Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Textbook Question
Chapter 4.5, Problem 11ES
Check the correctness of formula (4.5.1) given in Example 4.5.3 for the following values of DayT Example 4.5.3 for the following values of DayT and N.
- DayT =6 (Saturday) and N=15
- DayT =0 (Saturday) and N=7
- DayT =4 (Saturday) and N=12
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Chapter 4 Solutions
Discrete Mathematics With Applications
Ch. 4.1 - An integer is even if, and only if,_______.Ch. 4.1 - An integer is odd if, and only if,____Ch. 4.1 - An integer n is prime if, and only if,_______Ch. 4.1 - The most common way to disprove a universal...Ch. 4.1 - Prob. 5TYCh. 4.1 - To use the method of direct proof to prove a...Ch. 4.1 - In 1-4 justify your answer by using the...Ch. 4.1 - In 1-4 justify your answer by using by the...Ch. 4.1 - In 1-4 justify your answers by using the...Ch. 4.1 - In 1-4 justify your answers by using the...
Ch. 4.1 - Prove the statements in 5-11. There are integers m...Ch. 4.1 - Prove the statements in 5-11. There are distinct...Ch. 4.1 - Prove the statements in 5—11. 7. There are real...Ch. 4.1 - Prob. 8ESCh. 4.1 - Prove the statements in 5-11. There is a real...Ch. 4.1 - Prob. 10ESCh. 4.1 - Prove the statements in 5-11. There is an integer...Ch. 4.1 - In 12-13, (a) write a negation for the given...Ch. 4.1 - In 12-13, (a) write a negation for the given...Ch. 4.1 - Prob. 14ESCh. 4.1 - Disprove each of the statements in 14-16 by giving...Ch. 4.1 - Disprove each of the statements in 14-16 by giving...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - Prob. 21ESCh. 4.1 - Prove the statement is 21 and 22 by the method of...Ch. 4.1 - Prob. 23ESCh. 4.1 - Each of the statements in 23—26 is true. For each....Ch. 4.1 - Prob. 25ESCh. 4.1 - Prob. 26ESCh. 4.1 - Fill in the blanks in the following proof....Ch. 4.1 - In each of 28-31: a. Rewrite the theorem in three...Ch. 4.1 - In each of 28-31: a. Rewrite the theorem in three...Ch. 4.1 - In each of 28-31: a. Rewrite the theorem in three...Ch. 4.1 - Theorem 4,1-2: The sum of any even integer and...Ch. 4.2 - The meaning of every variable used in a proof...Ch. 4.2 - Proofs should be written in sentences that are...Ch. 4.2 - Every assertion in a proof should be supported by...Ch. 4.2 - Prob. 4TYCh. 4.2 - A new thought or fact that does not follow as an...Ch. 4.2 - Prob. 6TYCh. 4.2 - Displaying equations and inequalities increases...Ch. 4.2 - Some proof-writing mistakes are...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prob. 4ESCh. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prob. 7ESCh. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prob. 10ESCh. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove that the statements in 12—14 are false....Ch. 4.2 - Prove that the statements in 12—14 are false....Ch. 4.2 - Prove that the statements in 12-14 are false....Ch. 4.2 - Find the mistakes in the “proofs” shown in 15-19....Ch. 4.2 - Prob. 16ESCh. 4.2 - Prob. 17ESCh. 4.2 - Find the mistakes in the “proofs” show in 15-19....Ch. 4.2 - Find the mistakes in the “proofs” shown in 15-19....Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 23ESCh. 4.2 - Prob. 24ESCh. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 28ESCh. 4.2 - Prob. 29ESCh. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 32ESCh. 4.2 - Prob. 33ESCh. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 35ESCh. 4.2 - Prob. 36ESCh. 4.2 - Prob. 37ESCh. 4.2 - Prob. 38ESCh. 4.2 - Suppose that integers m and n are perfect squares....Ch. 4.2 - Prob. 40ESCh. 4.2 - Prob. 41ESCh. 4.3 - To show that a real number is rational, we must...Ch. 4.3 - Prob. 2TYCh. 4.3 - Prob. 3TYCh. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - Prob. 3ESCh. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The zero product property, says that if a product...Ch. 4.3 - Assume that a and b are both integers and that a0...Ch. 4.3 - Assume that m and n are both integers and that n0...Ch. 4.3 - Prove that every integer is a rational number.Ch. 4.3 - Prob. 12ESCh. 4.3 - Prob. 13ESCh. 4.3 - Consider the statement: The cube of any rational...Ch. 4.3 - Prob. 15ESCh. 4.3 - Determine which of the statements in 15—19 are...Ch. 4.3 - Prob. 17ESCh. 4.3 - Determine which of the statements in 15—19 are...Ch. 4.3 - Determine which of the statements in 15—19 are...Ch. 4.3 - Use the results of exercises 18 and 19 to prove...Ch. 4.3 - Prob. 21ESCh. 4.3 - Use the properties of even and odd integers that...Ch. 4.3 - Use the properties of even and odd integers that...Ch. 4.3 - Prob. 24ESCh. 4.3 - Derive the statements in 24-26 as corollaries of...Ch. 4.3 - Derive the statements in 24-26 as corollaries of...Ch. 4.3 - It is a fact that if n is any nonnegative integer,...Ch. 4.3 - Suppose a, b, c, and d are integers and ac ....Ch. 4.3 - Suppose a,b, and c are integers and x,y and z are...Ch. 4.3 - Prove that one solution for a quadratic equation...Ch. 4.3 - Prob. 31ESCh. 4.3 - Prove that for every real number c, if c is a root...Ch. 4.3 - Use the properties of even and odd integers that...Ch. 4.3 - Prob. 34ESCh. 4.3 - Prob. 35ESCh. 4.3 - In 35-39 find the mistakes in the “proofs” that...Ch. 4.3 - Prob. 37ESCh. 4.3 - In 35-39 find the mistakes in the "proofs” that...Ch. 4.3 - In 35-39 find the mistakes in the “proofs” that...Ch. 4.4 - TO show that a nonzero integer d divides an...Ch. 4.4 - To say that d divides n means the same as saying...Ch. 4.4 - Prob. 3TYCh. 4.4 - Prob. 4TYCh. 4.4 - Prob. 5TYCh. 4.4 - The transitivity of divisibility theorem says that...Ch. 4.4 - Prob. 7TYCh. 4.4 - Prob. 8TYCh. 4.4 - Prob. 1ESCh. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Prob. 3ESCh. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Prob. 6ESCh. 4.4 - Prob. 7ESCh. 4.4 - Prob. 8ESCh. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Prob. 10ESCh. 4.4 - Prob. 11ESCh. 4.4 - Prob. 12ESCh. 4.4 - Give a reason for your answer in each of 1—13....Ch. 4.4 - Fill in the blanks in the following proof that for...Ch. 4.4 - Prove statements 15 and 16 directly from the the...Ch. 4.4 - Prob. 16ESCh. 4.4 - Prob. 17ESCh. 4.4 - Consider the following statement: The negative of...Ch. 4.4 - Show that the following statement is false: For...Ch. 4.4 - Prob. 20ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - Prob. 22ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - Prob. 24ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - Prob. 26ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statements in 20-32, determine whether...Ch. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statement in 20—32, determine whether the...Ch. 4.4 - Prob. 33ESCh. 4.4 - Consider a string consisting of a’s, b’s, and c’s...Ch. 4.4 - Two athletes run a circular track at a steady pace...Ch. 4.4 - It can be shown (see exercises 44-48) that an...Ch. 4.4 - Use the unique factorization theorem to write the...Ch. 4.4 - Let n=8,424. Write the prime factorization for n....Ch. 4.4 - Prob. 39ESCh. 4.4 - Prob. 40ESCh. 4.4 - How many zeros are at the end of 458.885 ? Explain...Ch. 4.4 - Prob. 42ESCh. 4.4 - At a certain university 2/3 of the mathematics...Ch. 4.4 - Prove that if n is any nonnegative integer whose...Ch. 4.4 - Prove that if n is any nonnegative nonnegative...Ch. 4.4 - Prob. 46ESCh. 4.4 - Prob. 47ESCh. 4.4 - Prove that for any nonnegative integer n, if the...Ch. 4.4 - Prob. 49ESCh. 4.4 - The integer 123,123 has the form abc, abc, where...Ch. 4.5 - The quotient-remainder theorem says that for all...Ch. 4.5 - Prob. 2TYCh. 4.5 - Prob. 3TYCh. 4.5 - Prob. 4TYCh. 4.5 - Prob. 5TYCh. 4.5 - Prob. 6TYCh. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - Prob. 5ESCh. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - Evalute the expressions in 7-10 43div9 43mod9Ch. 4.5 - Evalute the expressions in7-10 50div7 50mod7Ch. 4.5 - Evalute the expressions in7-10 28div5 28mod5Ch. 4.5 - Prob. 10ESCh. 4.5 - Check the correctness of formula (4.5.1) given in...Ch. 4.5 - Justify formula (4.5.1) for general values of DayT...Ch. 4.5 - On a Monday a friend says he will meet you again...Ch. 4.5 - If today isTuesday, what day of the week will it...Ch. 4.5 - January 1,2000, was a Saturday, and 2000 was a...Ch. 4.5 - Prob. 16ESCh. 4.5 - Prove directky from the definitions that for every...Ch. 4.5 - Prove that the product of any two consecutive...Ch. 4.5 - Prove directly from the definitions that for all...Ch. 4.5 - Prob. 20ESCh. 4.5 - Suppose b is any integer. If bmod12=5 , what is...Ch. 4.5 - Suppose c is any integer. If c mod 15=3 , what is...Ch. 4.5 - Prove that for every integer n, if mod 5=3 then...Ch. 4.5 - Prove that for all integers m and n, if m mod 5=2...Ch. 4.5 - Prove that for all integrs a and b, if a mod 7=5...Ch. 4.5 - Prove that a necessary and sufficient and...Ch. 4.5 - Use the quotient-remainder theorem with divisor...Ch. 4.5 - Prove: Given any set of three consecutive...Ch. 4.5 - Use the quotient-remainder theorem with divisor...Ch. 4.5 - Use the quotient-remainder theorem with divisor...Ch. 4.5 - In 31-33, you may use the properties listed in...Ch. 4.5 - In 31-33, yoy may use the properties listed in...Ch. 4.5 - In 31-33, you may use the properties listed in...Ch. 4.5 - Given any integer n, if n3 , could n, n+2 , and...Ch. 4.5 - Prob. 35ESCh. 4.5 - Prove each of the statements in 35-43. The product...Ch. 4.5 - Prove each of the statements in 35-43. For any...Ch. 4.5 - Prove of the statements in 35-43. For every...Ch. 4.5 - Prove each of the statement in 35-43. Every prime...Ch. 4.5 - Prob. 40ESCh. 4.5 - Prob. 41ESCh. 4.5 - Prove each of the statements if 35-43. For all...Ch. 4.5 - Prob. 43ESCh. 4.5 - A matrix M has 3 rows and 4 columns. [ a 11 a 12 a...Ch. 4.5 - Prob. 45ESCh. 4.5 - Prob. 46ESCh. 4.5 - If m, n, and d are integers, d0 , and d(mn) , what...Ch. 4.5 - Prob. 48ESCh. 4.5 - Prob. 49ESCh. 4.5 - Prob. 50ESCh. 4.6 - Given any real number x, the floor of x is the...Ch. 4.6 - Prob. 2TYCh. 4.6 - Prob. 1ESCh. 4.6 - Compute x and x for each of the values of x in...Ch. 4.6 - Prob. 3ESCh. 4.6 - Compute x and x for each of the values of x in...Ch. 4.6 - Use the floor notation to express 259 div 11 and...Ch. 4.6 - If k is an integer, what is [k]? Why?Ch. 4.6 - If k is an integer, what is [k+12] ? Why?Ch. 4.6 - Prob. 8ESCh. 4.6 - Prob. 9ESCh. 4.6 - Prob. 10ESCh. 4.6 - Prob. 11ESCh. 4.6 - Prob. 12ESCh. 4.6 - Prob. 13ESCh. 4.6 - Prob. 14ESCh. 4.6 - Prob. 15ESCh. 4.6 - Some of the statements in 15-22 are true and some...Ch. 4.6 - Prob. 17ESCh. 4.6 - Prob. 18ESCh. 4.6 - Some of the statements is 15-22 are ture and some...Ch. 4.6 - Prob. 20ESCh. 4.6 - Prob. 21ESCh. 4.6 - Prob. 22ESCh. 4.6 - Prob. 23ESCh. 4.6 - Prob. 24ESCh. 4.6 - Prob. 25ESCh. 4.6 - Prob. 26ESCh. 4.6 - Prob. 27ESCh. 4.6 - Prob. 28ESCh. 4.6 - Prove each of the statements in 23-33. 29. For any...Ch. 4.6 - Prob. 30ESCh. 4.6 - Prob. 31ESCh. 4.6 - Prob. 32ESCh. 4.6 - Prob. 33ESCh. 4.7 - To prove a statement by contradiction, you suppose...Ch. 4.7 - Prob. 2TYCh. 4.7 - Prob. 3TYCh. 4.7 - Fill in the blanks in the following proof by...Ch. 4.7 - Is 10 an irrational numbre? Explain.Ch. 4.7 - Prob. 3ESCh. 4.7 - Use proof by contradiction to show that for every...Ch. 4.7 - Prob. 5ESCh. 4.7 - Prob. 6ESCh. 4.7 - Carefully formulate the negations of each of the...Ch. 4.7 - Fill in the blanks for the following proof that...Ch. 4.7 - a. When asked to prove that the difference of any...Ch. 4.7 - Let S be the statement: For all positive real...Ch. 4.7 - Let T be the statement: The sum of any two...Ch. 4.7 - Let R be the statement: The square root of any...Ch. 4.7 - Let S be the statement: The product of any...Ch. 4.7 - Let T be the statements: For every integer a, if...Ch. 4.7 - Do there exist integers a,b, and c such that a,b,...Ch. 4.7 - Prove each staement in 16-19 by contradiction. For...Ch. 4.7 - Prob. 17ESCh. 4.7 - Prove each statemtent in 16-19 by contradiction....Ch. 4.7 - Prove each statemet in 16-19 by contradiction. For...Ch. 4.7 - Fill in the blanks in the following proof by...Ch. 4.7 - Consider the statement “For everyinteger n, if n2...Ch. 4.7 - Consider the statement “For every real number r,...Ch. 4.7 - Prob. 23ESCh. 4.7 - Prove each of the statement in 23-24 in two ways:...Ch. 4.7 - Prob. 25ESCh. 4.7 - Use any method to prove the statements in 26-29....Ch. 4.7 - Use any method to prove the statements in 26-29....Ch. 4.7 - Use any method to prove the statements in 26-29....Ch. 4.7 - Prob. 29ESCh. 4.7 - Let n=53. Find an approximate value for n and...Ch. 4.7 - a. Prove by contraposition: For all positive...Ch. 4.7 - Prob. 32ESCh. 4.7 - The sieve of Eratosthenes, name after its...Ch. 4.7 - Prob. 34ESCh. 4.7 - Use proof by contradiction to show that every...Ch. 4.7 - Prob. 36ESCh. 4.8 - The ancient Greeks discovered that in a right...Ch. 4.8 - One way to prove that 2 is an irrational number is...Ch. 4.8 - One way to prove that there are infinitely many...Ch. 4.8 - Prob. 1ESCh. 4.8 - Prob. 2ESCh. 4.8 - Prob. 3ESCh. 4.8 - Prob. 4ESCh. 4.8 - Let S be the statement: The cube root of every...Ch. 4.8 - Prob. 6ESCh. 4.8 - Prob. 7ESCh. 4.8 - Prob. 8ESCh. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Prob. 10ESCh. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Prob. 14ESCh. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Prob. 16ESCh. 4.8 - Prob. 17ESCh. 4.8 - a. Prove that for every integer a, if a3 is even...Ch. 4.8 - Use proof by contradiction to show that for any...Ch. 4.8 - Prob. 20ESCh. 4.8 - Prob. 21ESCh. 4.8 - Prove that 5 is irrational.Ch. 4.8 - Prob. 23ESCh. 4.8 - Prob. 24ESCh. 4.8 - Use the proof technique illustrated in exercise 24...Ch. 4.8 - Prob. 26ESCh. 4.8 - Prob. 27ESCh. 4.8 - Prob. 28ESCh. 4.8 - Suppose a is an integer and p is a prime number...Ch. 4.8 - Let p1,p2,p3,... be a list of all prime numbers in...Ch. 4.8 - Prob. 31ESCh. 4.8 - Prob. 32ESCh. 4.8 - Prove that if p1,p2...., and pn are distinct prime...Ch. 4.8 - Prob. 34ESCh. 4.8 - Prob. 35ESCh. 4.8 - Prob. 36ESCh. 4.8 - Prob. 37ESCh. 4.8 - Prob. 38ESCh. 4.9 - The toatl degree of a graph is defined as_____Ch. 4.9 - Prob. 2TYCh. 4.9 - In any graph the number of vertices of odd degree...Ch. 4.9 - Prob. 4TYCh. 4.9 - Prob. 5TYCh. 4.9 - Prob. 6TYCh. 4.9 - Prob. 1ESCh. 4.9 - Prob. 2ESCh. 4.9 - A graph has vertices of degrees 0,2,2,3, and 9....Ch. 4.9 - A graph has vertices of degrees ,1,1,4,4, and 6....Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5—13 either draw a graph with the...Ch. 4.9 - Prob. 12ESCh. 4.9 - Prob. 13ESCh. 4.9 - Prob. 14ESCh. 4.9 - A small social network contains three people who...Ch. 4.9 - a. In a group of 15 people, is it possible for...Ch. 4.9 - In a group of 25 people, is it possible for each...Ch. 4.9 - Is there a simple graph, each of whose vertices...Ch. 4.9 - Prob. 19ESCh. 4.9 - Draw K6, a complete graph on six vertices. Use the...Ch. 4.9 - In a simple graph, must every vertex have degree...Ch. 4.9 - Prob. 22ESCh. 4.9 - Recall that Km,n denotes a complete bipartite...Ch. 4.9 - A (general) bipartite graph G is a simple graph...Ch. 4.9 - Prob. 25ESCh. 4.10 - When an algorithm statement of the form x:=e is...Ch. 4.10 - Consider an algorithm statement of the following...Ch. 4.10 - Prob. 3TYCh. 4.10 - Prob. 4TYCh. 4.10 - Given a nonnegative integer a and a positive...Ch. 4.10 - Prob. 6TYCh. 4.10 - If r is a positive integer, then gcd (r,0)=_____Ch. 4.10 - Prob. 8TYCh. 4.10 - Prob. 9TYCh. 4.10 - Find the value of z when each of the algorithm...Ch. 4.10 - Prob. 2ESCh. 4.10 - Consider the following algorithm segment:...Ch. 4.10 - Prob. 4ESCh. 4.10 - Prob. 5ESCh. 4.10 - Prob. 6ESCh. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Prob. 8ESCh. 4.10 - Prob. 9ESCh. 4.10 - Prob. 10ESCh. 4.10 - Prob. 11ESCh. 4.10 - Prob. 12ESCh. 4.10 - Prob. 13ESCh. 4.10 - Use the Euclidean algorithm to hand-calculate the...Ch. 4.10 - Use the Euclidean algorithm to hand-calculate the...Ch. 4.10 - Use the Euclidean algorithm to hand-calculate the...Ch. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Prob. 20ESCh. 4.10 - Prob. 21ESCh. 4.10 - Prove that for all positive integers a and b, a|b...Ch. 4.10 - Prove that if a and b are integers, not both zero,...Ch. 4.10 - Prob. 24ESCh. 4.10 - Prob. 25ESCh. 4.10 - Prob. 26ESCh. 4.10 - An alternative to the Euclidean algorithm uses...Ch. 4.10 - Prob. 28ESCh. 4.10 - Prob. 29ESCh. 4.10 - Prob. 30ESCh. 4.10 - Exercises 28—32 refer to the following definition....Ch. 4.10 - Prob. 32ES
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- For each of the time series, construct a line chart of the data and identify the characteristics of the time series (that is, random, stationary, trend, seasonal, or cyclical). Date IBM9/7/2010 $125.959/8/2010 $126.089/9/2010 $126.369/10/2010 $127.999/13/2010 $129.619/14/2010 $128.859/15/2010 $129.439/16/2010 $129.679/17/2010 $130.199/20/2010 $131.79 a. Construct a line chart of the closing stock prices data. Choose the correct chart below.arrow_forward1) Express these large and small numbers from the Read and Study section in scientific notation: (a) 239,000 miles (b) 3,800,000,000,000 sheets of paper (c) 0.0000000000000000000000167 grams 2) Find all values for the variable x that make these equations true. (a) 5x = 1 (b) 3x = 1/1 9 (c) 4* = 11/ 4 (e) 4* = 64 (g) 10x = 1,000,000 (d) 3x=-3 (f) 2x = = 8 (h) 10x = 0.001arrow_forward(b) 4) Find an equation to fit each of the following graphs: (a) 20 20 18 16 14 12 10 8 6 4 2 24 22 20 18 16 14 12 10 8 16 A 2 -3 -2 -1-0 2 3 4. -1 0 1 2 3. -2 -2arrow_forward
- 3) Which of the following are equivalent to 3? (There may be more than one that is equivalent!) -1 (a) (9)¯¹ 3. (b) (-3)-1 (c) (-3) -1 (d) -(¯3) (e) 11 3-1 (f) 3-4arrow_forwardY- ___b=_____ (X- )arrow_forwardFor each of the time series, construct a line chart of the data and identify the characteristics of the time series (that is, random, stationary, trend, seasonal, or cyclical) Date IBM9/7/2010 $125.959/8/2010 $126.089/9/2010 $126.369/10/2010 $127.999/13/2010 $129.619/14/2010 $128.859/15/2010 $129.439/16/2010 $129.679/17/2010 $130.199/20/2010 $131.79arrow_forward
- 5) State any theorems that you use in determining your solution. a) Suppose you are given a model with two explanatory variables such that: Yi = a +ẞ1x1 + ẞ2x2i + Ui, i = 1, 2, ... n Using partial differentiation derive expressions for the intercept and slope coefficients for the model above. [25 marks] b) A production function is specified as: Yi = α + B₁x1i + ẞ2x2i + Ui, i = 1, 2, ... n, u₁~N(0,σ²) where: y = log(output), x₁ = log(labor input), x2 = log(capital input) The results are as follows: x₁ = 10, x2 = 5, ỹ = 12, S11 = 12, S12= 8, S22 = 12, S₁y = 10, = 8, Syy = 10, S2y n = 23 (individual firms) i) Compute values for the intercept, the slope coefficients and σ². [20 marks] ii) Show that SE (B₁) = 0.102. [15 marks] iii) Test the hypotheses: ẞ1 = 1 and B2 = 0, separately at the 5% significance level. You may take without calculation that SE (a) = 0.78 and SE (B2) = 0.102 [20 marks] iv) Find a 95% confidence interval for the estimate ẞ2. [20 marks]arrow_forwardPage < 2 of 2 - ZOOM + The set of all 3 x 3 upper triangular matrices 6) Determine whether each of the following sets, together with the standard operations, is a vector space. If it is, then simply write 'Vector space'. You do not have to prove all ten vector space axioms. If it is not, then identify one of the ten vector space axioms with its number in the attached sheet that fails and also show that how it fails. a) The set of all polynomials of degree four or less. b) The set of all 2 x 2 singular matrices. c) The set {(x, y) : x ≥ 0, y is a real number}. d) C[0,1], the set of all continuous functions defined on the interval [0,1]. 7) Given u = (-2,1,1) and v = (4,2,0) are two vectors in R³-space. Find u xv and show that it is orthogonal to both u and v. 8) a) Find the equation of the least squares regression line for the data points below. (-2,0), (0,2), (2,2) b) Graph the points and the line that you found from a) on the same Cartesian coordinate plane.arrow_forward1. A consumer group claims that the mean annual consumption of cheddar cheese by a person in the United States is at most 10.3 pounds. A random sample of 100 people in the United States has a mean annual cheddar cheese consumption of 9.9 pounds. Assume the population standard deviation is 2.1 pounds. At a = 0.05, can you reject the claim? (Adapted from U.S. Department of Agriculture) State the hypotheses: Calculate the test statistic: Calculate the P-value: Conclusion (reject or fail to reject Ho): 2. The CEO of a manufacturing facility claims that the mean workday of the company's assembly line employees is less than 8.5 hours. A random sample of 25 of the company's assembly line employees has a mean workday of 8.2 hours. Assume the population standard deviation is 0.5 hour and the population is normally distributed. At a = 0.01, test the CEO's claim. State the hypotheses: Calculate the test statistic: Calculate the P-value: Conclusion (reject or fail to reject Ho): Statisticsarrow_forward
- Page < 1 of 2 - ZOOM + 1) a) Find a matrix P such that PT AP orthogonally diagonalizes the following matrix A. = [{² 1] A = b) Verify that PT AP gives the correct diagonal form. 2 01 -2 3 2) Given the following matrices A = -1 0 1] an and B = 0 1 -3 2 find the following matrices: a) (AB) b) (BA)T 3) Find the inverse of the following matrix A using Gauss-Jordan elimination or adjoint of the matrix and check the correctness of your answer (Hint: AA¯¹ = I). [1 1 1 A = 3 5 4 L3 6 5 4) Solve the following system of linear equations using any one of Cramer's Rule, Gaussian Elimination, Gauss-Jordan Elimination or Inverse Matrix methods and check the correctness of your answer. 4x-y-z=1 2x + 2y + 3z = 10 5x-2y-2z = -1 5) a) Describe the zero vector and the additive inverse of a vector in the vector space, M3,3. b) Determine if the following set S is a subspace of M3,3 with the standard operations. Show all appropriate supporting work.arrow_forwardFind the Laplace Transform of the function to express it in frequency domain form.arrow_forwardPlease draw a graph that represents the system of equations f(x) = x2 + 2x + 2 and g(x) = –x2 + 2x + 4?arrow_forward
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