Discrete Mathematics With Applications
5th Edition
ISBN: 9780357035283
Author: EPP
Publisher: Cengage
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Chapter 4.5, Problem 41ES
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Prove that for all real numbers x and y,
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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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- The cup on the 9th hole of a golf course is located dead center in the middle of a circular green which is 40 feet in radius. Your ball is located as in the picture below. The ball follows a straight line path and exits the green at the right-most edge. Assume the ball travels 8 ft/sec. Introduce coordinates so that the cup is the origin of an xy-coordinate system and start by writing down the equations of the circle and the linear path of the ball. Provide numerical answers below with two decimal places of accuracy. 50 feet green ball 40 feet 9 cup ball path rough (a) The x-coordinate of the position where the ball enters the green will be (b) The ball will exit the green exactly seconds after it is hit. (c) Suppose that L is a line tangent to the boundary of the golf green and parallel to the path of the ball. Let Q be the point where the line is tangent to the circle. Notice that there are two possible positions for Q. Find the possible x-coordinates of Q: smallest x-coordinate =…arrow_forwardDraw the unit circle and plot the point P=(8,2). Observe there are TWO lines tangent to the circle passing through the point P. Answer the questions below with 3 decimal places of accuracy. P L1 L (a) The line L₁ is tangent to the unit circle at the point (b) The tangent line L₁ has equation: X + (c) The line L₂ is tangent to the unit circle at the point ( (d) The tangent line 42 has equation: y= x + ).arrow_forwardIntroduce yourself and describe a time when you used data in a personal or professional decision. This could be anything from analyzing sales data on the job to making an informed purchasing decision about a home or car. Describe to Susan how to take a sample of the student population that would not represent the population well. Describe to Susan how to take a sample of the student population that would represent the population well. Finally, describe the relationship of a sample to a population and classify your two samples as random, systematic, cluster, stratified, or convenience.arrow_forward
- Answersarrow_forwardWhat is a solution to a differential equation? We said that a differential equation is an equation that describes the derivative, or derivatives, of a function that is unknown to us. By a solution to a differential equation, we mean simply a function that satisfies this description. 2. Here is a differential equation which describes an unknown position function s(t): ds dt 318 4t+1, ds (a) To check that s(t) = 2t2 + t is a solution to this differential equation, calculate you really do get 4t +1. and check that dt' (b) Is s(t) = 2t2 +++ 4 also a solution to this differential equation? (c) Is s(t)=2t2 + 3t also a solution to this differential equation? ds 1 dt (d) To find all possible solutions, start with the differential equation = 4t + 1, then move dt to the right side of the equation by multiplying, and then integrate both sides. What do you get? (e) Does this differential equation have a unique solution, or an infinite family of solutions?arrow_forwardthese are solutions to a tutorial that was done and im a little lost. can someone please explain to me how these iterations function, for example i Do not know how each set of matrices produces a number if someine could explain how its done and provide steps it would be greatly appreciated thanks.arrow_forward
- Q1) Classify the following statements as a true or false statements a. Any ring with identity is a finitely generated right R module.- b. An ideal 22 is small ideal in Z c. A nontrivial direct summand of a module cannot be large or small submodule d. The sum of a finite family of small submodules of a module M is small in M A module M 0 is called directly indecomposable if and only if 0 and M are the only direct summands of M f. A monomorphism a: M-N is said to split if and only if Ker(a) is a direct- summand in M & Z₂ contains no minimal submodules h. Qz is a finitely generated module i. Every divisible Z-module is injective j. Every free module is a projective module Q4) Give an example and explain your claim in each case a) A module M which has two composition senes 7 b) A free subset of a modale c) A free module 24 d) A module contains a direct summand submodule 7, e) A short exact sequence of modules 74.arrow_forward************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.arrow_forwardProve that Σ prime p≤x p=3 (mod 10) 1 Ρ = for some constant A. log log x + A+O 1 log x "arrow_forward
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