MML PRECALCULUS ENHANCED
7th Edition
ISBN: 9780134119250
Author: Sullivan
Publisher: INTER PEAR
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Textbook Question
Chapter 13.2, Problem 61AE
Basketball Teams A basketball team has 6 players who play guard (2 of 5 starting positions). How many different teams are possible, assuming that the remaining 3 positions are filled and it is not possible to distinguish a left guard from a right guard?
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A ladder 25 feet long is leaning against the wall of a building. Initially, the foot of the ladder is 7 feet from the wall. The foot of the ladder begins to slide at a rate of 2 ft/sec, causing the top of the ladder to slide down the wall. The location of the foot of the ladder, its x coordinate, at time t seconds is given by
x(t)=7+2t.
wall
y(1)
25 ft. ladder
x(1)
ground
(a) Find the formula for the location of the top of the ladder, the y coordinate, as a function of time t. The formula for y(t)= √ 25² - (7+2t)²
(b) The domain of t values for y(t) ranges from 0
(c) Calculate the average velocity of the top of the ladder on each of these time intervals (correct to three decimal places):
. (Put your cursor in the box, click and a palette will come up to help you enter your symbolic answer.)
time interval
ave velocity
[0,2]
-0.766
[6,8]
-3.225
time interval
ave velocity
-1.224
-9.798
[2,4]
[8,9]
(d) Find a time interval [a,9] so that the average velocity of the top of the ladder on this…
Total marks 15
3.
(i)
Let FRN Rm be a mapping and x = RN is a given
point. Which of the following statements are true? Construct counterex-
amples for any that are false.
(a)
If F is continuous at x then F is differentiable at x.
(b)
If F is differentiable at x then F is continuous at x.
If F is differentiable at x then F has all 1st order partial
(c)
derivatives at x.
(d) If all 1st order partial derivatives of F exist and are con-
tinuous on RN then F is differentiable at x.
[5 Marks]
(ii) Let mappings
F= (F1, F2) R³ → R² and
G=(G1, G2) R² → R²
:
be defined by
F₁ (x1, x2, x3) = x1 + x²,
G1(1, 2) = 31,
F2(x1, x2, x3) = x² + x3,
G2(1, 2)=sin(1+ y2).
By using the chain rule, calculate the Jacobian matrix of the mapping
GoF R3 R²,
i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)?
(iii)
[7 Marks]
Give reasons why the mapping Go F is differentiable at
(0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0).
[3 Marks]
5.
(i)
Let f R2 R be defined by
f(x1, x2) = x² - 4x1x2 + 2x3.
Find all local minima of f on R².
(ii)
[10 Marks]
Give an example of a function f: R2 R which is not bounded
above and has exactly one critical point, which is a minimum. Justify briefly
Total marks 15
your answer.
[5 Marks]
Chapter 13 Solutions
MML PRECALCULUS ENHANCED
Ch. 13.1 - Prob. 1AYPCh. 13.1 - Prob. 2AYPCh. 13.1 - True or false The intersection of two sets is...Ch. 13.1 - Prob. 4AYPCh. 13.1 - Prob. 5CVCh. 13.1 - If the number of elements in a set is a...Ch. 13.1 - Prob. 7CVCh. 13.1 - True or False If a task consists of a sequence of...Ch. 13.1 - Prob. 9SBCh. 13.1 - Prob. 10SB
Ch. 13.1 - If n( A )=15 , n( B )=20 , and n( AB )=10 , find...Ch. 13.1 - If n( A )=30 , n( B )=40 , and n( AB )=45 , find...Ch. 13.1 - If n( AB )=50 , n( AB )=10 , and n( B )=20 , find...Ch. 13.1 - If n( AB )=60 , n( AB )=40 , and n( A )=n( B ) ,...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - Shirts and Ties A man has 5 shirts and 3 ties. How...Ch. 13.1 - Blouses and Skirts A woman has 5 blouses and 8...Ch. 13.1 - Four-digit Numbers How many four-digit numbers can...Ch. 13.1 - Five-digit Numbers How many five-digit numbers can...Ch. 13.1 - Analyzing Survey Data In a consumer survey of 500...Ch. 13.1 - Analyzing Survey Data In a student survey, 200...Ch. 13.1 - Analyzing Survey Data In a survey of 100 investors...Ch. 13.1 - Prob. 30AECh. 13.1 - Demographics The following data represent the...Ch. 13.1 - Prob. 32AECh. 13.1 - Stock Portfolios As a financial planner, you are...Ch. 13.1 - Make up a problem different from any found in the...Ch. 13.1 - Problems 36-39 are based on material learned...Ch. 13.1 - Problems 36-39 are based on material learned...Ch. 13.1 - Problems 36-39 are based on material learned...Ch. 13.1 - Problems 36-39 are based on material learned...Ch. 13.2 - 0!= ; 1!= . (p. 642)Ch. 13.2 - True or False n!= ( n+1 )! n . (p. 642)Ch. 13.2 - A(n) __________ is an ordered arrangement of r...Ch. 13.2 - A(n) ___________ is an arrangement of r objects...Ch. 13.2 - P( n,r )= __________________.Ch. 13.2 - C( n,r )= _______________________.Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 7-14, find the value of each...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - In Problems 15-22, use formula (2) to find the...Ch. 13.2 - List all the ordered arrangements of 5 objects a ,...Ch. 13.2 - List all the ordered arrangements of 5 objects a ,...Ch. 13.2 - List all the ordered arrangements of 4 objects 1,...Ch. 13.2 - List all the ordered arrangements of 6 objects 1,...Ch. 13.2 - List all the combinations of 5 objects a , b , c ,...Ch. 13.2 - List all the combinationss of 5 objects a , b , c...Ch. 13.2 - List all the combinations of 4 objects 1, 2, 3,...Ch. 13.2 - List all the combinationss of 6 objects 1, 2, 3,...Ch. 13.2 - Forming Codes How many two-letter codes can be...Ch. 13.2 - Forming Codes How many two-letter codes can be...Ch. 13.2 - Forming Numbers How many three-digit numbers can...Ch. 13.2 - Forming Numbers How many three-digit numbers can...Ch. 13.2 - Lining People Up In how many ways can 4 people be...Ch. 13.2 - Stacking Boxes In how many ways can 5 different...Ch. 13.2 - Forming Codes How many different three-letter...Ch. 13.2 - Forming Codes How many different four-letter codes...Ch. 13.2 - Stocks on the NYSE Companies whose stocks are...Ch. 13.2 - Stocks on the NASDAQ Companies whose stocks are...Ch. 13.2 - Establishing Committees In how many ways can a...Ch. 13.2 - Establishing Committees In how many ways can a...Ch. 13.2 - Possible Answers on a True/False Test How many...Ch. 13.2 - Possible Answers on a Multiple-choice Test How...Ch. 13.2 - Arranging Books Five different mathematics books...Ch. 13.2 - Forming License Plate Numbers How many different...Ch. 13.2 - Birthday Problem In how many ways can 2 people...Ch. 13.2 - Birthday Problem In how many ways can 5 people all...Ch. 13.2 - Forming a Committee A student dance committee is...Ch. 13.2 - Forming a Committee The student relations...Ch. 13.2 - Forming Words How many different 9-letter words...Ch. 13.2 - Forming Words How many different 11-letter words...Ch. 13.2 - Selecting Objects An urn contains 7 white balls...Ch. 13.2 - Selecting Objects An urn contains 15 red balls and...Ch. 13.2 - Senate Committees The U.S. Senate has 100 members....Ch. 13.2 - Football Teams A defensive football squad consists...Ch. 13.2 - Baseball In the American Baseball League, a...Ch. 13.2 - Baseball In the National Baseball League, the...Ch. 13.2 - Baseball Teams A baseball team has 15 members....Ch. 13.2 - World Series In the World Series the American...Ch. 13.2 - Basketball Teams A basketball team has 6 players...Ch. 13.2 - Basketball Teams On a basketball team of 12...Ch. 13.2 - Combination Locks A combination lock displays 50...Ch. 13.2 - Create a problem different from any found in the...Ch. 13.2 - Create a problem different from any found in the...Ch. 13.2 - Explain the difference between a permutation and a...Ch. 13.2 - Problems 67-70 are based on material learned...Ch. 13.2 - Problems 67-70 are based on material learned...Ch. 13.2 - Problems 67-70 are based on material learned...Ch. 13.2 - Problems 67-70 are based on material learned...Ch. 13.3 - When the same probability is assigned to each...Ch. 13.3 - The _____________of an event E is the set of all...Ch. 13.3 - True or False The probability of an event can...Ch. 13.3 - True or False In a probability model, the sum of...Ch. 13.3 - In a probability model, which of the following...Ch. 13.3 - In a probability model, which of the following...Ch. 13.3 - Determine whether the following is a probability...Ch. 13.3 - Determine whether the following is a probability...Ch. 13.3 - Determine whether the following is a probability...Ch. 13.3 - Determine whether the following is a probability...Ch. 13.3 - In Problems 11-16, construct a probability model...Ch. 13.3 - In Problems 11-16, construct a probability model...Ch. 13.3 - In Problems 11-16, construct a probability model...Ch. 13.3 - In Problems 11-16, construct a probability model...Ch. 13.3 - In Problems 11-16, construct a probability model...Ch. 13.3 - In Problems 11-16, construct a probability model...Ch. 13.3 - In Problems 17-22, use the following spinners to...Ch. 13.3 - In Problems 17-22, use the following spinners to...Ch. 13.3 - In Problems 17-22, use the following spinners to...Ch. 13.3 - In Problems 17-22, use the following spinners to...Ch. 13.3 - In Problems 17-22, use the following spinners to...Ch. 13.3 - In Problems 17-22, use the following spinners to...Ch. 13.3 - In Problems 23-26, consider the experiment of...Ch. 13.3 - In Problems 23-26, consider the experiment of...Ch. 13.3 - Prob. 25SBCh. 13.3 - Prob. 26SBCh. 13.3 - Assigning Probabilities A coin is weighted so that...Ch. 13.3 - Assigning Probabilities A coin is weighted so that...Ch. 13.3 - Assigning Probabilities A die is weighted so that...Ch. 13.3 - Assigning Probabilities A die is weighted so that...Ch. 13.3 - For Problems 31-34, the sample space is S={...Ch. 13.3 - For Problems 31-34, the sample space is S={...Ch. 13.3 - For Problems 31-34, the sample space is S={...Ch. 13.3 - For Problems 31-34, the sample space is S={...Ch. 13.3 - For Problems 35 and 36, an urn contains 5 white...Ch. 13.3 - For Problems 35 and 36, an urn contains 5 white...Ch. 13.3 - In Problems 37-40, assume equally likely outcomes....Ch. 13.3 - In Problems 37-40, assume equally likely outcomes....Ch. 13.3 - In Problems 37-40, assume equally likely outcomes....Ch. 13.3 - In Problems 37-40, assume equally likely outcomes....Ch. 13.3 - For Problems 41-44, two fair dice are rolled....Ch. 13.3 - For Problems 41-44, two fair dice are rolled....Ch. 13.3 - For Problems 41-44, two fair dice are rolled....Ch. 13.3 - In Problems 45-48, find the probability of the...Ch. 13.3 - In Problems 45-48, find the probability of the...Ch. 13.3 - In Problems 45-48, find the probability of the...Ch. 13.3 - In Problems 45-48, find the probability of the...Ch. 13.3 - If P( A )=0.60 , P( AB )=0.85 , and P( AB )=0.05 ,...Ch. 13.3 - Prob. 50SBCh. 13.3 - Prob. 51SBCh. 13.3 - Prob. 52SBCh. 13.3 - Prob. 53SBCh. 13.3 - Doctorate Degrees According to the National...Ch. 13.3 - Prob. 55SBCh. 13.3 - For Problems 57-60, a golf ball is selected at...Ch. 13.3 - Prob. 57SBCh. 13.3 - Prob. 58SBCh. 13.3 - Prob. 59SBCh. 13.3 - Prob. 60SBCh. 13.3 - On The Price Is Right, there is a game in which a...Ch. 13.3 - Prob. 62SBCh. 13.3 - Prob. 63SBCh. 13.3 - Prob. 64SBCh. 13.3 - Prob. 65SBCh. 13.3 - Prob. 66SBCh. 13.3 - Prob. 67SBCh. 13.3 - Checkout Lines Through observation, it has been...Ch. 13.3 - Prob. 69SBCh. 13.3 - Prob. 70SBCh. 13.3 - Prob. 71SBCh. 13.3 - Birthday Problem What is the probability that at...Ch. 13.3 - Prob. 73SBCh. 13.3 - Prob. 74RYKCh. 13.3 - Prob. 75RYKCh. 13.3 - Prob. 76RYKCh. 13.3 - Solve the given system using matrices. { 3x+y+2z=1...Ch. 13.R - Prob. 1RECh. 13.R - If n( A )=8 , n( B )=12 , and n( AB )=3 , find n(...Ch. 13.R - Prob. 3RECh. 13.R - Prob. 4RECh. 13.R - Prob. 5RECh. 13.R - Prob. 6RECh. 13.R - Prob. 7RECh. 13.R - Prob. 8RECh. 13.R - Prob. 9RECh. 13.R - In Problems 10 and 11, compute the value of the...Ch. 13.R - In Problems 10 and 11, compute the value of the...Ch. 13.R - Stocking a Store A clothing store sells pure wool...Ch. 13.R - Baseball On a given day, the American Baseball...Ch. 13.R - Choosing Seats If 4 people enter a bus that has 9...Ch. 13.R - Choosing a Team In how many ways can a squad of 4...Ch. 13.R - Baseball In how many ways can 2 teams from 14...Ch. 13.R - Telephone Numbers Using the digits 0, 1, 2,...,9,...Ch. 13.R - Prob. 18RECh. 13.R - Binary Codes Using the digits 0 and 1, how many...Ch. 13.R - Arranging Flags How many different vertical...Ch. 13.R - Forming Committees A group of 9 people is going to...Ch. 13.R - Birthday Problem For this problem, assume that a...Ch. 13.R - Unemployment According to the U.S. Bureau of Labor...Ch. 13.R - Prob. 24RECh. 13.R - Each of the numbers 1, 2,..., 100 is written on an...Ch. 13.R - At the Milex tune-up and brake repair shop, the...
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- Total marks 15 4. : Let f R2 R be defined by f(x1, x2) = 2x²- 8x1x2+4x+2. Find all local minima of f on R². [10 Marks] (ii) Give an example of a function f R2 R which is neither bounded below nor bounded above, and has no critical point. Justify briefly your answer. [5 Marks]arrow_forward4. Let F RNR be a mapping. (i) x ЄRN ? (ii) : What does it mean to say that F is differentiable at a point [1 Mark] In Theorem 5.4 in the Lecture Notes we proved that if F is differentiable at a point x E RN then F is continuous at x. Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We want to show that F(xn) F(x), which means F is continuous at x. Denote hnxn - x, so that ||hn|| 0. Thus we find ||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) || (**) ||DF(x)hn||+||R(hn)||| → 0, because the linear mapping DF(x) is continuous and for all large nЄ N, (***) ||R(hn) || ||R(hn) || ≤ → 0. ||hn|| (a) Explain in details why ||hn|| → 0. [3 Marks] (b) Explain the steps labelled (*), (**), (***). [6 Marks]arrow_forward4. In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm is differentiable at x = RN then F is continuous at x. Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want F(x), which means F is continuous at x. to show that F(xn) Denote hn xnx, so that ||hn||| 0. Thus we find ||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)|| (**) ||DF(x)hn|| + ||R(hn) || → 0, because the linear mapping DF(x) is continuous and for all large n = N, |||R(hn) || ≤ (***) ||R(hn)|| ||hn|| → 0. Explain the steps labelled (*), (**), (***) [6 Marks] (ii) Give an example of a function F: RR such that F is contin- Total marks 10 uous at x=0 but F is not differentiable at at x = 0. [4 Marks]arrow_forward
- 3. Let f R2 R be a function. (i) Explain in your own words the relationship between the existence of all partial derivatives of f and differentiability of f at a point x = R². (ii) Consider R2 → R defined by : [5 Marks] f(x1, x2) = |2x1x2|1/2 Show that af af -(0,0) = 0 and -(0, 0) = 0, Jx1 მx2 but f is not differentiable at (0,0). [10 Marks]arrow_forward(1) Write the following quadratic equation in terms of the vertex coordinates.arrow_forwardThe final answer is 8/π(sinx) + 8/3π(sin 3x)+ 8/5π(sin5x)....arrow_forward
- Keity x२ 1. (i) Identify which of the following subsets of R2 are open and which are not. (a) A = (2,4) x (1, 2), (b) B = (2,4) x {1,2}, (c) C = (2,4) x R. Provide a sketch and a brief explanation to each of your answers. [6 Marks] (ii) Give an example of a bounded set in R2 which is not open. [2 Marks] (iii) Give an example of an open set in R2 which is not bounded. [2 Marksarrow_forward2. (i) Which of the following statements are true? Construct coun- terexamples for those that are false. (a) sequence. Every bounded sequence (x(n)) nEN C RN has a convergent sub- (b) (c) (d) Every sequence (x(n)) nEN C RN has a convergent subsequence. Every convergent sequence (x(n)) nEN C RN is bounded. Every bounded sequence (x(n)) EN CRN converges. nЄN (e) If a sequence (xn)nEN C RN has a convergent subsequence, then (xn)nEN is convergent. [10 Marks] (ii) Give an example of a sequence (x(n))nEN CR2 which is located on the parabola x2 = x², contains infinitely many different points and converges to the limit x = (2,4). [5 Marks]arrow_forward2. (i) What does it mean to say that a sequence (x(n)) nEN CR2 converges to the limit x E R²? [1 Mark] (ii) Prove that if a set ECR2 is closed then every convergent sequence (x(n))nen in E has its limit in E, that is (x(n)) CE and x() x x = E. [5 Marks] (iii) which is located on the parabola x2 = = x x4, contains a subsequence that Give an example of an unbounded sequence (r(n)) nEN CR2 (2, 16) and such that x(i) converges to the limit x = (2, 16) and such that x(i) # x() for any i j. [4 Marksarrow_forward
- 1. (i) which are not. Identify which of the following subsets of R2 are open and (a) A = (1, 3) x (1,2) (b) B = (1,3) x {1,2} (c) C = AUB (ii) Provide a sketch and a brief explanation to each of your answers. [6 Marks] Give an example of a bounded set in R2 which is not open. (iii) [2 Marks] Give an example of an open set in R2 which is not bounded. [2 Marks]arrow_forward2. if limit. Recall that a sequence (x(n)) CR2 converges to the limit x = R² lim ||x(n)x|| = 0. 818 - (i) Prove that a convergent sequence (x(n)) has at most one [4 Marks] (ii) Give an example of a bounded sequence (x(n)) CR2 that has no limit and has accumulation points (1, 0) and (0, 1) [3 Marks] (iii) Give an example of a sequence (x(n))neN CR2 which is located on the hyperbola x2 1/x1, contains infinitely many different Total marks 10 points and converges to the limit x = (2, 1/2). [3 Marks]arrow_forward3. (i) Consider a mapping F: RN Rm. Explain in your own words the relationship between the existence of all partial derivatives of F and dif- ferentiability of F at a point x = RN. (ii) [3 Marks] Calculate the gradient of the following function f: R2 → R, f(x) = ||x||3, Total marks 10 where ||x|| = √√√x² + x/2. [7 Marks]arrow_forward
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