrictions? (b) the girls walk through before the boys? In Exercises 23–28, evaluate n Pr . 23. 25. 27. 4P4 8P3 5P4 2... |
ences, Series, and Probability Exploration Toss two coins 100 times and write down the number of heads that occur on each toss (0, 1, or 2). How many times did two heads occur? How many times would you expect two heads to occur if you did the experiment 1000 times? Increasing likelihood of occurrence 0.0 0.5 1.0 Imposs... |
ment of the probability that at least two people have the same birthday is the probability that all 23 birthdays are different. So, first find the probability that all 23 people have different birthdays and then find the complement. Now, determine the proba- bility that in a room with 50 people at least two people have... |
ent on to graduate school. An alumni member is selected at random. What are the probabilities that the person is (a) female, (b) male, and (c) female and did not attend graduate school? 38. Education In a high school graduating class of 202 students, 95 are on the honor roll. Of these, 71 are going on to college, and o... |
ompound Interest A deposit of $10,000 is made in an account that earns 8% interest compounded monthly. The n balance in the account after months is given by n , n 1, 2, 3, . . . 10,0001 0.08 12 An (a) Write the first 10 terms of this sequence. (b) Find the balance in this account after 10 years by find- ing the 120th t... |
sider an idealized population with the characteristic that each member of the population produces one offspring at the end of every time period. If each member has a life span of three time periods and the population begins with 10 newborn members, then the following table shows the population during the first five tim... |
n outline of the proof is presented. n 1, x y1 x1 y1 and the formula is you have 1. If 1C0x 1C1y, valid. 2. Assuming that the formula is true for kCr k! k r!r! the coefficient of n k, kk 1k 2 . . . k r 1 r! . xkryr is To show that the formula is true for x k1ryr in the expansion of x yk1 x ykx y. n k 1, look at the coe... |
-axis to the line. (See Figure 10.1.) x ) y y y y = 0θ AP/Wide World Photos =θ π 2 x x θ x θ x Horizontal Line FIGURE 10.1 Vertical Line Acute Angle Obtuse Angle The inclination of a line is related to its slope in the following manner. Inclination and Slope If a nonvertical line has inclination and slope m, then m tan... |
tween the point and the line. (d) Is it possible for the distance to be 0? If so, what is the slope of the line that yields a distance of 0? (e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem. Skills Review In Exercises 61– 66, find all -intercepts and -intercepts of... |
1 4 b 5 4 b 1. So, the slope of the tangent line is m 1 1 1 0 2 and the equation of the tangent line in slope-intercept form is y 2x 1. Now try Exercise 55. W RITING ABOUT MATHEMATICS Television Antenna Dishes Cross sections of television antenna dishes are parabolic in shape. Use the figure shown to write a paragraph... |
Use a graphing utility to graph the function given by hx) 2x4 x3 19x 2 9x 9. Use the graph to approximate the zeros of h. In Exercises 83–90, use the information to solve the triangle. Round your answers to two decimal places. 85. 84. 83. 86. A 35, a 10, b 7 B 54, b 18, c 11 A 40, B 51, c 3 B 26, C 104, a 19 a 7, b 10... |
d use a graphing e 0.25, e and Discuss the changes in the shape of the ellipse as approaches 0. c. Make a conjecture about the shape of the graph in part (b) when e 0. What is the equation of this ellipse? What is another name for an ellipse with an eccentricity of 0? 333202_1003.qxd 12/8/05 9:01 AM Page 750 750 Chapte... |
equation of a hyperbola is similar to that of an ellipse. Note in the definition below that and are related differently for hyperbolas than for ellipses. b, a, c Standard Equation of a Hyperbola The standard form of the equation of a hyperbola with center h, k is x h2 a 2 y k 2 a2 y k2 b2 x h2 b 2 1 1. Transverse axis... |
the blanks. 1. A ________ is the set of all points x, y in a plane, the difference of whose distances from two distinct fixed points, called ________, is a positive constant. 2. The graph of a hyperbola has two disconnected parts called ________. 3. The line segment connecting the vertices of a hyperbola is called the... |
cos were developed to eliminate the xy -term in the rotated system. You can use this as a check on your work. In other words, if your final equation contains an xy -term, you know that you made a mistake. (x ′)2 2 ( 2( − (y ′)2 2 ( 2( = 1 y x ′ x 1 2 xy − 1 = 0 y ′ −2 −1 2 1 −1 -system: Vertices: xy In In xy-system: FI... |
ation x2 xy ky2 6x 10 0 where k is any constant less than 1 4, is a hyperbola. 60. After a rotation of axes is used to eliminate the xy -term from an equation of the form Ax2 Bxy Cy2 Dx Ey F 0 the coefficients of the respectively. x2 - and y2 -terms remain A and C, In Exercises 45–58, find any points of intersection of... |
of parametric equations to represent the graph of the following parameters. t 1 x t x b. a. y 1 x 2, using Solution a. Letting x t t x, you obtain the parametric equations and t 1 x, y 1 x 2 1 t 2. you obtain the parametric equations b. Letting x 1 t and y 1 x2 1 1 t2 2t t 2. In Figure 10.55, note how the resulting cu... |
83, you are asked to find multiple representations of polar coordinates. Introduction on the rectangular coordinate system, where and So far, you have been representing graphs of equations as collections of points x, y y represent the directed In this section, you will distances from the coordinate axes to the point st... |
inate system and find the distance between them. Then choose different polar representations of the same two points and apply the Distance Formula again. Discuss the result. 76. Exploration (a) Set the window format of your graphing utility on rectangular coordinates and locate the cursor at any position off the coordi... |
ou need to consider only the first of these two intervals. By finding a few additional points (see table below), you can obtain the graph shown in Figure 10.76. r ±3sin 2 0 0 12 ±3 2 4 ±3 5 12 ±3 2 2 0 Now try Exercise 39. 333202_1008.qxd 12/8/05 9:08 AM Page 791 10.8 Exercises Section 10.8 Graphs of Polar Equations 79... |
etermine that the asymptotes of the y 10 ± 3 4 x. The graph is shown in Figure 10.79. π 2 ( −16, )3 π 2 π ( ) 4, 2 r = 32 3 + 5 sin θ FIGURE 10.79 Te c h n o l o g y Use a graphing utility set in polar mode to verify the four orientations shown at the right. Remember that e must be positive, but p can be positive or ne... |
s of cos 2u, using the double-angle formulas. tan 2u and sin 2u , 83. sin u 4 5 , 2 < u < 84. tan u 3, 3 2 < u < 2 In Exercises 85–88, find a formula for sequence. an for the arithmetic 85. 87. a1 a3 0, d 1 4 27, a8 72 86. 88. a1 a1 13, d 3 5, a4 9.5 In Exercises 89–92, evaluate the expression. Do not use a calculator.... |
ses 103–106, identify the conic and sketch 10.9 its graph. 103. r 104. r 105. r 106. r 1 1 2 sin 2 1 sin 4 5 3 cos 16 4 5 cos In Exercises 73–76, a point in rectangular coordinates is given. Convert the point to polar coordinates. In Exercises 107–110, find a polar equation of the conic with its focus at the pole. 73. ... |
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