DISCRETE MATHEMATICS LOOSELEAF W/CONNECT
8th Edition
ISBN: 9781264309405
Author: ROSEN
Publisher: MCG
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 8.1, Problem 50E
To determine
To show:
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
18.9. Let denote the boundary of the rectangle whose vertices are
-2-2i, 2-21, 2+i and -2+i in the positive direction. Evaluate each of
the following integrals:
(a).
之一
dz, (b).
dz, (b).
COS 2
coz dz,
dz
(z+1)
(d).
z 2 +2
dz, (e).
(c). (2z+1)zdz,
z+
1
(f). £,
· [e² sin = + (2² + 3)²] dz.
(2+3)2
We consider the one-period model studied in class as an example. Namely, we assumethat the current stock price is S0 = 10. At time T, the stock has either moved up toSt = 12 (with probability p = 0.6) or down towards St = 8 (with probability 1−p = 0.4).We consider a call option on this stock with maturity T and strike price K = 10. Theinterest rate on the money market is zero.As in class, we assume that you, as a customer, are willing to buy the call option on100 shares of stock for $120. The investor, who sold you the option, can adopt one of thefollowing strategies: Strategy 1: (seen in class) Buy 50 shares of stock and borrow $380. Strategy 2: Buy 55 shares of stock and borrow $430. Strategy 3: Buy 60 shares of stock and borrow $480. Strategy 4: Buy 40 shares of stock and borrow $280.(a) For each of strategies 2-4, describe the value of the investor’s portfolio at time 0,and at time T for each possible movement of the stock.(b) For each of strategies 2-4, does the investor have…
eric
pez
Xte
in
z=
Therefore, we have
(x, y, z)=(3.0000,
83.6.1 Exercise
Gauss-Seidel iteration with
Start with (x, y, z) = (0, 0, 0). Use the convergent Jacobi i
Tol=10 to solve the following systems:
1.
5x-y+z = 10
2x-8y-z=11
-x+y+4z=3
iteration (x
Assi 2
Assi 3.
4.
x-5y-z=-8
4x-y- z=13
2x - y-6z=-2
4x y + z = 7
4x-8y + z = -21
-2x+ y +5z = 15
4x + y - z=13
2x - y-6z=-2
x-5y- z=-8
realme Shot on realme C30
2025.01.31 22:35
f
Chapter 8 Solutions
DISCRETE MATHEMATICS LOOSELEAF W/CONNECT
Ch. 8.1 - Use mathematical induction to verify the formula...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - A vending machine dispensing books of stamps...Ch. 8.1 - A country uses as currency coins with values of 1...Ch. 8.1 - How many was are there to pay a bill of 17 pesos...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...
Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - Messages are transmitted over a communications...Ch. 8.1 - A bus driver pays all tolls, using only nickels...Ch. 8.1 - a) Find the recurrence relation satisfied by Rn,...Ch. 8.1 - a) Find the recurrence relation satisfied by Rn,...Ch. 8.1 - a) Find the recurrence relation satisfied by Sn,...Ch. 8.1 - Find a recurrence relation for the number of bit...Ch. 8.1 - How many bit sequences of length seven contain an...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - a) Find a recurrence relation for the number of...Ch. 8.1 - Show that the Fibonacci numbers satisfy the...Ch. 8.1 - Prob. 29ECh. 8.1 - Prob. 30ECh. 8.1 - a) Use the recurrence relation developed in...Ch. 8.1 - In the Tower of Hanoi puzzle, suppose our goal is...Ch. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Prob. 35ECh. 8.1 - Exercises 33-37 deal with a variation of the...Ch. 8.1 - Prob. 37ECh. 8.1 - Prob. 38ECh. 8.1 - Show that the Reve’s puzzle with four disks can be...Ch. 8.1 - Prob. 40ECh. 8.1 - Show that if R(n) is the number of moves used by...Ch. 8.1 - Prob. 42ECh. 8.1 - Prob. 43ECh. 8.1 - Prob. 44ECh. 8.1 - Prob. 45ECh. 8.1 - Prob. 46ECh. 8.1 - Prob. 47ECh. 8.1 - Prob. 48ECh. 8.1 - Show that an2=an2an+2an .Ch. 8.1 - Prob. 50ECh. 8.1 - Prob. 51ECh. 8.1 - Prob. 52ECh. 8.1 - Construct the algorithm described in the text...Ch. 8.1 - Use Algorithm 1 to determine the maximum number of...Ch. 8.1 - For each part of Exercise 54, use your algorithm...Ch. 8.1 - In this exercise we will develop a dynamic...Ch. 8.1 - Dynamic programming can be used to develop an...Ch. 8.2 - Determine which of these are linear homogeneous...Ch. 8.2 - Determine which of these are linear homogeneous...Ch. 8.2 - Solve these recurrence relations together with the...Ch. 8.2 - Solve these recurrence relations together with the...Ch. 8.2 - Prob. 5ECh. 8.2 - Prob. 6ECh. 8.2 - Prob. 7ECh. 8.2 - A model for the number of lobsters caught per year...Ch. 8.2 - Prob. 9ECh. 8.2 - Prob. 10ECh. 8.2 - The Lucas numbers satisfy the recurrence relation...Ch. 8.2 - Find the solution to an=2an1+an2+2an3 for n = 3,4,...Ch. 8.2 - Find the solution to an=7an2+6an3 with a0=9,a1=10...Ch. 8.2 - Find the solution to an=5an24an4 with...Ch. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prove this identity relating the Fibonacci numbers...Ch. 8.2 - Solve the recurrence relation an=6an112an2+8an3...Ch. 8.2 - Prob. 19ECh. 8.2 - Prob. 20ECh. 8.2 - Prob. 21ECh. 8.2 - What is the general form of the solutions of a...Ch. 8.2 - Consider the nonhomogeneous linear recurrence...Ch. 8.2 - Consider the nonhomogeneous linear recurrence...Ch. 8.2 - a) Determine values of the constants A and B such...Ch. 8.2 - What is the general form of the particular...Ch. 8.2 - What is the general form of the particular...Ch. 8.2 - a) Find all solutions of the recurrence relation...Ch. 8.2 - Prob. 29ECh. 8.2 - Prob. 30ECh. 8.2 - Find all solutions of the recurrence relation...Ch. 8.2 - Find the solution of the recurrence relation...Ch. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Find the solution of the recurrence relation...Ch. 8.2 - Prob. 36ECh. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.2 - Prob. 39ECh. 8.2 - Solve the simultaneous recurrence relations...Ch. 8.2 - Prob. 41ECh. 8.2 - Prob. 42ECh. 8.2 - Prob. 43ECh. 8.2 - Prob. 44ECh. 8.2 - Prob. 45ECh. 8.2 - Suppose that there are two goats on an island...Ch. 8.2 - Prob. 47ECh. 8.2 - Prob. 48ECh. 8.2 - Use Exercise 48 to solve the recurrence relation...Ch. 8.2 - It can be shown that Cn, the average number of...Ch. 8.2 - Prob. 51ECh. 8.2 - Prob. 52ECh. 8.2 - Prob. 53ECh. 8.3 - How many comparisons are needed for a binary...Ch. 8.3 - Prob. 2ECh. 8.3 - Multiply (1110)2 and (1010)2 using the fast...Ch. 8.3 - Express the fast multiplication algorithm in...Ch. 8.3 - Determine a value for the constant C in Example...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Suppose that f(n)=2f(n/2)+3 when is an even...Ch. 8.3 - Prob. 9ECh. 8.3 - Find f(n) when n=2k , where f satisfies the...Ch. 8.3 - Give a big-O estimate for the function f in...Ch. 8.3 - Find f(n) when n=3k , where f satisfies the...Ch. 8.3 - Give a big-O estimate for the function f in...Ch. 8.3 - Suppose that there are n=2k terms in an...Ch. 8.3 - How many rounds are in the elimination tournament...Ch. 8.3 - Prob. 16ECh. 8.3 - Suppose that the votes of n people for different...Ch. 8.3 - Suppose that each person in a group of n people...Ch. 8.3 - a) Set up a divide-and-conquer recurrence relation...Ch. 8.3 - a) Set up a divide-and-conquer recurrence relation...Ch. 8.3 - Suppose that the function f satisfies the...Ch. 8.3 - Suppose that the function f satisfies the...Ch. 8.3 - This exercise deals with the problem of finding...Ch. 8.3 - Apply the algorithm described in Example 12 for...Ch. 8.3 - Apply the algorithm described in Example 12 for...Ch. 8.3 - Use pseudocode to describe the recursive algorithm...Ch. 8.3 - Prob. 27ECh. 8.3 - Prob. 28ECh. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - Prob. 30ECh. 8.3 - Prob. 31ECh. 8.3 - Prob. 32ECh. 8.3 - Prob. 33ECh. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.3 - In Exercises 29-33, assume that f is an increasing...Ch. 8.4 - Find the generating function for the finite...Ch. 8.4 - Find the generating function for the finite...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - Prob. 5ECh. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - In Exercises 3-8, by a closed form we mean an...Ch. 8.4 - Find the coefficient of x10in the power series of...Ch. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to determine the number...Ch. 8.4 - Use generating functions to find the number of...Ch. 8.4 - In how many ways can 25 identical donuts be...Ch. 8.4 - Use generating functions to find the number of...Ch. 8.4 - Prob. 19ECh. 8.4 - Prob. 20ECh. 8.4 - Prob. 21ECh. 8.4 - Prob. 22ECh. 8.4 - Prob. 23ECh. 8.4 - Prob. 24ECh. 8.4 - Explain how generating functions can be used to...Ch. 8.4 - Explain how generating functions can be used to...Ch. 8.4 - Prob. 27ECh. 8.4 - Prob. 28ECh. 8.4 - Use generating functions (and a computer algebra...Ch. 8.4 - Use generating functions (and a computer algebra...Ch. 8.4 - Prob. 31ECh. 8.4 - If G(x) is the generating function for the...Ch. 8.4 - Prob. 33ECh. 8.4 - Prob. 34ECh. 8.4 - Prob. 35ECh. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Prob. 37ECh. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Use generating functions to solve the recurrence...Ch. 8.4 - Prob. 40ECh. 8.4 - Prob. 41ECh. 8.4 - Prob. 42ECh. 8.4 - (Calculus required) Let {Cn}be the sequence of...Ch. 8.4 - Use generating functions to prove Pascal’s...Ch. 8.4 - Use generating functions to prove Vandermonde’s...Ch. 8.4 - Prob. 46ECh. 8.4 - Prob. 47ECh. 8.4 - Prob. 48ECh. 8.4 - Find the sequence with each of these functions as...Ch. 8.4 - Find the sequence with each of these functions as...Ch. 8.4 - A coding system encodes messages using strings of...Ch. 8.4 - A coding system encodes messages using strings of...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Prob. 55ECh. 8.4 - Prob. 56ECh. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Generating functions are useful in studying the...Ch. 8.4 - Suppose that X is a random variable on a sample...Ch. 8.4 - Prob. 60ECh. 8.4 - Prob. 61ECh. 8.4 - Show that if X and Y are independent random...Ch. 8.5 - How many elements are in A1A2 if there are 12...Ch. 8.5 - There are 345 students at a college who have taken...Ch. 8.5 - A survey of households in the United States...Ch. 8.5 - A marketing report concerning personal computers...Ch. 8.5 - Find the number of elements A1A2A3 if there are...Ch. 8.5 - Prob. 6ECh. 8.5 - There are 2504 computer science students at a...Ch. 8.5 - In a survey of 270 college students, it is found...Ch. 8.5 - How many students are enrolled in a course either...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - Find the number of positive integers not exceeding...Ch. 8.5 - How many swings of length eight do not contain six...Ch. 8.5 - How many permutations of the 26 letters of the...Ch. 8.5 - How many permutations of the 10 digits either...Ch. 8.5 - Prob. 18ECh. 8.5 - Prob. 19ECh. 8.5 - How many terms are there in the formula for the...Ch. 8.5 - Prob. 21ECh. 8.5 - Prob. 22ECh. 8.5 - Prob. 23ECh. 8.5 - Prob. 24ECh. 8.5 - Let E1, E2 ,and E3 be three events from a sample...Ch. 8.5 - Prob. 26ECh. 8.5 - Find the probability that when four numbers from 1...Ch. 8.5 - Prob. 28ECh. 8.5 - Prob. 29ECh. 8.5 - Prob. 30ECh. 8.5 - Prob. 31ECh. 8.6 - Suppose that in a bushel of 100 apples there are...Ch. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 8.6 - Prob. 4ECh. 8.6 - Find the number of primes less than 200 using the...Ch. 8.6 - Prob. 6ECh. 8.6 - How many positive integers less than 10,000 are...Ch. 8.6 - Prob. 8ECh. 8.6 - How many ways are there to distribute six...Ch. 8.6 - In how many ways can eight distinct balls be...Ch. 8.6 - In how many ways can seven different jobs be...Ch. 8.6 - List all the derangements of {1, 2,3, 4}.Ch. 8.6 - Prob. 13ECh. 8.6 - Prob. 14ECh. 8.6 - A machine that inserts letters into envelopes goes...Ch. 8.6 - A group of n students is assigned seats for each...Ch. 8.6 - Prob. 17ECh. 8.6 - Prob. 18ECh. 8.6 - Prob. 19ECh. 8.6 - Prob. 20ECh. 8.6 - For which positive integers n is Dn, the number of...Ch. 8.6 - Prob. 22ECh. 8.6 - Prob. 23ECh. 8.6 - Prob. 24ECh. 8.6 - Prob. 25ECh. 8.6 - How many derangements of {1, 2, 3, 4, 5, 6} end...Ch. 8.6 - Prove Theorem 1.Ch. 8 - a) What is a recurrence re1aon? b) Find a...Ch. 8 - Explain how the Fibonacci numbers are used to...Ch. 8 - a) Find a recurrence relation for the number of...Ch. 8 - Prob. 6RQCh. 8 - a) Explain how to solve linear homogeneous...Ch. 8 - Prob. 8RQCh. 8 - Prob. 9RQCh. 8 - a) Give a formula for the number of elements in...Ch. 8 - a) Give a formula for the number of elements in...Ch. 8 - Prob. 12RQCh. 8 - Explain how the principle of inclusion-exclusion...Ch. 8 - Prob. 14RQCh. 8 - Prob. 15RQCh. 8 - a) Define a derangement. b) Why is counting the...Ch. 8 - A group of 10 people begin a chain letter, with...Ch. 8 - A nuclear reactor has created 18 grams of a...Ch. 8 - Every hour the U.S. government prints 10,000 more...Ch. 8 - Suppose that every hour there are two new bacteria...Ch. 8 - Messages are sent over a communications channel...Ch. 8 - Prob. 6SECh. 8 - How many ways are there to form these postages...Ch. 8 - Prob. 8SECh. 8 - Solve the recurrence relation an=a2n1/bn2 if a0=1...Ch. 8 - Prob. 10SECh. 8 - Find the solution of the recurrence relation...Ch. 8 - Prob. 12SECh. 8 - Prob. 13SECh. 8 - Prob. 14SECh. 8 - Prob. 15SECh. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - In Exercises 15-18 we develop a dynamic...Ch. 8 - Find the solution to the recurrence relation...Ch. 8 - Find the solution to the recurrence relation...Ch. 8 - Give a big-O estimate for the size of f in...Ch. 8 - Find a recurrence relation that describes the...Ch. 8 - Prob. 23SECh. 8 - Prob. 24SECh. 8 - Prob. 25SECh. 8 - Find an where a) an=3 . b) an=4n+7 . c) an=n2+n+1Ch. 8 - Prob. 27SECh. 8 - Prob. 28SECh. 8 - Prob. 29SECh. 8 - Prob. 30SECh. 8 - Prob. 31SECh. 8 - Prob. 32SECh. 8 - Prob. 33SECh. 8 - Prob. 34SECh. 8 - Prob. 35SECh. 8 - How many terms are needed when the...Ch. 8 - How many solutions in positive integers are there...Ch. 8 - How many positive integers less than 1,000,000 are...Ch. 8 - How many positive integers less than 200 are a)...Ch. 8 - How many ways are there to assign six different...Ch. 8 - What is the probability that exactly one person is...Ch. 8 - How many bit stings of length six do not contain...Ch. 8 - What is the probability that a bit string of...Ch. 8 - Prob. 1CPCh. 8 - Prob. 2CPCh. 8 - Prob. 3CPCh. 8 - Prob. 4CPCh. 8 - Prob. 5CPCh. 8 - Prob. 6CPCh. 8 - Prob. 7CPCh. 8 - Prob. 8CPCh. 8 - Prob. 9CPCh. 8 - Prob. 10CPCh. 8 - Prob. 11CPCh. 8 - Prob. 12CPCh. 8 - Given a positive integer n, list all the...Ch. 8 - Prob. 1CAECh. 8 - Prob. 2CAECh. 8 - Find as many prime Fibonacci numbers as you can....Ch. 8 - Prob. 4CAECh. 8 - Prob. 5CAECh. 8 - Prob. 6CAECh. 8 - Prob. 7CAECh. 8 - Prob. 8CAECh. 8 - Prob. 9CAECh. 8 - List all the derangements of 1,2,3,4,5,6,7,8 .Ch. 8 - Prob. 11CAECh. 8 - Find the original source where Fibonacci presented...Ch. 8 - Explain how the Fibonacci numbers arise in a...Ch. 8 - Prob. 3WPCh. 8 - Discuss as mans different problems as possible...Ch. 8 - Prob. 5WPCh. 8 - Prob. 6WPCh. 8 - Prob. 7WPCh. 8 - Prob. 8WPCh. 8 - Describe the solution of Ulam’s problem (see...Ch. 8 - Discuss variations of Ulam’s problem (see Exercise...Ch. 8 - Prob. 11WPCh. 8 - Describe how sieve methods are used in number...Ch. 8 - Look up the rules of the old French card game of...Ch. 8 - Prob. 14WPCh. 8 - Describe the Polyá theory of counting and the kind...Ch. 8 - The problème des ménages (the problem of the...Ch. 8 - Explain how rook polynomials can be used to solve...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- Negate the following compound statement using De Morgans's laws.arrow_forwardNegate the following compound statement using De Morgans's laws.arrow_forwardQuestion 6: Negate the following compound statements, using De Morgan's laws. A) If Alberta was under water entirely then there should be no fossil of mammals.arrow_forward
- Negate the following compound statement using De Morgans's laws.arrow_forward18.10. Let f be analytic inside and on the unit circle 7. Show that, for 0<|z|< 1, f(E) f(E) 2πif(z) = --- d.arrow_forwardCharacterize (with proof) all connected graphs that contain no even cycles in terms oftheir blocks.arrow_forward
- 18.4. Let f be analytic within and on a positively oriented closed contoury, and the point zo is not on y. Show that L f(z) (-20)2 dz = '(2) dz. 2-20arrow_forward18.9. Let denote the boundary of the rectangle whose vertices are -2-2i, 2-21,2+i and -2+i in the positive direction. Evaluate each of the following integrals: (a). rdz, (b). dz (b). COS 2 coz dz, (z+1) (d). 之一 z 2 +2 dz, (e). dz (c). (2z + 1)2dz, (2z+1) 1 (f). £, · [e² sin = + (2² + 3)²] dz. z (22+3)2arrow_forwardLet G be a connected graph that does not have P4 or C3 as an induced subgraph (i.e.,G is P4, C3 free). Prove that G is a complete bipartite grapharrow_forward
- 18.8. (a). Let be the contour z = e-≤0≤ traversed in the า -dz = 2xi. positive direction. Show that, for any real constant a, Lex dzarrow_forwardProve sufficiency of the condition for a graph to be bipartite that is, prove that if G hasno odd cycles then G is bipartite as follows:Assume that the statement is false and that G is an edge minimal counterexample. That is, Gsatisfies the conditions and is not bipartite but G − e is bipartite for any edge e. (Note thatthis is essentially induction, just using different terminology.) What does minimality say aboutconnectivity of G? Can G − e be disconnected? Explain why if there is an edge between twovertices in the same part of a bipartition of G − e then there is an odd cyclearrow_forwardLet G be a connected graph that does not have P4 or C4 as an induced subgraph (i.e.,G is P4, C4 free). Prove that G has a vertex adjacent to all othersarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
03 - The Cartesian coordinate system; Author: Technion;https://www.youtube.com/watch?v=hOgKEplCx5E;License: Standard YouTube License, CC-BY
What is the Cartesian Coordinate System? | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=mgx0kT5UbKk;License: Standard YouTube License, CC-BY