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of the lines. Two lines may intersect in a single point, they may be parallel, or they may coincide, as shown in Figure 1. So there are three possible outcomes in solving such a system. NUMBER OF SOLUTIONS OF A LINEAR SYSTEM IN TWO VARIABLES For a system of linear equations in two variables, exactly one of the followi... |
4. 26. 28. 30. 32. 34. 36. b 4x 12y 0 12x 4y 160 0.2x 0.2y 1.8 0.3x 0.5y 3.3 4x 2y 16 x 5y 70 b 3x 5y 2 9x 15y 6 b 2x 3y 8 14x 21y 3 b 25x 75y 100 10x 30y 40 b u 30√ 5 3u 80√ 5 b 2 x 1 3 y 3 2x 1 2 y 1 b 26x 10y 4 0.6x 1.2y 3 b 1 2 y 4 2x 10y 80 b 10 x 1 1 2 2 b 37–40 ■ Use a graphing device to graph both lines in the ... |
z 1 3x 2y z 13 3x 2y 5z 3 Equation 1 Equation 2 Equation 3 ▼ SO LUTI O N We need to change this to a triangular system, so we begin by eliminating the x-term from the second equation. c x 2y 0z 13 x 2y 3z 1 4y 4z 12 Equation 2 Equation 1 Equation 2 + (–1) Equation 1 = new Equation 2 This gives us a new, equivalent syst... |
5 c y 4z 0 2z 1 8. x y 3z 8 y 3z 5 z 1 c 10. x 2y 3z 10 2y z 2 3z 12 c 12. 4x 3z 10 2y 3z 6 2z 4 1 c 13–16 ■ Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. c 13. x 2y z 4 x y 3z 0 2x y z 0 Eliminate the x-term from the second equation. c 15. 2x y 3z 2 x... |
he denominator Q of r, we can express of partial fractions of the form x r 1 2 as a sum A ax b 1 i 2 and Ax B ax2 bx c 1 j 2 This sum is called the partial fraction decomposition of r. Let’s examine the details of the four possible cases. CASE 1: THE DENOMINATOR IS A PRODUCT OF DISTINCT LINEAR FACTORS x Suppose that we... |
ality. We already know that the graph 2 , for example, is the parabola in Figure 1. If we replace the equal sign by the symof bol , we obtain the inequality y x FIGURE 1 y x 2 SE CTI O N 6.5 | Systems of Inequalities 475 Its graph consists of not just the parabola in Figure 1, but also every point whose y-coordinate is... |
on, this production plan is not legal. It violates the CO restriction, although it does not violate the SO2 restriction (see Figure 9). 60, 160 1 2 (b) Since the point (see Figure 9). (c) Since the point ✎ Practice what you’ve learned: Do Exercise 51. ▲ 6. ▼ CONCE PTS 1. To graph an inequality, we first graph the corres... |
tial fraction decomposition of a rational expression in the above cases Section 6.5 ■ Graph the solution of an inequality ■ Graph the solution of a system of inequalities ■ Graph the solution of a system of linear inequalities 31–32 33–34 35–36 37–38 31–38 Review Exercises 39–44 45–46, 49–50 47–48, 51–52 ▼ E X E RC I S... |
is means that we need to check the profit only at the vertices. The largest value of P occurs at the point , where P $560. Thus, the manufacturer should make 16 pairs of oxfords and 16, 16 1 2 16 pairs of loafers, for a maximum daily profit of $560. Vertex 0, 0 2 1 0, 24 1 2 16, 16 1 32, 0 1 2 2 P 15x 20y 0 15 15 15 0 1 ... |
cipal bonds paying 7% interest per year, bank investment certificates paying 8%, and high-risk bonds paying 12%. For tax reasons she wants the amount invested in municipal bonds to be at least three times the amount invested in bank certificates. To keep her level of risk manageable, she will invest no more than $2000 in... |
nted Matrix. Write the augmented matrix of the system. 2. Row-Echelon Form. Use elementary row operations to change the augmented matrix to row-echelon form. 3. Back-Substitution. Write the new system of equations that corresponds to the row-echelon form of the augmented matrix and solve by back-substitution. E X AM P ... |
even thousands of variables, occur frequently in the applications of algebra to the sciences and to other fields. For now, let’s consider an example that involves only three variables. E X AM P L E 8 | Nutritional Analysis Using a System of Linear Equations A nutritionist is performing an experiment on student volunteer... |
equal if they have the same entries in the same positions. EQUALITY OF MATRICES The matrices A ”aij’ and B ”bij’ are equal if and only if they have the same dimension m n, and corresponding entries are equal, that is, for i 1, 2, . . . , m and j 1, 2, . . . , n. aij bij Equal matrices 24 0.5 c 22 1 e0 Unequal matrices... |
late the product AB. (b) How many males are registered as Democrats in this city? (c) How many females are registered as Republicans? 514 CHAPTER 7 | Matrices and Determinants ▼ SO LUTI O N (a) AB 0.30 0.50 0.20 0.60 0.35 0.05 0.50 0.25 0.25 5,000 10,000 12,000 6,000 12,000 15,000 13,500 9,000 4,500 16,500 10,950 5,550... |
fined? What must be true about the dimensions of the matrices A and B if both products AB and BA are defined? 51. Powers of a Matrix Let A 1 0 c 1 1 d Calculate A2, A3, A4, . . . until you detect a pattern. Write a general formula for An. A3, A4, . . . until you detect a pattern. Write a general formula for An. 53. Squar... |
you’ve learned: Do Exercise 17A]-1 Frac [[ -3 2 0 ] [ -4 1 -2/3] [1 0 1/3 ]] Graphing calculators are also able to calculate matrix inverses. On the TI-82 and TI-83 calculators, matrices are stored in memory using names such as [A], [B], [C], . . . . To find the inverse of [A], we key in FIGURE 1 [A] 1 X ENTER For the ... |
1 T 2 43–46 ■ Find the inverse of the matrix. For what value(s) of x, if any, does the matrix have no inverse? 43. 2 x c x x2 d 45. ex 1 ex e2x 0 0 0 0 2 44. ex e2x e3x e2x d c x 46 ▼ APPLICATIONS 47. Nutrition A nutritionist is studying the effects of the ✎ nutrients folic acid, choline, and inositol. He has three ty... |
erminant gives us 3 det A 1 2 4 0 25 0 3 2 4 5 11 2 1 Expand this by column 1 4 4 3 1 1 25 2 3 2 4 2 1 25 600 Since the determinant of A is not zero, A does have an inverse. ✎ Practice what you’ve learned: Do Exercise 27 Emmy Noether (1882–1935) was one of the foremost mathematicians of the early 20th century. Her grou... |
oints and Determinants (a) If three points lie on a line, what is the area of the “triangle” that they determine? Use the answer to this question, together with the determinant formula for the a1, b12 area of a triangle, to explain why the points , 1 a2, b22 are collinear if and only if 1 , and a3, b32 1 a1 a2 a3 b1 b2... |
al. c d 21 22. A C c 225 0 1 S 21 d B 5 log 1 c e0 1 2 d 23–34 ■ Let Carry out the indicated operation, or explain why it cannot be performed. A B C D 2C 3D 23. 25. 24. 26. 5B 2C 29. BC 32. FC 27. GA 30. CB 28. AG 31. BF 33. 1 C D E 2 34. F 1 2C D 2 35–36 ■ Verify that the matrices A and B are inverses of each other by... |
a single point (see page 558). Many such parabolic mirrors, assembled into “solar farms,” can concentrate sunlight from a large area. The concentrated sunlight can heat water or other liquids to thousands of degrees, driving steam turbines for generating electricity. Two such solar power plants have been built in the M... |
ations. Focus p Equation x2 4py Form of the equation for graphing calculator F1A F2A F31 F41 0, 1 8B 0, 1 2B 0, 1 2 0 x2 1 2 y x 2 2y x 2 4y x 2 16y y 2x 2 y 0.5x 2 y 0.25x 2 y 0.0625x 2 Archimedes (287–212 b.c.) was the greatest mathematician of the ancient world. He was born in Syracuse, a Greek colony on Sicily, a g... |
the shape of the parabola? 5. Try rolling the ball up the ramp from the bottom to the spot you marked in Step 1, so that it rolls up and then back down again. If you perform the experiment this way instead of just letting the ball roll down, how does your graph in Step 3 change? 562 SECTION 8.2 | Ellipses 563 8.2 Elli... |
s. High-intensity sound waves generated at the other focus are reflected to the stone and destroy it with minimal damage to surrounding tissue. The patient is spared the trauma of surgery and recovers within days instead of weeks. The reflection property of ellipses is also used in the construction of whispering gallerie... |
f the hyperbola. This leads to a hyperbola with a vertical transverse axis. ■ Equations and Graphs of Hyperbolas The main properties of hyperbolas are listed in the following box. HYPERBOLA WITH CENTER AT THE ORIGIN The graph of each of the following equations is a hyperbola with center at the origin and having the giv... |
¤(0, _13) _12 y 2 (4, 4) 2œ∑3 x 27–30 ■ Use a graphing device to graph the hyperbola. 3y2 4x2 24 28. 27. x2 2y2 8 2 2 y 2 x 6 1 29. 30. 2 x 100 2 y 64 1 31–42 ■ Find an equation for the hyperbola that satisfies the given conditions. ✎ 31. Foci 32. Foci 33. Foci 34. Foci 5, 0 2 0, 10 0, 2 6, 0 2 1 1 1 1 , vertices 1 , ve... |
a). y 0 (4, 2) F⁄(_1, _1) (0, _1) (8, _1) x F¤(9, _1) _5 5 _7 y = –1 + 0.75 x2 – 8x œ∑∑∑∑∑∑∑ 13 y = –1 – 0.75 x2 – 8x œ∑∑∑∑∑∑∑ 3 y=_ x+2 4 (b) (4, _1) (4, _4) (a) 3 y= x-4 4 FIGURE 6 9x2 72x 16y2 32y 16 SECT IO N 8.4 | Shifted Conics 585 (c) To draw the graph using a graphing calculator, we need to solve for y. The giv... |
TER 8 | Review 589 y 1 0 Eccentricity of an Ellipse (p. 567) The eccentricity of an ellipse with equation x2 a2 (where a b 0) is the number x2 b2 y2 a2 1 x y2 b2 1 or e c a c 2a2 b2 . The eccentricity e of any ellipse is a where number between 0 and 1. If e is close to 0, then the ellipse is nearly circular; the closer... |
conic sections in their design. Roman Amphitheater in Alexandria, Egypt (circle) © Nick Wheeler/CORBIS Ceiling of Statuary Hall in the U.S. Capitol (ellipse) Courtesy of The Architect of the Capitol Roof of the Skydome in Toronto, Canada (parabola) © Stone/Getty Images Roof of Washington Dulles Airport (hyperbola and p... |
called the first term, a2 is the second term, and in general an is the nth term. Since for every natural number n there is a corresponding number an, we can define a sequence as a function. DEFINITION OF A SEQUENCE A sequence is a function f whose domain is the set of natural numbers. The values f are called the terms of... |
calculus we are often interested in adding the terms of a sequence. This leads to the following definition. THE PARTIAL SUMS OF A SEQUENCE For the sequence the partial sums are a1, a2, a3, a4, . . . , an, . . . S1 S2 S3 S4 Sn a1 a1 a1 a1 . . . a1 . . . a2 a2 a2 a3 a3 a4 a2 a3 . . . an S1 is called the first partial sum, ... |
. Salary Increases A newly hired salesman is promised a beginning salary of $30,000 a year with a $2000 raise every year. Let Sn be his salary in his nth year of employment. (a) Find a recursive definition of Sn. (b) Find his salary in his fifth year of employment. 78. Concentration of a Solution A biologist is trying to... |
the sum S50 50 2 2 3 1 3950 30 2 49 2 1 2 4 Sn n 2 3 2a n 1 1 d 4 2 Stage Thus, the amphitheater has 3950 seats. ✎ Practice what you’ve learned: Do Exercise 65. ▲ E X AM P L E 7 | Finding the Number of Terms in a Partial Sum How many terms of the arithmetic sequences 5, 7, 9, . . . must be added to get 572? ▼ SO LUTI ... |
nsecutive terms. 618 CHAPTER 9 | Sequences and Series For instance, r 45 15 3 . Thus, 3 The eighth term is 2 ✎ Practice what you’ve learned: Do Exercise 27. a8 3 2 1 1 5 81 5 an n1 5 3 1 7 10,935 . 2 ▲ E X AM P L E 3 | Finding Terms of a Geometric Sequence The third term of a geometric sequence is 63 4 , and the sixth ... |
atio, and express the nth term of the sequence in the standard form an an ar n1. 2 1 1 4n ln an an 21. 23. 25. 3 n 2 22. an 4 3n 5n1 2 24. an 26. an n2n 2 1 1 n n 1 27–36 ■ Determine the common ratio, the fifth term, and the nth term of the geometric sequence. ✎ 27. 2, 6, 18, 54, . . . 28. 29. 0.3, 0.09, 0.027, 0.0081, ... |
ables for Cn Do the same for Fn n4. n3. Which difference sequence is constant? 4. Make up a polynomial of degree 5, and construct a difference table. Which dif- ference sequence is constant? 5. The first few terms of a polynomial sequence are 1, 2, 4, 8, 16, 31, 57, . . . . Construct a difference table, and use it to fin... |
$405 per month for 5 years. What interest rate is this car dealer charging? $18,000, ▼ SO LUTI O N The payments form an annuity with present value Ap R 405, and n 12 5 60. To find the interest rate, we must solve for i in the equation R 1 iAp 1 i n 1 A little experimentation will convince you that it is not possible to ... |
to n 40. It might seem reasonable at this point to conjecis prime for every natural ture that number n. But that conjecture would be too hasty, because it is easily seen that p is not prime. This illustrates that we cannot be certain of the truth of a statement no matter how many special cases we check. We need a conv... |
ow that 5n 1 is divisible by 4 for all natural numbers n. 17. Show that n 2 n 41 is odd for all natural numbers n. 18. Show that n 3 n 3 is divisible by 3 for all natural 4 numbers n. 19. Show that 8n 3n is divisible by 5 for all natural numbers n. 20. Show that 32n 1 is divisible by 8 for all natural numbers n. 21. Pr... |
e. We now state this property in terms of the binomial coefficients. KEY PROPERTY OF THE BINOMIAL COEFFICIENTS For any nonnegative integers r and k with r k Notice that the two terms on the left side of this equation are adjacent entries in the kth row of Pascal’s triangle and the term on the right side is the entry dia... |
erm of the sequence is an a n 1 1 d 2 CHAPTER 9 | Review 647 10 10 5 1 ? On the basis of the pattern you have found, find the sum of the nth row Prove your result by expanding Theorem. 1 1 1 n 2 using the Binomial 57. Alternating Sums of Binomial Coefficients Find the sum by finding a pattern as in Exercise 56. Prove your... |
. . . . (a) Find the common ratio r for this sequence. (b) Find a formula for the nth term an of the sequence. (c) Find the tenth term of the sequence. 1 5. The first term of a geometric sequence is 25, and the fourth term is . 5 (a) Find the common ratio r and the fifth term. (b) Find the partial sum of the first eight ... |
of the first year, $1000 at the end of the second, $1500 at the end of the third, and so on. (a) Explain why the recursive formula displayed below gives the amount Vn in Victoria’s CD when she reinvests at the end of the nth year. 1.05Vn1 Vn 500n (b) Using the Seq (“sequence”) mode on your graphing calculator, enter th... |
letters (a) is allowed? (b) is not allowed? 5. Three-Letter Words How many three-letter “words” (strings of letters) can be formed by using the letters WXYZ if repetition of letters (a) is allowed? (b) is not allowed? 6. Horse Race Eight horses are entered in a race. (a) How many different orders are possible for comp... |
rs ABCDWXYZ are XAYBZWCD ZAYBCDWX DBWAZXYC YDXAWCZB How many such permutations are possible? Since there are eight choices for the first position, seven for the second (after the first has been chosen), six for the third (after the first two have been chosen), and so on, the Fundamental Counting Principle tells us that th... |
In how many ways can the four winners be chosen? ▼ SO LUTI O N We need to find the number of ways of choosing four winners from 20 entries. The order in which the tickets are chosen doesn’t matter, because the same prize is awarded to each of the four winners. So we want the number of combinations of 20 objects (the tic... |
pping Pizzas A pizza parlor offers a choice of 16 different toppings. How many three-topping pizzas are possible? 53. Violin Recital A violinist has practiced 12 pieces. In how many ways can he choose eight of these pieces for a recital? 54. Choosing Clothing If a woman has eight skirts, in how many ways can she choose... |
r rolling a die, the outcomes are 1, 2, 3, 4, 5, and 6. The sample space of an experiment is the set of all possible outcomes. If we let H stand for heads and T for tails, then the sample space . of the coin-tossing experiment is S H, T 5 The table lists some experiments and the corresponding sample spaces. 6 The mathe... |
a face card or a spade? ▼ SO LUTI O N We let E and F denote the following events: E: The card is a face card. F: The card is a spade. Face cards K Q J K Q J K Q J Spades There are 12 face cards and 13 spades in a 52-card deck, so 10 E P 1 2 12 52 and P 13 52 F 1 2 Since 3 cards are simultaneously face cards and spades... |
nd the probability that the couple has only boys. (c) Find the probability that the couple has two boys and two girls. (d) Find the probability that the couple has four children of the same sex. (e) Find the probability that the couple has at least two girls. 28. Bridge Hands What is the probability that a 13-card brid... |
Birthday Problem What is the probability that in a group of six students at least two have birthdays in the same month? 69. Combination Lock A student has locked her locker with a combination lock, showing numbers from 1 to 40, but she has forgotten the three-number combination that opens the lock. To open the lock, sh... |
we flip a balanced coin eight times? What is the number of heads that is most likely to show up? To find out, we need to find the probability of getting no heads, one head, two heads, and so on. 0 head 1 head P P 1 1 2 heads , 0 2 a 8, 1 2 a 8.003906 0.03125 0.109375 The probabilities for any number of heads (from 0 to 8... |
h shows the probabilities of getting any number of heads from 0 to 9. (a) Find the probabilities of getting exactly one head, exactly two heads, and so on, to confirm the probabilities given by the graph. (b) What is the most likely outcome(s) (the number of heads with the greatest probability of occurring)? Compare you... |
cannot be obtained from the p nk n2 n n1 other simply by interchanging the positions of elements of the same kind. (In other words, the permutations “look” different.) The number of distinguishable permutations of these objects is n! n1!n2! p nk! Combinations (p. 666) A combination of r objects from a set is any subse... |
ck. (a) An ace (b) An ace or a jack (c) An ace or a spade (d) A red ace 29. A card is drawn from a 52-card deck, a die is rolled, and a coin is tossed. Find the probability of each outcome. (a) The ace of spades, a six, and heads (b) A spade, a six, and heads (c) A face card, a number greater than 3, and heads 30. Two ... |
l of one to five spots. The probability that a randomly selected insect has n spots is 1 4B n 1, 2, 3, 4, or 5 n A . 1 2 (a) What event has probability 5 a n1 A n ? Calculate this sum. 1 4B (b) What is the probability that a randomly selected insect has no spots THE MONTE CARLO METHOD A good way to familiarize ourselves... |
or p. To implement this method, we use a random-number generator to obtain the coordinates 2 x, y of a random point in the square, and then check to see if it lies inside the circle (that is, 1 we check if x 2 y 2 1). Note that we need to use only points in the first quadrant, since the ratio of areas is the same in eac... |
8 (f) x4 9x2 9. (a) (b) x 3 x 2 1 x 2 2x c) 2 2 3x 9 x 1 (e) 2 (d) 2 1 3x1/ 2x 5 2x 5 2 2 1 Answers to Section 1.3 A3 (f) (d) x 2 x y xy 1 1 x 2 2 11. (a) 2 1 2 10. (a) 313 2 (b) x 1 x 3 (c) 1 x 2 x 2 x 2 (b) 216 3 12 25 32 FOCUS ON PROBLEM SOLVING ■ page 62 1. 37.5 mi/h 3. 150 mi 11. 2p 15. 15,999,999,999,992,000,000,... |
0 29. y 5 35. 5x 2y 1 0 31. x 2y 11 0 37. x y 6 0 (b) 3x 2y 8 0 −10 2 −2 15. −20 19. −4 5 −1 100 −50 23. No 25. Yes, 2 −4 17. −50 21. 20 6 −3 10 150 5 −10 2000 −2000 5 −1 y 5 (−2, 1) 0 1 x −3 41. They all have the same slope. 43. They all have the same x-intercept. 8 5 m = 1.5 m = 0.75 m = 0.25 m = 0 8 m = −0.25 m = −0... |
cal minimum 73.32 when x 3.21 decreasing on 5.66 when x 4.00 on local minimum 0.38 when x 1.73 q, 1.73 1 4 43. (a) 500 MW, 725 MW (b) Between 3:00 A.M. and 4:00 A.M. (c) Just before noon 30, 32 decreasing on weight, only to regain it again later. 3.21, q 39. (a) Local maximum q, 4.00 ; decreasing 1 41. (a) Local maximu... |
s 3 113 ; y-intercept 4 6 −2 2 x 2 1 2 1, 1 2 1 x 1 ; no x-intercept; y-intercept 3 2 x f 17. (a) 1 (b) Vertex (c) 1 2 y 3 23. (a) (bc) Minimum f 1 1 2 2 3 0 _3 _2 (_1, _2) x 2 25. (a) (bc) Minimum f 2 1 1 2 2 0 1 (1, −2) x 27. (a) (b) f x 1 2 x 3 2B y A 2 21 4 21 3 !_ , @ 4 2 (c) Maximum f 3 2B A 21 4 3 0 _2 _3 x 3 _3... |
7. (a) 1, 2, 3, 6, 41. (a) 2, 0 (multiplicity 2), 1 (b) y 25. x 2 3x 23, 94 31. 3 (b) 2 or 0 positive, 3 or 1 negative 29. 2x 3, 12 75. 30 _6 6 _30 x-intercept 2 y-intercept 4 vertical x 1, x 2 slant y x 1 local maximum local minimum 0.425, 3.599 1 4.216, 7.175 1 2 2 30 0 _4 _30 4 x 4 0 _4 _2 1 x 43. (a) 2, 1, 2, 3 (b)... |
pt 0; horizontal asymptote y 3; vertical asymptotes x 2 and x 1 y 2 2 8 2x 3 1 2 (b) 9. (a) 4 10. (a) 4 (b) After 6.23 years 12. (a) t P 1 2 5 log x 1 log x 1 2 log 11. (a) $29,396.15 1 2 (b) ln 2, ln 4 120e0.0565t (c) 12.837 years (b) 917 (c) After 49.8 months FOCUS ON MODELING ■ page 437 1. (a) 290 1780 0 2020 0 25 (... |
nswers to Selected Exercises and Chapter Tests 12. 2 3 2 1 d 1 c 15. 3 16. (a) 4 3 3 2 x y 17. A 0 0 0, B 0 0 2, B1 B B R R 19. 1.2 lb almonds, 1.8 lb walnuts C 10 30 b) 70, 90 1 2 18. 1 5, 5, 4 2 0 0 1 S FOCUS ON MODELING ■ page 549 3. (a) Shear to the right (b) T 1 1 1.5 1 0 (c) Shear to the left (d) We get back the ... |
45. 441 47. 3280 49. 6141 1024 3. $13,180.79 11. $13,007.94 SECTION 9.4 ■ page 630 1. amount 9. $572.34 17. $733.76, $264,153.60 23. (a) $859.15 25. 18.16% 27. 11.68% 5. $360,262.21 13. $2,601.59 19. $583,770.65 7. $5,591.79 15. $307.24 21. $9020.60 (b) $309,294.00 (c) $1,841,519.29 SECTION 9.5 ■ page 637 1. natural; P... |
60 27. 120 37. 15 47. 2300 57. 1,560,780 (b) 792 67. 104,781,600 (b) 8640 59. (a) 56 65. 69. 6600 75. 17,813,250 49. 2,598,960 (c) 6160 41. 2,522,520 51. 120 (b) 256 20 # 19 # C 77. 182 7. 7920 9. 100 11. 60 21. 100 23. 2730 33. 997,002,000 13. 60 25. 151,200 35. 24 43. 168 45. 20 53. 495 55. 2,035,800 61. 1024 63. (a)... |
5–98 confocal, 588–89 constructing, 595–96 degenerate conic, 585–86 general equation of shifted conic, 585–86 shifted, 581–89 Conjecture, 632–33 Conjugate, complex, 99, 102 Conjugate hyperbolas, 579 Conjugate radical, 50–51 Conjugate Zeros Theorem, 340–41 Constant, 66 growth, 263–64 Constant coefficient (constant term),... |
–55 reflecting, 246–47 shifted, 581 stretching and shrinking, horizontal, 248–49 stretching and shrinking, vertical, 247–48 transformations of functions and, 243–54 vertical shift of, 243–44, 246 Graphical addition of functions, 256 Graphical method, 446–47 Graphing calculator ellipse, 565–66 exponential functions, 371 ... |
reflector, 558–59, 560, 561 Parameter, 74 of system of solutions, 462 Pareto, Vilfredo, 399 Pareto’s Principle, 399 Partial fraction decomposition, 469–73 distinct linear factors, 469–70 distinct quadratic factors, 471–72 repeated irreducible quadratic factor, 472–73 repeated linear factors, 471 Partial fractions, 469–7... |
d, 246 Vertical stretching and shrinking of graphs, 247–48 Viète, François, 89 Virus, exponential model for spread of, 376 Visualizing formula, 38 Visual representation of function, 209–11 Volume of box, modeling, 280–81 of pyramid, 63 Von Neumann, John, 246 Voting methods, 674 Weber-Fechner Law, 420 Weight of astronau... |
e sets. Example. With n = 3, the singletons {1}, {2}, {3} form a 1-uniform cover, and so does {1}, {2, 3}. Also, {1, 2}, {1, 3} and {2, 3} form a uniform 2-cover. However, {1, 2} and {2, 3} do not form a uniform cover of [3]. Note that we allow repetitions. Example. {1}, {1}, {2, 3}, {2}, {3} is a 2-uniform cover of [3... |
that the coefficient of X |A|−1Y |B|−1 is |C| |A|−1 which is non-zero in Zp, since C < p. This contradicts Alon’s combinatorial Nullstellensatz. We can also use this to prove Erd¨os–Ginzburg–Ziv again. Theorem (Erd¨os–Ginzburg–Ziv). Let p be a prime and a1, . . . , a2p+1 ∈ Zp. Then there exists I ∈ [2p − 1](p) such that ... |
lly, we get to the integration part. Suppose we picked all our γw to be the fixed straight line segment from a0. Then for antiderivative to be differentiable, we needed f (z) dz = f (z) dz. γw∗δh γw+h In other words, we needed to the integral along the path γw ∗ δh ∗ (−γw+h) to vanish. This is a rather simple kind of pat... |
e so that it is. Hence the integral of f (w) w−z around the half-contour vanishes by Cauchy’s theorem. Adding these together, we get ∂B(z0,r) f (w) w − z dw = ∂B(z,δ) f (w) w − z dw, where the balls are both oriented anticlockwise. Now we have f (z) − 1 2πi ∂B(z0,r) f (w) w − z dw = f (z) − 1 2πi ∂B(z,δ) f (w) w − z dw... |
ay. One of the applications is the following: Lemma (Principle of isolated zeroes). Let f : B(a, r) → C be holomorphic and not identically zero. Then there exists some 0 < ρ < r such that f (z) = 0 in the punctured neighbourhood B(a, ρ) \ {a}. Proof. If f (a) = 0, then the result is obvious by continuity of f . The oth... |
omorphic and has (at worst) poles on S is said to be meromorphic on U . The requirement that S is discrete is so that each pole in S is actually an isolated singularity. Example. A rational function P (z) Q(z) , where P, Q are polynomials, is holomorphic on C \ {z : Q(z) = 0}, and meromorphic on C. More is true — it is... |
near z = 0. So f has sin2(πz) also has a double pole at each 0, f has a double pole, since f (z) = 1 a double pole at each k ∈ Z. Note that k ∈ Z. 1 Now, consider the principal parts of our functions — at k ∈ Z, f (z) has principal part 1 (z−k)2 . Looking at our previous Laurent series for cosec(z), if g(z) = π 2 sin π... |
x open sets C1, · · · , Cn ⊆ U such that for xi−1 ≤ t ≤ xi, we have φ(t) and ψ(t) in Ci. It was a rather unnatural definition, since we have to make reference to this arbitrarily constructed dissection of [a, b] and convex sets Ci. Moreover, this definition fails to be transitive (e.g. on R \ {0}, rotating a circle about... |
rately described as “Integrals, integrals, integrals”. Our main objective is to evaluate real integrals, but to do so, we will pretend they are complex integrals, and apply the residue theorem. 55 3 Residue calculus IB Complex Analysis Before that, we first come up with some tools to compute residues, since we will have... |
e theorem says 2 γN f (z) dz = 2πi 2 N n=1 1 n2 − π2 3 . 64 3 Residue calculus IB Complex Analysis We can thus get the desired series if we can show that γN f (z) dz → 0 as n → ∞. We first note that γN f (z) dz π cot πz z2 4(2N + 1) | cot πz| 4(2N + 1)π 2 N + 1 2 | cot πz|O(N −1). ≤ sup γN ≤ sup γN = sup γN So everythin... |
inciple. This is just equal to I(f ◦ γ, f (a)) = deg(f, a), by the invariance of I(Γ, ∗) as we move ∗ in a component C \ Γ. Now if w = f (a), since f (z) = 0 on B(a, r) \ {a}, all roots of f (z) − w must be simple. So there are exactly deg(f ; a) distinct zeros. The local degree theorem says the equation f (z) = w has ... |
and limits indexed on D are coproducts and products indexed on the set D. Coproducts are disjoint unions in S or U , wedges (or one-point unions) in T , free products in G , and direct sums in A b. Products are Cartesian products in all of these categories; more precisely, they are Cartesian products of underlying set... |
shall later use the following application of the van Kampen theorem to prove that any group is the fundamental group of some space. We need a definition. Definition. A space X is said to be simply connected if it is path connected and satisfies π1(X) = 0. Proposition. Let X = U ∪V , where U , V , and U ∩V are path connec... |
of G such that gs = s′. Equivalently, S consists of a single orbit. If H is a subgroup of G, the set G/H of cosets gH is a transitive G-set. When G acts transitively on a set S, we obtain an isomorphism of G-sets between S and the G-set G/Gs for any fixed s ∈ S by sending gs to the coset gGs. The following lemma descri... |
clear that p(π(E (G/H), e)) = H. ⊂ B(p(f H), p(f ′H)). E (G/H)(f H, f ′H) = This defines the object function of the functor E : O(G) −→ Cov(B). To define E on morphisms, consider α : G/H −→ G/K. If α(eH) = gK, then g−1Hg ⊂ K and α(f H) = f gK. The functor E (α) : E (G/H) −→ E (G/K) sends the object f H to the object α(f... |
omeomorphism. The functoriality on O(G) of our construction of general covers will be immediate from the following observation. Lemma. Let X be a G-space. Then passage to orbit spaces defines a functor X/(−) : O(G) −→ U . Proof. The functor sends G/H to X/H and sends a map α : G/H −→ G/K to the map X/H −→ X/K that sends... |
escribed behavior on the boundary of the square makes it clear W 5. APPLICATIONS TO GROUPS 37 that H exists: −1 a(v) j a(v) cv j a(v ′ ) ****************** cv′ a(v ′ ) 4. Covers of graphs and Euler characteristics Define the Euler characteristic χ(X) of a finite graph X to be V − E, where V is the number of vertices of X... |
ly well be viewed as a map X −→ Y I . These adjoint, or “dual,” points of view will play an important role in the next two chapters. (1) PROBLEMS (a) Any subspace of a weak Hausdorff space is weak Hausdorff. (b) Any closed subspace of a k-space is a k-space. (c) An open subset U of a compactly generated space X is compac... |
us homotopy inverse ι : X −→ M i has ι(x) = (x, 0) and is thus very far from being a map under A. The proposition ensures that ι is homotopic to a map under A that is homotopy inverse to r under A. The following generalization asserts that, for inclusions that are cofibrations, a pair of homotopy equivalences is a homot... |
< q. Certainly s(e, β) = e and (p ◦ s)(e, β) = β. It is not hard to check that s is well defined and continuous, hence it is a path lifting function for p. P 5. Fiber homotopy equivalence It is often important to study fibrations over a given base space B, working in the category of spaces over B. A space over B is a map... |
called the reduced cylinder on X, and a based homotopy X × I −→ Y is the same thing as a based map X ∧ I+ −→ Y . We change notations and write M f for the based mapping cylinder Y ∪f (X ∧ I+) of a based map f . As in the unbased case, we conclude that a based map i : A −→ X is a cofibration if and only if M i is a retra... |
X −→ Y be a map of based spaces. Then the following diagram is homotopy commutative, where j : X −→ M f is the inclusion, r : M f −→ Y is the retraction, and π is induced by the quotient map M f −→ Cf : F j = X ×j P M f F r=id ×P r X ×f P Y = F f 'OOOOOOOOOOO π wppppppppppp η ΩCf. (1) Prove the two lemmas stated at the... |
given path component of A. Corollary. A homotopy equivalence of spaces or of pairs of spaces induces an isomorphism on all homotopy groups. We shall soon show that the converse holds for a quite general class of spaces, namely the class of CW complexes, but we first need a few preliminaries. 6. n-Equivalences, weak equi... |
se for immediate use. Of course, the unit interval is a graph with two vertices and one edge. Lemma. For a CW complex X, X × I is a CW complex that contains X × ∂I as a subcomplex and, in addition, has one (n + 1)-cell for each n-cell of X. A “cellular homotopy” h : f ≃ f ′ between cellular maps X −→ Y of CW complexes ... |
X together with subspaces A and B. This must not be confused with a triple (X, A, B), which would require B ⊂ A ⊂ X. A triad (X; A, B) is said to be excisive if X is the union of the interiors of A and B. Such triads play a fundamental role in homology and cohomology theory, and some version of the arguments to follow... |
ows of which are homotopy equivalences of pairs. The hypothesis on f and the long exact sequence of the pair (M f, X) imply that (M f, X) and therefore also (A, C) are (n − 1)-connected. In view of the connecting isomorphism ∂ : πq+1(CX, X) −→ πq(X) and the evident homotopy equivalence of pairs (B, C) ≃ (CX, X), (B, C)... |
c to a map f ′ that has image in (X − {y}; A, X − {x, y}). This will imply that f is null homotopic. Let Dm 1/2 and f (I q α such that f (I q 1/2 ⊂ Dm and Dn α) is contained in the interior of Dn if it intersects Dn 1/2 and whose restriction to the (m − 1)-skeleton of I q does not cover Dm 1/2 ⊂ Dn be the subdisks of r... |
π). • ADDITIVITY If (X, A) is the disjoint union of a set of pairs (Xi, Ai), then the inclusions (Xi, Ai) −→ (X, A) induce an isomorphism iH∗(Xi, Ai; π) −→ H∗(X, A; π). • WEAK EQUIVALENCE If f : (X, A) −→ (Y, B) is a weak equivalence, P then f∗ : H∗(X, A; π) −→ H∗(Y, B; π) is an isomorphism. 95 96 AXIOMATIC AND CELLUL... |
IC AND CELLULAR HOMOLOGY THEORY as canonical basis elements of ˜H ′ n(X n/X n−1). Lemma. The differential dn : Cn(X) −→ Cn−1(X) can be identified with the composite −1 (∂n)∗ −−−→ ˜H ′ n(X n/X n−1) n(Σ(X n−1/X n−2)) Σ n : ˜H ′ d′ Proof. The identification of the groups is clear: we let the basis element [j] of Cn(X) corres... |
pper hemispheres to the basepoint. Of course, these are homotopy equivalences. We define homeomorphisms ± : Dq −→ Eq jq ± ⊂ Sq by jq ±(x1, . . ., xq) = (±x1, . . ., ±xq, ±(1 − i )1/2). x2 This decomposes Sq as the the union of the images of two q-cells. The intersection of these images is Sq−1 since P (x1, . . ., xq, (1... |
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