في هذا الدرس نتعرف على العلاقات الثنائية او ما يسمي بقوانين التركيب الداخلية. نقدم التعاريف الضرورية ثم نحل مجموعة من التنارين المتنوعة
We present several examples of binary realtions. We will also solve many questions related to this defintion
The document provides an overview of basic math concepts for computer graphics, including:
- Sets, mappings, and Cartesian coordinates are introduced to represent vectors and points in 2D and 3D space.
- Linear interpolation is described as a fundamental operation in graphics used to connect data points.
- Parametric and implicit equations are discussed for representing common 2D curves and lines.
- Concepts like the dot product, cross product, and gradient are covered, which are important for calculations involving vectors.
1. The document provides information about an upcoming lecture and test on conic sections, specifically parabolas.
2. It defines a parabola geometrically as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. Examples are given of finding the equation of a parabola given its geometric properties, and finding the focus and directrix of a parabola given its equation. Applications of parabolas as reflectors are also mentioned.
The document discusses parabolas and their geometric and algebraic properties. It includes:
1. A summary of the geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
2. Examples of finding the equation of a parabola given its geometric features, and finding the focus and directrix from a parabola's equation.
3. A discussion of parabolas with vertical and horizontal axes of symmetry and their standard equations.
4. An application of parabolic reflectors for lamps and telescopes.
This document provides an agenda and notes for a math class that is reviewing quadratic functions and equations. It includes warm-up problems to solve quadratic equations algebraically and graphically, identifies an upcoming test on sections 10.1-10.3, and provides class work where students must show the steps to graph quadratic functions using a table of values.
Linear algebra power of abstraction - LearnDay@Xoxzo #5Xoxzo Inc.
LearnDay@Xoxzo is a monthly online seminar initiated by the Xoxzo team. We will have speakers from the team or guest speakers which will talk for 20 minutes each, on a subject of their choosing.
Linear algebra power of abstraction by Akira.
XOXZO Learn day
2018/12/21
======================
We have recorded sessions of our previous LearnDay here: https://www.youtube.com/channel/UCiV-bQprArQxKBSzaKY1vQg
For updates and news on our future LearnDays, follow us on Twitter (https://twitter.com/xoxzocom/) or sign up for our Exchange Newsletter (https://info.xoxzo.com/en/exchange-mailing-list/)
The document discusses quadratic equations and functions. It provides objectives of solving quadratic equations by factoring and graphing. It defines the zero of a function as where the graph crosses the x-axis. Examples are given of solving quadratic equations by factoring using the zero product property. Another example solves a quadratic equation graphically. Homework problems from the text are assigned.
Complex numbers are used to solve quadratic equations that have no real solutions, such as x2 + 1 = 0. Euler introduced the symbol i to represent the square root of -1, allowing numbers of the form a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be represented graphically on a plane with real numbers on the x-axis and imaginary numbers on the y-axis. They can also be expressed in polar form as r(cosθ + i sinθ) or in exponential form as reiθ. Operations like addition, subtraction, multiplication and division can be performed with complex numbers.
An ellipse is defined as the set of all points such that the sum of the distances from the point to two fixed foci points is a constant. The major axis is the longest side of the ellipse, and the two fixed points are called the foci. The standard form of the ellipse equation is (x/a)^2 + (y/b)^2 = 1, where a is half the length of the major axis and b is half the length of the minor axis.
The document provides an overview of basic math concepts for computer graphics, including:
- Sets, mappings, and Cartesian coordinates are introduced to represent vectors and points in 2D and 3D space.
- Linear interpolation is described as a fundamental operation in graphics used to connect data points.
- Parametric and implicit equations are discussed for representing common 2D curves and lines.
- Concepts like the dot product, cross product, and gradient are covered, which are important for calculations involving vectors.
1. The document provides information about an upcoming lecture and test on conic sections, specifically parabolas.
2. It defines a parabola geometrically as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. Examples are given of finding the equation of a parabola given its geometric properties, and finding the focus and directrix of a parabola given its equation. Applications of parabolas as reflectors are also mentioned.
The document discusses parabolas and their geometric and algebraic properties. It includes:
1. A summary of the geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
2. Examples of finding the equation of a parabola given its geometric features, and finding the focus and directrix from a parabola's equation.
3. A discussion of parabolas with vertical and horizontal axes of symmetry and their standard equations.
4. An application of parabolic reflectors for lamps and telescopes.
This document provides an agenda and notes for a math class that is reviewing quadratic functions and equations. It includes warm-up problems to solve quadratic equations algebraically and graphically, identifies an upcoming test on sections 10.1-10.3, and provides class work where students must show the steps to graph quadratic functions using a table of values.
Linear algebra power of abstraction - LearnDay@Xoxzo #5Xoxzo Inc.
LearnDay@Xoxzo is a monthly online seminar initiated by the Xoxzo team. We will have speakers from the team or guest speakers which will talk for 20 minutes each, on a subject of their choosing.
Linear algebra power of abstraction by Akira.
XOXZO Learn day
2018/12/21
======================
We have recorded sessions of our previous LearnDay here: https://www.youtube.com/channel/UCiV-bQprArQxKBSzaKY1vQg
For updates and news on our future LearnDays, follow us on Twitter (https://twitter.com/xoxzocom/) or sign up for our Exchange Newsletter (https://info.xoxzo.com/en/exchange-mailing-list/)
The document discusses quadratic equations and functions. It provides objectives of solving quadratic equations by factoring and graphing. It defines the zero of a function as where the graph crosses the x-axis. Examples are given of solving quadratic equations by factoring using the zero product property. Another example solves a quadratic equation graphically. Homework problems from the text are assigned.
Complex numbers are used to solve quadratic equations that have no real solutions, such as x2 + 1 = 0. Euler introduced the symbol i to represent the square root of -1, allowing numbers of the form a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be represented graphically on a plane with real numbers on the x-axis and imaginary numbers on the y-axis. They can also be expressed in polar form as r(cosθ + i sinθ) or in exponential form as reiθ. Operations like addition, subtraction, multiplication and division can be performed with complex numbers.
An ellipse is defined as the set of all points such that the sum of the distances from the point to two fixed foci points is a constant. The major axis is the longest side of the ellipse, and the two fixed points are called the foci. The standard form of the ellipse equation is (x/a)^2 + (y/b)^2 = 1, where a is half the length of the major axis and b is half the length of the minor axis.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
This document contains past paper questions from the IITJEE mathematics exam in 2010. It includes 4 sections with different types of questions - multiple choice, integer answer, paragraph and matrix matching. The questions cover topics in mathematics like planes, functions, trigonometry, complex numbers and matrices. Some questions provide contexts involving triangles, signals, parallelograms and other geometric shapes.
This document discusses various concepts related to vectors and 3D geometry including dot products, cross products, planes, lines, and their relationships. Dot products can be used to find the angle between vectors and determine if vectors are perpendicular. Cross products give a vector perpendicular to both input vectors. Plane equations can be defined using a point and normal vector, three points, or two vectors in the plane. Lines are defined by two points or a point and direction vector. The intersection of planes and lines, parallelism, and distances between lines and points and planes are also covered.
This document discusses graphing parabolas using squares. It explains that parabolas can be graphed by comparing their equations to standard forms and using squares of length 2p, where p is determined by the equation. Examples are given of graphing parabolas from equations x^2=6y and y^2=8x. The focus, vertex, directrix, and latus rectum are identified for each parabola by comparing the equations to the standard forms.
The document discusses graph coloring through the example of coloring a map with 6 regions. It defines a graph as G = (V,E) where V is the set of vertices and E is the set of edges. Colorings assign each vertex a color such that adjacent vertices have different colors. The chromatic number is the minimum number of colors needed. It shows that coloring the map is equivalent to coloring its corresponding graph. The chromatic number of the example map graph is 3, and there are 6 possible proper colorings using 3 colors.
1) The document discusses the distributive law for cross products and how it applies to vectors in the x-y plane.
2) It also explains how to find the components of a cross product vector C=A×B if the components of A and B are known using properties like the cross product of parallel vectors being zero and the direction of perpendicular vectors following the right hand rule.
3) Key identities for the cross product of unit vectors are given according to the right hand rule.
The study guide provides 3 methods for proving that the power set of a set A with n elements has 2^n elements: 1) Using the product rule of counting, 2) Applying the binomial theorem, and 3) Using induction and defining a function between subsets of A with and without a particular element x.
1. The document discusses three geometry problems involving vectors and their solutions:
2. It shows that the midpoint lines of a parallelogram trisect its diagonal lines.
3. It proves that if two pairs of opposite edges in a tetrahedron are perpendicular, then the third pair is also perpendicular.
4. It demonstrates that the midpoint line between two sides of a triangle is parallel to the third side and half its length.
Math 1300: Section 5- 3 Linear Programing in Two Dimensions: Geometric ApproachJason Aubrey
This document discusses linear programming problems. It begins by defining a linear programming problem as finding the optimal value of a linear objective function subject to linear constraints. It notes that linear programming has been widely used as a decision-making tool in management science. The document then provides an example of a linear programming problem to maximize profit based on labor constraints and profit levels for different product types. It indicates that solving such a problem involves constructing a mathematical model and applying the fundamental theorem of linear programming.
The document provides instructions for graphing and writing equations for absolute value functions. It discusses key aspects like the vertex, line of symmetry, slope, and general form of the equation. Examples are provided to demonstrate how to graph an absolute value function given its equation, and how to write the equation of an absolute value function given its graph by finding the vertex and another point.
This document provides instructions for solving quadratic equations by completing the square. It explains that you can complete the square by adding a constant to both sides of the equation to make the quadratic term a perfect square. You then take the square root of both sides to solve for x. It also notes that when the leading coefficient is not 1, you must first divide both sides by the coefficient a. Finally, it shows that completing the square allows you to derive the quadratic formula, which can be used to solve any quadratic equation in standard form.
The document defines X and Y intercepts and describes how to find them. The Y-intercept is the value of Y when the line crosses the Y-axis, and can be found by plugging the slope and a point into the linear equation y=mx+b and solving for b. The X-intercept is the value of X when the line crosses the X-axis, and can be found by plugging 0 for Y into the linear equation and solving for X. An example demonstrates finding the Y-intercept of 7 and the X-intercept of 3.5 for the equation y=-2x+7.
The dot product of two vectors a and b is a scalar value defined as |a||b|cosθ, where |a| and |b| are the lengths of the vectors and θ is the angle between them. The dot product will be 0 if the vectors are perpendicular, positive if the angle between them is less than 90 degrees, and negative if the angle is greater than 90 degrees. If one vector is a unit vector, the dot product equals the length of the other vector projected onto the direction of the unit vector.
Complex numbers are numbers of the form a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added, subtracted, multiplied, and divided. When multiplying complex numbers, the real parts and imaginary parts are multiplied separately and combined. The conjugate of a + bi is a - bi. When a complex number is multiplied by its conjugate, the result is a real number equal to the modulus (magnitude) of the complex number squared. Complex numbers can also be expressed in polar form as r(cosθ + i sinθ), where r is the modulus and θ is the argument.
1006 segment and angle addition postulate updated 2013jbianco9910
The document provides information about an upcoming geometry class including:
- Homework will be collected on Monday and students should complete an independent drill without talking.
- The class will define key geometry terms like length, segment addition postulate, and how to measure angles using a protractor.
- Vocabulary and concepts are explained such as the ruler postulate for finding distances between points on a line, and how to notate angle measures.
- Angle addition and adjacent angle definitions are given.
The document discusses matrix multiplication and properties of matrices such as identity matrices, transposes, and powers of matrices. It also covers zero-one matrices, which have entries of only 0 and 1. Operations on zero-one matrices include join, meet, Boolean product, and Boolean powers. Boolean operations like AND and OR are used in place of multiplication and addition for zero-one matrix operations. Examples are provided to illustrate matrix multiplication and operations on zero-one matrices.
Osama Tahir's presentation introduces complex numbers. [1] Complex numbers consist of a real and imaginary part and can be written in the form a + bi, where i = -1. [2] Complex numbers were introduced to solve equations like x^2 = -1 that have no real number solutions. [3] Key topics covered include addition, subtraction, multiplication, and division of complex numbers, representing them in polar form using De Moivre's theorem, and applications in fields like electric circuits and root locus analysis.
An introduction to Abstract algebra in Arabic
مدخل للجبر المجرد من خلال مجموعة من التعاريف
We introduce binary operations and homomorphisms law.
More courses will be available
The document discusses finding a particular integral, which is any solution to an inhomogeneous differential equation. It describes trial solutions that can be used, including solutions with the same form as the right-hand side term. Examples are provided of finding particular integrals through substitution and equating coefficients. The general solution to an inhomogeneous equation is the sum of the particular integral and complementary function.
The document is a problem set that contains 8 questions:
1. Name 3 elements and 3 subsets of the set A = {∅, a, b, c, d, e, f} and explain why a repeated element is irrelevant.
2. Find the union of the sets E = {2n : n is an integer} and O = {2n + 1 : n is an integer}.
3. Find the intersection of E and O.
4. Solve several linear equations.
5. Solve several quadratic equations.
6. Solve a cubic equation.
7. Generalize the distributive property to higher orders.
8. Pro
The document provides information about sets, relations, and functions. It defines what a set is and how sets can be described in roster form or set-builder form. It discusses different types of sets such as the empty set, singleton sets, finite sets, and infinite sets. It also covers topics like subsets, power sets, operations on sets such as union, intersection, difference, complement, and applications of these concepts to solve problems involving cardinality of sets.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
“ help.mbaassignments@gmail.com ”
or
Call us at : 08263069601
This document contains past paper questions from the IITJEE mathematics exam in 2010. It includes 4 sections with different types of questions - multiple choice, integer answer, paragraph and matrix matching. The questions cover topics in mathematics like planes, functions, trigonometry, complex numbers and matrices. Some questions provide contexts involving triangles, signals, parallelograms and other geometric shapes.
This document discusses various concepts related to vectors and 3D geometry including dot products, cross products, planes, lines, and their relationships. Dot products can be used to find the angle between vectors and determine if vectors are perpendicular. Cross products give a vector perpendicular to both input vectors. Plane equations can be defined using a point and normal vector, three points, or two vectors in the plane. Lines are defined by two points or a point and direction vector. The intersection of planes and lines, parallelism, and distances between lines and points and planes are also covered.
This document discusses graphing parabolas using squares. It explains that parabolas can be graphed by comparing their equations to standard forms and using squares of length 2p, where p is determined by the equation. Examples are given of graphing parabolas from equations x^2=6y and y^2=8x. The focus, vertex, directrix, and latus rectum are identified for each parabola by comparing the equations to the standard forms.
The document discusses graph coloring through the example of coloring a map with 6 regions. It defines a graph as G = (V,E) where V is the set of vertices and E is the set of edges. Colorings assign each vertex a color such that adjacent vertices have different colors. The chromatic number is the minimum number of colors needed. It shows that coloring the map is equivalent to coloring its corresponding graph. The chromatic number of the example map graph is 3, and there are 6 possible proper colorings using 3 colors.
1) The document discusses the distributive law for cross products and how it applies to vectors in the x-y plane.
2) It also explains how to find the components of a cross product vector C=A×B if the components of A and B are known using properties like the cross product of parallel vectors being zero and the direction of perpendicular vectors following the right hand rule.
3) Key identities for the cross product of unit vectors are given according to the right hand rule.
The study guide provides 3 methods for proving that the power set of a set A with n elements has 2^n elements: 1) Using the product rule of counting, 2) Applying the binomial theorem, and 3) Using induction and defining a function between subsets of A with and without a particular element x.
1. The document discusses three geometry problems involving vectors and their solutions:
2. It shows that the midpoint lines of a parallelogram trisect its diagonal lines.
3. It proves that if two pairs of opposite edges in a tetrahedron are perpendicular, then the third pair is also perpendicular.
4. It demonstrates that the midpoint line between two sides of a triangle is parallel to the third side and half its length.
Math 1300: Section 5- 3 Linear Programing in Two Dimensions: Geometric ApproachJason Aubrey
This document discusses linear programming problems. It begins by defining a linear programming problem as finding the optimal value of a linear objective function subject to linear constraints. It notes that linear programming has been widely used as a decision-making tool in management science. The document then provides an example of a linear programming problem to maximize profit based on labor constraints and profit levels for different product types. It indicates that solving such a problem involves constructing a mathematical model and applying the fundamental theorem of linear programming.
The document provides instructions for graphing and writing equations for absolute value functions. It discusses key aspects like the vertex, line of symmetry, slope, and general form of the equation. Examples are provided to demonstrate how to graph an absolute value function given its equation, and how to write the equation of an absolute value function given its graph by finding the vertex and another point.
This document provides instructions for solving quadratic equations by completing the square. It explains that you can complete the square by adding a constant to both sides of the equation to make the quadratic term a perfect square. You then take the square root of both sides to solve for x. It also notes that when the leading coefficient is not 1, you must first divide both sides by the coefficient a. Finally, it shows that completing the square allows you to derive the quadratic formula, which can be used to solve any quadratic equation in standard form.
The document defines X and Y intercepts and describes how to find them. The Y-intercept is the value of Y when the line crosses the Y-axis, and can be found by plugging the slope and a point into the linear equation y=mx+b and solving for b. The X-intercept is the value of X when the line crosses the X-axis, and can be found by plugging 0 for Y into the linear equation and solving for X. An example demonstrates finding the Y-intercept of 7 and the X-intercept of 3.5 for the equation y=-2x+7.
The dot product of two vectors a and b is a scalar value defined as |a||b|cosθ, where |a| and |b| are the lengths of the vectors and θ is the angle between them. The dot product will be 0 if the vectors are perpendicular, positive if the angle between them is less than 90 degrees, and negative if the angle is greater than 90 degrees. If one vector is a unit vector, the dot product equals the length of the other vector projected onto the direction of the unit vector.
Complex numbers are numbers of the form a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added, subtracted, multiplied, and divided. When multiplying complex numbers, the real parts and imaginary parts are multiplied separately and combined. The conjugate of a + bi is a - bi. When a complex number is multiplied by its conjugate, the result is a real number equal to the modulus (magnitude) of the complex number squared. Complex numbers can also be expressed in polar form as r(cosθ + i sinθ), where r is the modulus and θ is the argument.
1006 segment and angle addition postulate updated 2013jbianco9910
The document provides information about an upcoming geometry class including:
- Homework will be collected on Monday and students should complete an independent drill without talking.
- The class will define key geometry terms like length, segment addition postulate, and how to measure angles using a protractor.
- Vocabulary and concepts are explained such as the ruler postulate for finding distances between points on a line, and how to notate angle measures.
- Angle addition and adjacent angle definitions are given.
The document discusses matrix multiplication and properties of matrices such as identity matrices, transposes, and powers of matrices. It also covers zero-one matrices, which have entries of only 0 and 1. Operations on zero-one matrices include join, meet, Boolean product, and Boolean powers. Boolean operations like AND and OR are used in place of multiplication and addition for zero-one matrix operations. Examples are provided to illustrate matrix multiplication and operations on zero-one matrices.
Osama Tahir's presentation introduces complex numbers. [1] Complex numbers consist of a real and imaginary part and can be written in the form a + bi, where i = -1. [2] Complex numbers were introduced to solve equations like x^2 = -1 that have no real number solutions. [3] Key topics covered include addition, subtraction, multiplication, and division of complex numbers, representing them in polar form using De Moivre's theorem, and applications in fields like electric circuits and root locus analysis.
An introduction to Abstract algebra in Arabic
مدخل للجبر المجرد من خلال مجموعة من التعاريف
We introduce binary operations and homomorphisms law.
More courses will be available
The document discusses finding a particular integral, which is any solution to an inhomogeneous differential equation. It describes trial solutions that can be used, including solutions with the same form as the right-hand side term. Examples are provided of finding particular integrals through substitution and equating coefficients. The general solution to an inhomogeneous equation is the sum of the particular integral and complementary function.
The document is a problem set that contains 8 questions:
1. Name 3 elements and 3 subsets of the set A = {∅, a, b, c, d, e, f} and explain why a repeated element is irrelevant.
2. Find the union of the sets E = {2n : n is an integer} and O = {2n + 1 : n is an integer}.
3. Find the intersection of E and O.
4. Solve several linear equations.
5. Solve several quadratic equations.
6. Solve a cubic equation.
7. Generalize the distributive property to higher orders.
8. Pro
The document provides information about sets, relations, and functions. It defines what a set is and how sets can be described in roster form or set-builder form. It discusses different types of sets such as the empty set, singleton sets, finite sets, and infinite sets. It also covers topics like subsets, power sets, operations on sets such as union, intersection, difference, complement, and applications of these concepts to solve problems involving cardinality of sets.
1 of 11UMGC College Algebra MATH 107 6980 - Fall 2020 – Instruct.docxteresehearn
1 of 11
UMGC College Algebra MATH 107 6980 - Fall 2020 – Instructor: Timothy J. Elsner
Page 1 of 11
MATH 107 FINAL EXAMINATION - Nov 15, 2020 - Due Tue Nov 17 11:59 pm
This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may
use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed.
MAKE CERTAIN YOUR SUBMITTAL IS CLEARLY READABLE. FOR THE SHORT ANSWER SECTIONS make sure your ANSWER IS CIRCLED
There are 30 problems. Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown. Also read:
Mathematics in Montessori
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function. 1._______
A. Domain [ -5, 5]; Range [- 6, 6]
B. Domain [- 4, 5]; Range [- 6, 6]
C. Domain [- 6, 5]; Range [- 4, 6]
D. Domain [- 6, 6]; Range [- 4, 5]
2. Solve: x = √−8x + 9 and check your solution(s) 2.________
A. x = - 9
B. x = 1
C. x = {-9, 1}
D. No
Solution
2 of 11
3. Determine the x interval(s) on which the function is increasing. 3.__________
A. (−4, 0] and [4, ∞)
B. [0, 4]
C. (−∞, 3) ∪ [−1, 5 ]
D. (−∞, −4] and [0, 4 ]
4. Determine whether the graph of Y = | x | - 3 is symmetric with respect 4. _________
to the origin, the x-axis, or the y-axis.
A. symmetric with respect to the x-axis only
B. symmetric with respect to the y-axis only
C. symmetric with respect to the origin only
D. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis,
and not symmetric with respect to the origin
5. Find the solution to the inequality : | 6 – x | + 3 < 8 5. ___________
A. (????, ∞)
B. (???? , ???????? )
C. (−∞, ????) ∪ (????????, ∞)
D. (−1, −????????)
3 of 11
6. Which of the following represents the graph of −3x + 5y = 15 ? __________
A. B.
C. D.
7. Write a slope-intercept equation for a line perpendicular to the line −3x + 5y = 15
which passes through the point (6, – 5).
A. y = − ????
???? ???? + ????
B. y = ????
???? ???? − ????????
C. y = − ????
???? ???? + ????
D. y = ????
???? ???? − ????????
4 of 11
8. Choose what type of graph is below ? 8.___________
A. It is not a function.
B. It is a function and it is one-to-one.
C. It is a function but it is not one-to-one.
D. It is not a function and it is not one-to-one.
9. Express as a single logarithm: log (2x + 1) + log 2x - 4 log x 9.__________
A. log ( 4x+1
4x )
B. log ( 2x(2x+1)
4x )
C. log ( 4x2 - 2x)
D. log ( 2???? (2???? + 1)
????4 )
10. Which of the functions correspond to the graph? 10.__________
A. f(x) = e x
B. f(x) = e x – 1
C. f(x) = log(x)
D. f(x) = log(x) – 1
5 of 11
11. Suppose that for a function f(x), that it has exactly 1 zero (or 1 X-intercept)
Which of the following statements MUST true? (only one answer is correct) 11. _________
A. f(x) is linear and has a positive slope.
B.
Dynamic programming is an algorithm design paradigm that can be applied to problems exhibiting optimal substructure and overlapping subproblems. It works by breaking down a problem into subproblems and storing the results of already solved subproblems, rather than recomputing them multiple times. This allows for an efficient bottom-up approach. Examples where dynamic programming can be applied include the matrix chain multiplication problem, the 0-1 knapsack problem, and finding the longest common subsequence between two strings.
The document discusses key concepts about sets, including:
1) Intervals are subsets of real numbers that can be open, closed, or half-open/half-closed. Intervals are represented visually on a number line.
2) The power set of a set A contains all possible subsets of A. Its size is 2 to the power of the size of A.
3) The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements common to both sets.
4) Practical problems can use formulas involving the sizes of unions and intersections of finite sets.
The document defines and describes sets. Some key points include:
- A set is a collection of well-defined objects. Sets can be described in roster form or set-builder form.
- There are different types of sets such as the empty/null set, singleton sets, finite sets, and infinite sets.
- Set operations include union, intersection, difference, symmetric difference, and complement. Properties like subsets and the power set are also discussed.
- Cardinality refers to the number of elements in a set. Formulas are given for finding the cardinality of sets under different operations.
Tìm tòi sáng tạo - SOME POLYNOMIAL PROBLEMS ARE BUILT ON IDENTITIESTùng Thanh
The document discusses three outstanding results from identities used to create polynomial problems. The first result finds all polynomials satisfying an equality for triples satisfying ab + bc + ca = 0. The solution is polynomials of the form ax^2. The second result finds polynomials satisfying an equality involving binomial coefficients, with the solution kx^n. The third result finds polynomials satisfying an equality for triples satisfying (a+b)(b+c)(c+a) = 8c^3, with solution kx^3.
3D Representation
Read chapter 10 in Computer Science: note especially section 10.2. Create a 2-page document which will summarize the three steps involved in producing an image using 3D graphics. After you describe each step, give a good example of each. The examples should be different from the one given in the text.
Find a recent news article (not a tutorial or description) that relates to 3D graphics. Explain how any aspect of the news article relates to one of the steps you summarized above.
The document should be clear and concise, free from syntax and semantic errors.
Please submit the document on time please.
College Algebra MATH 107 Spring, 2017, V.1.3
Page 1 of 11
MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function. 1. ______
A. Domain [– 3, 3]; Range [– 1, 3]
B. Domain [– 3, 1]; Range [– 3, 3]
C. Domain [– 1, 0.5]; Range [–1, 0]
D. Domain (–∞, 3]; Range [–1, ∞)
2. Solve: 10 3x x− = − 2. ______
A. –5, 2
B. 5/2
C. –5
D. No solution
2 4 -4
-2
-4
2
4
-2
College Algebra MATH 107 Spring, 2017, V.1.3
Page 2 of 11
3. Determine the interval(s) on which the function is increasing. 3. ______
A. (–∞, –1)
B. (– 2, 2)
C. (–∞, – 3) and (1, ∞)
D. (– 4.5, – 1) and (2.5, ∞)
4. Determine whether the graph of ( )
2
4xy −= is symmetric with respect to the origin,
the x-axis, or the y-axis. 4. ______
A. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. symmetric with respect to the origin only
5. Solve, and express the answer in interval notation: | 6 – 5x | ≤ 14. 5. ______
A. (–∞, −8/5] ∪ [4, ∞)
B. (–∞, –8/5]
C. [4, ∞)
D. [–8/5, 4]
College Algebra MATH 107 Spring, 2017, V.1.3
Page 3 of 11
6. Which of the following represents the graph of 7x + 4y = 28 ? 6. ______
A. B.
C. D.
College Algebra MATH 107 Spring, 2017, V.1.3
Page 4 of 11
7. Write a slope-intercept equation for a line parallel to the line x – 3y = 5 which passes through
the point (6, –8). 7. ______
A.
1
8
3
y x= −
B.
1
10
3
y x= −
C.
1
6
3
y x= − −
...
This document provides an outline and introduction to the topic of number theory. It begins with definitions and properties of various number sets, including natural numbers, integers, rational numbers, irrational numbers, and real numbers. It discusses how rational numbers can be represented as fractions and irrational numbers cannot. The document also states that the set of rational numbers is dense in the set of real numbers and presents the Archimedean property. The overall summary is an introduction to number sets and basic concepts in number theory.
Software product selectionSelecting a software product for use i.docxwhitneyleman54422
Software product selection
Selecting a software product for use in a healthcare facility can be a complicated and laborious process, although the things you've read this week about gathering and preparing requirements can be a terrific aid in making the task easier. Based on your experiences and the reading assignment for this week, what other activities are required in order to make the right decision about what system to purchase?
Instructions:
This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator.
Record your answers and work in this document.
There are 30 problems.
Problems #1-12 are multiple choice. Record your choice for each problem.
Problems #13-21 are short answer. Record your answer for each problem.
Problems #22-30 are short answer with work required. When requested, show all work and write all answers in the spaces allotted on the following pages. You may type your work using plain-text formatting or an equation editor, or you may hand-write your work and scan it. In either case, show work neatly and correctly, following standard mathematical conventions. Each step should follow clearly and completely from the previous step. If necessary, you may attach extra pages.
You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Your exam will receive a zero grade unless you complete the following honor statement.
(
Please sign (or type) your name below the following honor statement:
I have completed this
final examination
myself,
working independently and not consulting anyone except the instructor.
I have neither given nor received help on this final examination.
Name ____________
______
___
Date___________________
)
MULTIPLE CHOICE. Record your answer choices.
1.7.
2.8.
3.9.
4.10.
5.11.
6.12.
SHORT ANSWER. Record your answers below.
13.
14.
15.
16.
17.
18.
19. (a)
(b)
(c)
20. (a)
(b)
(c)
(d)
21. (a)
(b)
(c)
(d)
SHORT ANSWER with Work Shown. Record your answers and work.
Problem Number
Solution
22
Answers:
(a)
(b)
Work/for part (a) and explanation for part (b):
23
Answers:
(a)
(b)
(c)
Work for part (a):
24
Answer:
Work:
25
Answer:
Work:
26
Answers:
(a)
(b)
Work for part (a) and for part (b):
27
Answer:
Work:
28
Answer:
Work:
29
Answers:
(a)
(b)
Work for (b):
30
Answer:
Work:
College Algebra MATH 107 Spring, 2017, V3.2
Page 1 of 11
MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate .
This document provides an informal introduction to set theory concepts including:
- Defining elements, subsets, unions, intersections, differences, and the empty set
- Examples of finite and infinite sets like natural numbers, integers, and real numbers
- Equality of sets and the power set
- Using sets to represent natural numbers
- Defining pairs and Cartesian products of sets
- Types of binary relations like reflexive, symmetric, and transitive relations
- Infinite cardinal numbers like aleph-null and exploring paradoxes in set theory.
This document provides an overview of sets and set operations including:
- Four ways to define sets: by listing elements, with a condition, Venn diagrams, or verbal description
- Examples of set operations like union, intersection, complement, and difference
- Examples of how to express sets using these operations
- Diagrams explaining set relationships and operations
The document is a reference for the basic concepts and notation of sets and set operations in mathematics. It includes examples and exercises to demonstrate expressing sets in different ways using operations like union, intersection, complement, and difference.
This document contains a 100 problem inequalities marathon collected from MathLinks.ro. It provides the problems, solutions submitted by participants, and important definitions seen in the problems. The document includes the list of participants, real names if provided, and thanks those who contributed problems and solutions. Readers are invited to submit additional solutions or feedback to the editor.
(1) The question asks to find the last two digits of 2^1000. (2) Computing exponents modulo 100, it is shown that 2^20 = 24 (mod 100) and 2^20 = 76 (mod 100). (3) By induction, 2^n = 76 (mod 100) for any exponent n. Therefore, the last two digits of 2^1000 are 76.
College Algebra MATH 107 Spring, 2020Page 1 of 11 MA.docxmary772
College Algebra MATH 107 Spring, 2020
Page 1 of 11
MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function. 1. ______
A. Domain [– 4, 2]; Range [– 2, 4]
B. Domain [– 2, 4]; Range [– 2, 1]
C. Domain [– 2, 4]; Range [– 4 , 2]
D. Domain [0, 2]; Range [– 2, 0]
2. Solve: 3 10x x+ = − 2. ______
A. No solution
B. –2, 5
C. 5
D. –2
-2 2 4 -4
-2
-4
2
4
College Algebra MATH 107 Spring, 2020
Page 2 of 11
3. Determine the interval(s) on which the function is decreasing. 3. ______
A. (–1, 3)
B. (–2, 4)
C. (–3.6, 0) and (6.7, )
D. (–, –2) and (4, )
4. Determine whether the graph of 2y x= + is symmetric with respect to the origin,
the x-axis, or the y-axis. 4. ______
A. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. symmetric with respect to the origin only
5. Solve, and express the answer in interval notation: | 5 – 6x | 13. 5. ______
A. (–, 3] [−4/3, )
B. (–, −4/3] [3, )
C. [4/3, )
D. [–4/3, 3]
College Algebra MATH 107 Spring, 2020
Page 3 of 11
6. Which of the following represents the graph of 8x − 3y = 24 ? 6. ______
A. B.
C. D.
College Algebra MATH 107 Spring, 2020
Page 4 of 11
7. Write a slope-intercept equation for a line parallel to the line x + 7y = 9 which passes through
the point (28, –3). 7. ______
A. 25y x=− +
B.
1
1
7
y x= − +
C.
1
3
7
y x= − −
D.
1
7
7
y x= −
8. Which of the following best describes the graph? 8. ______
A. It is the graph of a function but not one-to-one.
B. It is the graph of a function and it is one-to-one.
C. It is not the graph of a function.
D. It is the graph of an absolute value relation.
.
College Algebra MATH 107 Spring, 2020
Page 5 of 11
9. Write as an equivalent expression: log (x – 3) – 8 log y + log 1 9. ______
A.
log( 3)
log(8 )
x
y
−
B.
2
log
8
x
y
−
C.
8
3
log
x
y
−
D. ( )log 2 8x y− −
10. Which of the functions corresponds to the graph? 10. ______
A. ( ) 2 xf x e−=
B. ( ) 2 xf x e−= +
C. ( ) 2 xf x e= −
D. ( ) 2xf x e−=
College Algebra MATH 107 Spring, 2020
Page 6 of 11
11. Suppose that for a function f, the equation f (x) = 0 has no real-number solution.
Which of the following statements MUST be tr.
This document discusses lattices and Boolean algebra. It defines lattices as algebraic systems that satisfy certain axioms involving binary operations of join (∨) and meet (⋅). Boolean algebras are lattices that also have a unary complement operation and satisfy additional axioms such as distributivity and the existence of complements. Examples of Boolean algebras include the power set of a set under set operations and vectors of 0s and 1s under component-wise operations. Boolean functions and logical expressions are then introduced and their evaluation is discussed.
This document provides an overview of linear equations and inequalities. It begins by stating the learning objectives which are to solve linear equations and inequalities, and apply them to word problems. Several examples are then shown of solving linear equations by clearing fractions and combining like terms. The concepts of equivalent equations, solving formulas, and solving linear inequalities are also explained. Interval notation is introduced to describe solutions to inequalities. Finally, a multi-step word problem on break-even analysis is presented and solved to demonstrate applying linear equations to applications.
I am Tahira H. I am a Complex Variables Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from La Trobe University. I have been helping students with their assignments for the past 15 years. I solve assignments related to Complex Variables.
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The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
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E
b a f (a, b) E b a
a⊥b a ∧ b aTb a ∗ b
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f : N × N → N
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f : Q × Q → Q
(a, b) → |a| − |b|
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T
f : T × T → T
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C2 C1 C
f
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P P
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(C(a, b), R, <)
R (a, b)
(C(a, b), R, <)
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9. Abstract Algebra
E = {n2, n ∈ N}
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(N, ×) E
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N N2 N1
E = {2k + 1k ∈ N}
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(N2, ×) (N1, ×) E
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N N2 N1
f : N1 × N1 → N1
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N N2 N1
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13. Abstract Algebra
∆ E P(E)
A∆B = (A − B) ∪ (B − A)
φ : P(E) × P(E) → P(E)
(A, B) → A∆B
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α E P(E)
AαB = (A − B)
ψ : P(E) × P(E) → P(E)
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AαB = (A − B)
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