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This document covers topics in mathematics including: 1. Linear algebra concepts like partitioned matrices and block multiplication 2. Properties of summation in number theory 3. Simplifying rational algebraic expressions using factoring methods 4. Evaluating integrals using trigonometric substitutions 5. Identifying parts of a circle graph for trigonometric functions.

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002#pedagogy math test

The document discusses various math concepts related to numbers and quantities for grades 3-12, including:
1) Prime and composite numbers, with prime numbers being only divisible by 1 and itself, and composite numbers being divisible by other numbers besides 1 and itself.
2) Prime factorization, which is expressing a number as a product of its prime factors.
3) Place value, with examples testing identification of digits in the ones, tens, hundreds, and thousands places.
4) Order relationships and properties of relations between sets.

Mathematics-Form 3-Revision © By Kelvin

This document contains a mathematics revision exercise for Form 2 students with questions in three sections:
1) Simple calculation questions involving addition, subtraction, multiplication, and division of fractions, decimals, and integers. Students are asked to show their work.
2) Distance problems calculating the distance between points on a coordinate plane using the distance formula. Students are asked to use a calculator and show their steps.
3) Circumference and diameter problems for circles using pi and the circumference formula. Students are asked to calculate values and find values using the appropriate formulas, showing their work.
The final section contains word problems involving ratios, rates, and coordinate geometry finding the midpoint of a line segment. Students must show their

Lesson 2b - scalar multiplication

Students will learn that a scalar is a single number that can scale each value in a matrix. They will practice multiplying matrices by scalars and performing scalar addition and subtraction of matrices. For a matrix A and scalar c, the scalar product cA means multiplying each element of A by c. Examples show multiplying matrices by scalars like 2A and -1/2A, and adding or subtracting matrices with scalar coefficients like 5C - B.

Calculus

This document discusses partial fractions. It begins by defining partial fractions as decomposing a proper rational expression into a sum of two or more rational expressions. It then provides three examples of decomposing rational functions into partial fractions. Finally, it discusses some applications of partial fractions, such as using them to integrate rational functions in calculus and finding inverse Laplace transforms, as well as giving a real-world example of using partial fractions to calculate the area under a curve to determine the amount of paint needed.

Dividing integers

The document discusses dividing integers and the rules for doing so. It states that the sign of the quotient depends on the signs of the dividend and divisor, with different signs producing a negative quotient and same signs a positive one. It also notes that dividing by zero is undefined. Examples are provided to demonstrate applying these rules.

Mathematics form 3-chapter 11 & 12 Linear Equation + Linear Inequalities © By...

This document contains notes for a mathematics chapter covering linear equations and inequalities. It introduces key topics like conversions between units of length, mass, time, and money. It also covers solving linear equations in two variables, simultaneous linear equations using substitution and elimination methods, and solving inequalities in one and two variables. Examples of each type of problem are provided.

Numbers 2º eso

This document contains a math problem involving classifying different types of numbers and performing calculations with integers, rational numbers, whole numbers, and natural numbers. It asks the reader to:
1) Classify various numbers into the different number categories, identify numbers that don't fit into any category, and draw a map showing the relationships between the categories.
2) Write out the first line of operations and results in English for 4 calculations involving integers, rational numbers, parentheses, and order of operations.

3.5 write and graph equations of lines

This geometry document covers writing equations of lines in slope-intercept form. It reviews the concepts of slope and y-intercept and teaches that an equation of a line needs a slope and y-intercept. Examples are given of writing equations given these components. The document also addresses writing equations of lines parallel or perpendicular to given lines and passing through given points. Students are provided practice problems to write equations of lines in different scenarios.

002#pedagogy math test

The document discusses various math concepts related to numbers and quantities for grades 3-12, including:
1) Prime and composite numbers, with prime numbers being only divisible by 1 and itself, and composite numbers being divisible by other numbers besides 1 and itself.
2) Prime factorization, which is expressing a number as a product of its prime factors.
3) Place value, with examples testing identification of digits in the ones, tens, hundreds, and thousands places.
4) Order relationships and properties of relations between sets.

Mathematics-Form 3-Revision © By Kelvin

This document contains a mathematics revision exercise for Form 2 students with questions in three sections:
1) Simple calculation questions involving addition, subtraction, multiplication, and division of fractions, decimals, and integers. Students are asked to show their work.
2) Distance problems calculating the distance between points on a coordinate plane using the distance formula. Students are asked to use a calculator and show their steps.
3) Circumference and diameter problems for circles using pi and the circumference formula. Students are asked to calculate values and find values using the appropriate formulas, showing their work.
The final section contains word problems involving ratios, rates, and coordinate geometry finding the midpoint of a line segment. Students must show their

Lesson 2b - scalar multiplication

Students will learn that a scalar is a single number that can scale each value in a matrix. They will practice multiplying matrices by scalars and performing scalar addition and subtraction of matrices. For a matrix A and scalar c, the scalar product cA means multiplying each element of A by c. Examples show multiplying matrices by scalars like 2A and -1/2A, and adding or subtracting matrices with scalar coefficients like 5C - B.

Calculus

This document discusses partial fractions. It begins by defining partial fractions as decomposing a proper rational expression into a sum of two or more rational expressions. It then provides three examples of decomposing rational functions into partial fractions. Finally, it discusses some applications of partial fractions, such as using them to integrate rational functions in calculus and finding inverse Laplace transforms, as well as giving a real-world example of using partial fractions to calculate the area under a curve to determine the amount of paint needed.

Dividing integers

The document discusses dividing integers and the rules for doing so. It states that the sign of the quotient depends on the signs of the dividend and divisor, with different signs producing a negative quotient and same signs a positive one. It also notes that dividing by zero is undefined. Examples are provided to demonstrate applying these rules.

Mathematics form 3-chapter 11 & 12 Linear Equation + Linear Inequalities © By...

This document contains notes for a mathematics chapter covering linear equations and inequalities. It introduces key topics like conversions between units of length, mass, time, and money. It also covers solving linear equations in two variables, simultaneous linear equations using substitution and elimination methods, and solving inequalities in one and two variables. Examples of each type of problem are provided.

Numbers 2º eso

This document contains a math problem involving classifying different types of numbers and performing calculations with integers, rational numbers, whole numbers, and natural numbers. It asks the reader to:
1) Classify various numbers into the different number categories, identify numbers that don't fit into any category, and draw a map showing the relationships between the categories.
2) Write out the first line of operations and results in English for 4 calculations involving integers, rational numbers, parentheses, and order of operations.

3.5 write and graph equations of lines

This geometry document covers writing equations of lines in slope-intercept form. It reviews the concepts of slope and y-intercept and teaches that an equation of a line needs a slope and y-intercept. Examples are given of writing equations given these components. The document also addresses writing equations of lines parallel or perpendicular to given lines and passing through given points. Students are provided practice problems to write equations of lines in different scenarios.

Lecture 5 (solving simultaneous equations)

This document provides examples for solving simultaneous linear equations using three methods: substitution, elimination, and graphical approach. It also presents sample problems for students to practice solving simultaneous equations using these three methods. The document aims to teach students how to solve simultaneous linear equations after covering the three solution methods.

Lesson 4 b special matrix multiplication

The document discusses using matrix multiplication to manipulate matrices. It provides an example of a matrix representing the number of TVs sold at different stores each day of the week. It shows how to use matrix multiplication to calculate the total number of TVs sold each week at each store and each day across all stores. The document also discusses using diagonal matrices to change parts of other matrices, like increasing one store's prices by 30% and giving another store a 50% discount.

Lesson 5 b solving matrix equations

The document provides examples for solving matrix equations using inverses. It introduces the concept of pre-multiplying both sides of a matrix equation AX=B by the inverse of A to solve for X. Example 1 demonstrates this process, finding the inverse of the coefficient matrix and multiplying both sides to isolate X. Example 2 shows that the inverse does not exist when the coefficient matrix is singular, meaning there may be no solution or infinite solutions depending on whether the rows of B satisfy the relationship of the rows of the singular A.

E. math

This document contains information about several mathematicians and geometry concepts:
- Euclid compiled known geometry works in his famous treatise "Elements", which influenced geometry for generations. He assumed certain properties as either postulates (assumptions specific to geometry) or axioms (assumptions used throughout mathematics).
- Heron of Alexandria was a Greek engineer and mathematician who derived Heron's formula for calculating the area of a triangle using the lengths of its three sides. He invented early machines including a steam turbine and rocket-like device propelled by steam.
- The document also provides Heron's formula and definitions of terms like postulates, axioms, and the parts of a coordinate plane.

Mathematics magazine

This document contains information about several mathematicians and geometry concepts:
- Euclid compiled known geometry works in his treatise "Elements", establishing foundations for geometry for generations. He assumed certain properties as axioms or postulates.
- Heron of Alexandria derived Heron's formula for calculating the area of a triangle using only the lengths of its three sides. He invented early machines including a steam turbine and rocket-like device propelled by steam.
- The document also provides Heron's formula and definitions of terms like postulates, axioms, and parts of a coordinate plane.

Evaluating an Algebraic Expression

This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.

Graphing Simple Quadratic Functions

The document discusses graphing simple quadratic functions by comparing graphs of equations of the form y = ax^2 + b and y = -1/2x^2 + c. It asks how changing the values of a, b, and c affects the parabola. Specifically, it asks how the graphs are alike and different, and how to identify the vertex of each. It notes that being able to identify the vertex from the equation will be helpful for graphing using a table of values. Finally, it discusses choosing appropriate domains and ranges when dealing with very small or large numbers.

A1 Chapter 5 Study Guide

This document provides a study guide for Chapter 5 of Algebra 1 covering slope-intercept form, point-slope form, standard form, writing equations of lines given points or slope, finding parallel and perpendicular lines, and using linear models to fit data. Key concepts covered include the definitions of slope, forms of linear equations, using various forms to write equations of lines, and interpreting linear models from data.

(8) Lesson 3.2

This document provides instruction on determining the slope of a line from graphs, tables, and point pairs. It includes examples of finding the slope from a graph showing patients seen per day based on hygienists working, a table showing pages read over time, and using the slope formula with point pairs. Key steps shown are finding the rise over run and using the slope formula m=(y2-y1)/(x2-x1).

Discrete_Matrices

A matrix is an arrangement of items (usually numbers) in rows and columns. Matrices can be added, subtracted, and multiplied following specific rules. The determinant of a matrix is calculated using a calculator function. The inverse of a matrix is found using a calculator function and can be multiplied by the original matrix to yield the identity matrix.

Matrices - Discrete Structures

The document discusses matrix multiplication. Matrix multiplication involves multiplying the elements of each row of the first matrix by the elements of each column of the second matrix and adding the products. The number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be defined. The result of multiplying an m×n matrix by an n×p matrix is an m×p matrix. Matrix multiplication is not generally commutative.

7-1 Exploring Quadratic Functions

This document discusses key concepts about quadratic functions including that they form parabolas with an axis of symmetry, how to identify the a, b, and c coefficients, and that the vertex represents the minimum or maximum value depending on whether a is positive or negative. It also explains how to graph quadratic functions using a table of values on a calculator or by hand and how the width of the parabola relates to the value of a.

Adding and subtracting matrices unit 3, lesson 2

To add or subtract matrices, they must have the same dimensions. When adding, corresponding entries are added, while when subtracting, negatives are subtracted correctly. Algebraic expressions within matrices can also be added or subtracted provided the matrices have matching dimensions, otherwise the result is undefined.

Activity 5 (answer key)

This document provides an answer key for an activity involving parallel lines cut by a transversal. It identifies corresponding, alternate interior, and alternate exterior angles formed, as well as interior and exterior angles on the same side of the transversal. Given two parallel lines cut by a transversal, it names angles congruent to and supplementary to a given angle. It also shows examples of finding the value of x when given information about corresponding angles formed by two parallel lines cut by a transversal.

Mat feb monthly paper 2

This document contains a 20 question mathematics test for Year 5 students. It includes questions on topics like place value, addition, subtraction, multiplication, division, fractions, mixed numbers, word problems, and more. The test provides spaces for students to write their answers.

Lesson 4a - permutation matrices

Permutation matrices can be used to rearrange the rows or columns of a matrix. A permutation matrix is a square matrix with ones in exactly one position of each row and column and zeros elsewhere. Multiplying a matrix A on the left by a permutation matrix P rearranges the rows of A, while multiplying A on the right by P rearranges the columns. The position of the one in each row/column of P determines which row/column of A will be moved.

8th alg -l6.3

This document contains notes from a math lesson on solving systems of linear equations by graphing and elimination. It includes examples of using both graphing and elimination to solve 5 systems of 2 equations with 2 unknowns. The steps for using elimination are outlined as writing the system with like terms aligned, eliminating one variable by adding or subtracting equations, solving for the remaining variable, and substituting back into one equation to find the other variable.

Material Didactico

This document summarizes three methods for solving systems of linear equations with two unknowns:
1) The method of substitution substitutes the equation into the other.
2) The method of elimination eliminates one variable by adding a multiple of one equation to the other.
3) An example problem is worked out using the three methods.

Rational exponents and radicals

Rational exponents are exponents that are ratios or fractions. There are three different ways to write a rational exponent: as a ratio of exponents, as a root of a root, and with a variable exponent. Rational exponents can be rewritten between exponential and radical forms. They follow the standard exponent rules when simplifying expressions, distributing exponents over division and applying negative exponent rules.

Matrix multiplication

To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Each entry in a row of the first matrix is multiplied by the corresponding entry in a column of the second matrix and summed. This process is repeated for each row and column to populate the final matrix. The dimensions of the final matrix will be the number of rows in the first matrix and the number of columns in the second.

Unit 7.5

This document discusses systems of inequalities in two variables and linear programming. It begins with an overview of graphing inequalities and systems of inequalities. Examples are provided of graphing linear inequalities and solving systems of inequalities graphically. Linear programming is then introduced as a method for finding maximum and minimum values of an objective function subject to constraints. An example linear programming problem is worked through to find the maximum and minimum values.

6.6 Graphing Inequalities In Two Variables

This document discusses graphing linear inequalities in two variables. It provides definitions of key terms like half-plane and boundary. It also gives helpful hints for graphing different types of inequalities based on whether the sign is >, <, ≥, or ≤ and whether it involves just x and y or a slanted line. Examples are provided to demonstrate how to graph inequalities like y > 3, x - 2, y - 3x + 2, and an applied problem involving nickels and dimes. The key steps for graphing a linear inequality on the coordinate plane are outlined.

Lecture 5 (solving simultaneous equations)

This document provides examples for solving simultaneous linear equations using three methods: substitution, elimination, and graphical approach. It also presents sample problems for students to practice solving simultaneous equations using these three methods. The document aims to teach students how to solve simultaneous linear equations after covering the three solution methods.

Lesson 4 b special matrix multiplication

The document discusses using matrix multiplication to manipulate matrices. It provides an example of a matrix representing the number of TVs sold at different stores each day of the week. It shows how to use matrix multiplication to calculate the total number of TVs sold each week at each store and each day across all stores. The document also discusses using diagonal matrices to change parts of other matrices, like increasing one store's prices by 30% and giving another store a 50% discount.

Lesson 5 b solving matrix equations

The document provides examples for solving matrix equations using inverses. It introduces the concept of pre-multiplying both sides of a matrix equation AX=B by the inverse of A to solve for X. Example 1 demonstrates this process, finding the inverse of the coefficient matrix and multiplying both sides to isolate X. Example 2 shows that the inverse does not exist when the coefficient matrix is singular, meaning there may be no solution or infinite solutions depending on whether the rows of B satisfy the relationship of the rows of the singular A.

E. math

This document contains information about several mathematicians and geometry concepts:
- Euclid compiled known geometry works in his famous treatise "Elements", which influenced geometry for generations. He assumed certain properties as either postulates (assumptions specific to geometry) or axioms (assumptions used throughout mathematics).
- Heron of Alexandria was a Greek engineer and mathematician who derived Heron's formula for calculating the area of a triangle using the lengths of its three sides. He invented early machines including a steam turbine and rocket-like device propelled by steam.
- The document also provides Heron's formula and definitions of terms like postulates, axioms, and the parts of a coordinate plane.

Mathematics magazine

This document contains information about several mathematicians and geometry concepts:
- Euclid compiled known geometry works in his treatise "Elements", establishing foundations for geometry for generations. He assumed certain properties as axioms or postulates.
- Heron of Alexandria derived Heron's formula for calculating the area of a triangle using only the lengths of its three sides. He invented early machines including a steam turbine and rocket-like device propelled by steam.
- The document also provides Heron's formula and definitions of terms like postulates, axioms, and parts of a coordinate plane.

Evaluating an Algebraic Expression

This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.

Graphing Simple Quadratic Functions

The document discusses graphing simple quadratic functions by comparing graphs of equations of the form y = ax^2 + b and y = -1/2x^2 + c. It asks how changing the values of a, b, and c affects the parabola. Specifically, it asks how the graphs are alike and different, and how to identify the vertex of each. It notes that being able to identify the vertex from the equation will be helpful for graphing using a table of values. Finally, it discusses choosing appropriate domains and ranges when dealing with very small or large numbers.

A1 Chapter 5 Study Guide

This document provides a study guide for Chapter 5 of Algebra 1 covering slope-intercept form, point-slope form, standard form, writing equations of lines given points or slope, finding parallel and perpendicular lines, and using linear models to fit data. Key concepts covered include the definitions of slope, forms of linear equations, using various forms to write equations of lines, and interpreting linear models from data.

(8) Lesson 3.2

This document provides instruction on determining the slope of a line from graphs, tables, and point pairs. It includes examples of finding the slope from a graph showing patients seen per day based on hygienists working, a table showing pages read over time, and using the slope formula with point pairs. Key steps shown are finding the rise over run and using the slope formula m=(y2-y1)/(x2-x1).

Discrete_Matrices

A matrix is an arrangement of items (usually numbers) in rows and columns. Matrices can be added, subtracted, and multiplied following specific rules. The determinant of a matrix is calculated using a calculator function. The inverse of a matrix is found using a calculator function and can be multiplied by the original matrix to yield the identity matrix.

Matrices - Discrete Structures

The document discusses matrix multiplication. Matrix multiplication involves multiplying the elements of each row of the first matrix by the elements of each column of the second matrix and adding the products. The number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be defined. The result of multiplying an m×n matrix by an n×p matrix is an m×p matrix. Matrix multiplication is not generally commutative.

7-1 Exploring Quadratic Functions

This document discusses key concepts about quadratic functions including that they form parabolas with an axis of symmetry, how to identify the a, b, and c coefficients, and that the vertex represents the minimum or maximum value depending on whether a is positive or negative. It also explains how to graph quadratic functions using a table of values on a calculator or by hand and how the width of the parabola relates to the value of a.

Adding and subtracting matrices unit 3, lesson 2

To add or subtract matrices, they must have the same dimensions. When adding, corresponding entries are added, while when subtracting, negatives are subtracted correctly. Algebraic expressions within matrices can also be added or subtracted provided the matrices have matching dimensions, otherwise the result is undefined.

Activity 5 (answer key)

This document provides an answer key for an activity involving parallel lines cut by a transversal. It identifies corresponding, alternate interior, and alternate exterior angles formed, as well as interior and exterior angles on the same side of the transversal. Given two parallel lines cut by a transversal, it names angles congruent to and supplementary to a given angle. It also shows examples of finding the value of x when given information about corresponding angles formed by two parallel lines cut by a transversal.

Mat feb monthly paper 2

This document contains a 20 question mathematics test for Year 5 students. It includes questions on topics like place value, addition, subtraction, multiplication, division, fractions, mixed numbers, word problems, and more. The test provides spaces for students to write their answers.

Lesson 4a - permutation matrices

Permutation matrices can be used to rearrange the rows or columns of a matrix. A permutation matrix is a square matrix with ones in exactly one position of each row and column and zeros elsewhere. Multiplying a matrix A on the left by a permutation matrix P rearranges the rows of A, while multiplying A on the right by P rearranges the columns. The position of the one in each row/column of P determines which row/column of A will be moved.

8th alg -l6.3

This document contains notes from a math lesson on solving systems of linear equations by graphing and elimination. It includes examples of using both graphing and elimination to solve 5 systems of 2 equations with 2 unknowns. The steps for using elimination are outlined as writing the system with like terms aligned, eliminating one variable by adding or subtracting equations, solving for the remaining variable, and substituting back into one equation to find the other variable.

Material Didactico

This document summarizes three methods for solving systems of linear equations with two unknowns:
1) The method of substitution substitutes the equation into the other.
2) The method of elimination eliminates one variable by adding a multiple of one equation to the other.
3) An example problem is worked out using the three methods.

Rational exponents and radicals

Rational exponents are exponents that are ratios or fractions. There are three different ways to write a rational exponent: as a ratio of exponents, as a root of a root, and with a variable exponent. Rational exponents can be rewritten between exponential and radical forms. They follow the standard exponent rules when simplifying expressions, distributing exponents over division and applying negative exponent rules.

Matrix multiplication

To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Each entry in a row of the first matrix is multiplied by the corresponding entry in a column of the second matrix and summed. This process is repeated for each row and column to populate the final matrix. The dimensions of the final matrix will be the number of rows in the first matrix and the number of columns in the second.

Lecture 5 (solving simultaneous equations)

Lecture 5 (solving simultaneous equations)

Lesson 4 b special matrix multiplication

Lesson 4 b special matrix multiplication

Lesson 5 b solving matrix equations

Lesson 5 b solving matrix equations

E. math

E. math

Mathematics magazine

Mathematics magazine

Evaluating an Algebraic Expression

Evaluating an Algebraic Expression

Graphing Simple Quadratic Functions

Graphing Simple Quadratic Functions

A1 Chapter 5 Study Guide

A1 Chapter 5 Study Guide

(8) Lesson 3.2

(8) Lesson 3.2

Discrete_Matrices

Discrete_Matrices

Matrices - Discrete Structures

Matrices - Discrete Structures

7-1 Exploring Quadratic Functions

7-1 Exploring Quadratic Functions

Adding and subtracting matrices unit 3, lesson 2

Adding and subtracting matrices unit 3, lesson 2

Activity 5 (answer key)

Activity 5 (answer key)

Mat feb monthly paper 2

Mat feb monthly paper 2

Lesson 4a - permutation matrices

Lesson 4a - permutation matrices

8th alg -l6.3

8th alg -l6.3

Material Didactico

Material Didactico

Rational exponents and radicals

Rational exponents and radicals

Matrix multiplication

Matrix multiplication

Unit 7.5

This document discusses systems of inequalities in two variables and linear programming. It begins with an overview of graphing inequalities and systems of inequalities. Examples are provided of graphing linear inequalities and solving systems of inequalities graphically. Linear programming is then introduced as a method for finding maximum and minimum values of an objective function subject to constraints. An example linear programming problem is worked through to find the maximum and minimum values.

6.6 Graphing Inequalities In Two Variables

This document discusses graphing linear inequalities in two variables. It provides definitions of key terms like half-plane and boundary. It also gives helpful hints for graphing different types of inequalities based on whether the sign is >, <, ≥, or ≤ and whether it involves just x and y or a slanted line. Examples are provided to demonstrate how to graph inequalities like y > 3, x - 2, y - 3x + 2, and an applied problem involving nickels and dimes. The key steps for graphing a linear inequality on the coordinate plane are outlined.

Math Proj

The document discusses the student's experience in their math class. It covers several topics:
1) The student reviewed many topics from high school algebra including expressions, equations, functions, and more. Word problems were the hardest.
2) New topics included distance/midpoint formulas, graphing inequalities, and the difference between linear and quadratic functions.
3) The student enjoyed learning everything and found the lab class helpful for asking questions and reviewing lessons. They discovered their ability in math and gained knowledge from their teachers.

Linear inequalities in two variables

The document discusses linear inequalities in two variables and their graphical representations. It introduces the Cartesian coordinate system developed by Rene Descartes and its importance. It explains how to graph linear inequalities by first drawing the line as an equation, then determining whether to shade above or below the line based on whether a test point satisfies the inequality. Students are assigned to bring graphing paper, coloring materials, and a ruler to class on Monday to graph linear inequalities.

Q2 l2 the centipede

1) The narrator finds his dog Biryuk injured after his sister Delia beats him with a stick.
2) The narrator later throws a dead centipede onto his sister's lap in retaliation for her mistreatment of him and his dog.
3) His sister shrieks in fear at the centipede, and the narrator feels momentary guilt for frightening her due to her weak heart, though he remains angry at her past actions.

Q2 l4 the wedding dance

Custom dictates social norms and behaviors, controlling people's feelings and manners like a despotic ruler. The document provides biographical information about Amador T. Daguio, including his birthdate in 1912 in Laoag, Ilocos Norte, graduating with honors from U.P. in 1932, obtaining his M.A. in English from Stanford University in 1952 as a Fulbright scholar, and writing his first work "Man of Earth" at the young age of 20 in 1932. The document also prompts the reader to watch a related video.

Solving of system of linear inequalities

This document discusses linear inequalities in two variables and their graphical representations. It can be summarized as:
1) A linear inequality in two variables has infinitely many solutions that can be represented on a coordinate plane as all points on one side of a boundary line.
2) The graph of a linear inequality consists of all points in a region called a half-plane, bounded by the boundary line. Points on one side of the line are solutions while points on the other side are not solutions.
3) To solve a system of linear inequalities, the inequalities are graphed on the same grid. The solution set contains all points in the region where the graphs overlap, and any points on solid boundary

Linear Inequalities

This document contains the solutions to two systems of linear inequalities graphically. For the first system, the common shaded region represents the solution where x + y ≤ 10, 3x – 4y ≤ 26, x≥0, and y≥0. For the second system, the solution is represented by the region enclosed by the lines 3x+2y = 150, 3x+2y ≥ 150, x + 4y = 80, and x + 4y ≥ 80.

Bandasan, renalyn. philippine literature

Francisco Arcellana was a prominent Filipino writer known for pioneering the modern Filipino short story in English. He wrote many short stories that were included in anthologies and received several literary awards. One of his most famous works is the short story "The Mats", which tells the story of Mr. Angeles bringing home woven mats for each of his family members, including two that represent children who have passed away. When he presents the mats, both he and his wife are overcome with emotion at the memory of their deceased children. The mats symbolize the strong, enduring bonds of the Filipino family.

Figurative language

This document defines and provides examples of various types of figurative language. It discusses figurative language categories such as simile, metaphor, personification, onomatopoeia, hyperbole, litotes, metonymy, synecdoche, epithet, allegory, symbol, irony, oxymoron, and puns. Examples are provided for each type to illustrate how figurative language utilizes creative comparisons and non-literal meanings to enrich writing.

Figurative Language

Figurative language uses comparisons between two unlike things through devices such as similes, metaphors, and personification. Similes directly compare two things using "like" or "as", while metaphors imply a comparison without using those words. Personification gives human traits to non-human objects. Other figures of speech include imagery, which creates mental pictures, and hyperbole, which exaggerates for effect.

Figurative language commericals

The document provides information and examples about different types of figurative language including similes, metaphors, hyperboles, personification, alliteration, onomatopoeia and imagery. It includes instructions for making a figurative language flip chart and examples of identifying different types of figurative language in sentences. Examples of figurative language used in poems, stories, music and advertisements are also provided.

Figurative Language

Figurative Language, basic definitions and examples of terms such as simile, metaphor, personification, irony, etc.

Figures of speech : Basics

This document defines and provides examples of various literary devices and poetic forms, including figures of speech like oxymoron, simile, metaphor and puns. It also outlines poetic forms such as sonnets, epics, elegies and ballads, with details on their line structure and rhyme schemes. Meter is discussed in terms of iambic pentameter. Literary devices covered include imagery, onomatopoeia, alliteration, and allegory.

Figuresof speech

1) The document discusses various figures of speech such as simile, metaphor, alliteration, onomatopoeia, and hyperbole.
2) It provides definitions and examples for each figure of speech.
3) At the end, it includes a short quiz to test the reader's knowledge of identifying these rhetorical devices.

The wedding dance summary

The document summarizes a short story called "Wedding Dance". It describes the characters of Lumnay, a woman left by her husband Awiyao to marry another woman named Madulimay, as Awiyao and Lumnay were unable to have children after 7 years of marriage. On the night of Awiyao and Madulimay's wedding, Awiyao goes to personally invite Lumnay, the best dancer in the tribe, to the traditional wedding dance. However, Lumnay refuses to attend. It is revealed they still love each other but their tribe's customs force them to separate so Awiyao can have a child.

Unit 7.5

Unit 7.5

6.6 Graphing Inequalities In Two Variables

6.6 Graphing Inequalities In Two Variables

Math Proj

Math Proj

Linear inequalities in two variables

Linear inequalities in two variables

Q2 l2 the centipede

Q2 l2 the centipede

Q2 l4 the wedding dance

Q2 l4 the wedding dance

Solving of system of linear inequalities

Solving of system of linear inequalities

Linear Inequalities

Linear Inequalities

Bandasan, renalyn. philippine literature

Bandasan, renalyn. philippine literature

Figurative language

Figurative language

Figurative Language

Figurative Language

Figurative language commericals

Figurative language commericals

Figurative Language

Figurative Language

Figures of speech : Basics

Figures of speech : Basics

Figuresof speech

Figuresof speech

The wedding dance summary

The wedding dance summary

Rational Expressions

The document discusses rational expressions and operations involving them. It begins with an introduction to rational expressions, noting they are algebraic expressions with both the numerator and denominator being polynomials. It then outlines the lessons that will be covered in the module, including illustrating, simplifying, and performing operations on rational expressions. Several examples are then provided of simplifying rational expressions by factoring the numerator and denominator and cancelling common factors. The document also discusses multiplying rational expressions by using the same process as multiplying fractions, and provides examples of multiplying rational expressions.

Rational Expressions Module

This document provides instruction on rational expressions. It begins with a definition of rational expressions as algebraic expressions where both the numerator and denominator are polynomials. The document then outlines the key lessons to be covered: illustrating, simplifying, and performing operations on rational expressions. Examples are provided of simplifying rational expressions by factoring and canceling common factors between the numerator and denominator. The document also demonstrates multiplying rational expressions using the same rules as multiplying fractions, as well as canceling common factors.

Solucao_Marion_Thornton_Dinamica_Classic (1).pdf

This document is the preface to the instructor's manual for Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion. It provides an overview of the contents of the manual, which contains solutions to the end-of-chapter problems from the textbook. The preface notes there are now 509 problems and the solutions range from straightforward to challenging. It stresses the solutions are only for instructors and should not be shared with students.

P1-Chp13-Integration.pptx

1. The document discusses integration and areas under curves. It provides examples of indefinite integration by finding antiderivatives given derivative functions.
2. Definite integration allows finding the exact area under a curve between limits, unlike indefinite integration which has a constant of integration. Examples are worked through of evaluating definite integrals.
3. Problem solving with definite integrals is demonstrated, such as finding the possible values of a constant P given the value of a definite integral involving P.

LEARNING PLAN IN MATH 9 Q1W1

This document outlines a learning plan for a 9th grade mathematics class on patterns and algebra for the first quarter. It includes standards, competencies, lessons, activities and a performance task on solving quadratic equations using different algebraic methods. Students will compare the methods and apply them to design classroom fixtures using measurements and equations. The plan provides instruction, practice and assessments to help students master solving quadratic equations and transfer their knowledge to real-world problems.

Lesson plan in mathematics 9 (illustrations of quadratic equations)

The lesson plan outlines a lesson on quadratic equations. It introduces quadratic equations and their standard form of ax2 + bx + c = 0. Examples are provided to illustrate how to write quadratic equations in standard form given values of a, b, and c or when expanding multiplied linear expressions. Students complete an activity identifying linear and quadratic equations. They are then assessed by writing equations in standard form and identifying the values of a, b, and c.

P2-Chp12-Vectors.pptx

This document provides an overview of vectors in 3 dimensions. It discusses finding the distance between points using Pythagoras' theorem, writing vectors in i, j, k notation, finding the magnitude of a vector, and finding the angle between a vector and the coordinate axes. It also provides examples of using vectors to solve geometric problems by drawing diagrams and comparing coefficients of vectors. The exercises involve finding distances, directions of vectors, parallel vectors, and using vector operations to find missing points to form geometric shapes like parallelograms.

MATRICES-MATHED204.pptx

The document discusses matrices and matrix operations. It defines a matrix as a rectangular array of elements arranged in rows and columns. It provides examples of matrix addition, multiplication, and properties such as commutativity and associativity of addition. Matrix multiplication is defined as the sum of the products of corresponding elements of the first matrix's rows and second matrix's columns. For multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second.

Direct solution of sparse network equations by optimally ordered triangular f...

Triangular factorization method of a power network problem (in form of matrix). Direct solution can be found without calculating inverse matrix which usually considered an exhaustive method, especially in large scale network.

P1-Chp2-Quadratics.pptx

The document provides an overview of key topics in quadratic equations, including solving quadratic equations by factorizing, completing the square, and using the quadratic formula. It discusses why quadratics are important, such as in modeling projectile motion or summations, and provides examples of solving quadratic equations and completing the square to put them in standard form. The document also includes interactive tests and exercises to help students practice these skills in working with quadratic equations.

Unidad 3 tarea 3 grupo208046_379

This document contains a homework assignment on vector spaces from a Linear Algebra course. It includes 4 exercises for students to complete. Exercise 1 involves creating an infographic conceptualizing either combination linear and generated space or axioms and properties of vector spaces. Exercise 2 involves calculating linear combinations of vectors. Exercise 3 involves determining if sets of vectors generate all of R3 or are linearly dependent. Exercise 4 involves calculating the rank of a matrix using Gaussian elimination and determinants, and determining linear dependence or independence. The document provides the questions and tasks for each exercise for two students to complete as part of their group assignment.

Unit-1 Basic Concept of Algorithm.pptx

The document discusses various topics related to algorithms including algorithm design, real-life applications, analysis, and implementation. It specifically covers four algorithms - the taxi algorithm, rent-a-car algorithm, call-me algorithm, and bus algorithm - for getting from an airport to a house. It also provides examples of simple multiplication methods like the American, English, and Russian approaches as well as the divide and conquer method.

CP2-Chp2-Series.pptx

The document discusses power series representations of functions. Power series allow functions that cannot be integrated, like ex2, to be approximated by polynomials which can be integrated. This allows integrals to be approximated to any degree of accuracy. Power series also allow irrational numbers like π to be expressed as an infinite decimal expansion. Power series approximations can be used to find approximate solutions to difficult differential equations.

Chapter 2 - Types of a Function.pdf

This document discusses different types of functions including polynomials, rational functions, radical functions, absolute value functions, exponential functions, logarithmic functions, and trigonometric functions. It provides examples of how to sketch graphs of various functions by completing the square, reflecting, shifting, compressing/expanding, and using properties of exponentials, logarithms, and trigonometric functions. Key aspects like periodicity and domains/ranges are also covered.

Advanced-Differentiation-Rules.pdf

The document discusses rules for differentiating exponential and logarithmic functions with base e. It states that the derivative of the natural exponential function ex is itself, or dex/dx = ex. It proves this by examining limiting values of (1 + x)1/x as x approaches 0, showing it approaches e. For any function f(x) = ex, the derivative is defined as the limit of (f(x+h) - f(x))/h as h approaches 0, which simplifies to dex/dx = ex. Other rules covered include the derivative of the natural logarithm function ln(x) and logarithmic differentiation.

2023 St Jospeh's College Geelong

Students are reminded to show clear working when answering specialist math exam questions. Common errors include unclear working, algebraic or numerical slips, and poor graphing skills. Time management is important as most questions cannot be fully answered within the time allocated. Exam preparation requires practicing past exams, understanding marking rubrics, and developing strategies for question selection and time planning.

Other operations with exponents

This document discusses operations with exponents such as addition, subtraction, and order of operations. It provides examples of adding terms with the same base but different exponents (a^m + a^n) and shows that they cannot be combined. It also gives examples of subtracting and multiplying terms with exponents through substitution to prove the rules. The document demonstrates applying order of operations, like first completing multiplication before addition when terms are combined.

GraphTransformations.pptx

1. This document discusses graph transformations of functions, including translations, stretches, and reflections. It provides rules for how modifications inside and outside the function f(x) will affect the x-values and y-values of the graph.
2. Examples are given of applying transformation rules to specific points on a graph and determining the new coordinates. The document also demonstrates sketching a transformed graph using key points.
3. An exercise section provides multiple choice and short answer questions to test understanding of describing transformations and finding coordinates of transformed points.

TABREZ KHAN.ppt

The document is a maths project report for class 12th student Tabrez Khan on the topic of determinants. It contains definitions and properties of determinants of order 1, 2 and 3 matrices. It discusses minors, cofactors and applications of determinants like solving systems of linear equations using Cramer's rule. It also contains examples of evaluating determinants and applying properties of determinants to simplify expressions.

Rational Expressions

Rational Expressions

Rational Expressions Module

Rational Expressions Module

Solucao_Marion_Thornton_Dinamica_Classic (1).pdf

Solucao_Marion_Thornton_Dinamica_Classic (1).pdf

P1-Chp13-Integration.pptx

P1-Chp13-Integration.pptx

LEARNING PLAN IN MATH 9 Q1W1

LEARNING PLAN IN MATH 9 Q1W1

Lesson plan in mathematics 9 (illustrations of quadratic equations)

Lesson plan in mathematics 9 (illustrations of quadratic equations)

CP1-Chp6-Matrices (2).pptx used for revision

CP1-Chp6-Matrices (2).pptx used for revision

P2-Chp12-Vectors.pptx

P2-Chp12-Vectors.pptx

MATRICES-MATHED204.pptx

MATRICES-MATHED204.pptx

Direct solution of sparse network equations by optimally ordered triangular f...

Direct solution of sparse network equations by optimally ordered triangular f...

P1-Chp2-Quadratics.pptx

P1-Chp2-Quadratics.pptx

Unidad 3 tarea 3 grupo208046_379

Unidad 3 tarea 3 grupo208046_379

Unit-1 Basic Concept of Algorithm.pptx

Unit-1 Basic Concept of Algorithm.pptx

CP2-Chp2-Series.pptx

CP2-Chp2-Series.pptx

Chapter 2 - Types of a Function.pdf

Chapter 2 - Types of a Function.pdf

Advanced-Differentiation-Rules.pdf

Advanced-Differentiation-Rules.pdf

2023 St Jospeh's College Geelong

2023 St Jospeh's College Geelong

Other operations with exponents

Other operations with exponents

GraphTransformations.pptx

GraphTransformations.pptx

TABREZ KHAN.ppt

TABREZ KHAN.ppt

220711130100 udita Chakraborty Aims and objectives of national policy on inf...

Aims and objectives of national policy on information and communication technology(ICT) in school education in india

Bonku-Babus-Friend by Sathyajith Ray (9)

Bonku-Babus-Friend by Sathyajith Ray for class 9 ksb

Accounting for Restricted Grants When and How To Record Properly

In this webinar, member learned how to stay in compliance with generally accepted accounting principles (GAAP) for restricted grants.

Pharmaceutics Pharmaceuticals best of brub

First year pharmacy
Best for u

Standardized tool for Intelligence test.

ASSESSMENT OF INTELLIGENCE USING WITH STANDARDIZED TOOL

How to Download & Install Module From the Odoo App Store in Odoo 17

Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.

HYPERTENSION - SLIDE SHARE PRESENTATION.

IT WILL BE HELPFULL FOR THE NUSING STUDENTS
IT FOCUSED ON MEDICAL MANAGEMENT AND NURSING MANAGEMENT.
HIGHLIGHTS ON HEALTH EDUCATION.

Ch-4 Forest Society and colonialism 2.pdf

All the best

How to Fix [Errno 98] address already in use

This slide will represent the cause of the error “[Errno 98] address already in use” and the troubleshooting steps to resolve this error in Odoo.

Observational Learning

Simple Presentation

Data Structure using C by Dr. K Adisesha .ppsx

Data Structure using C ppt by Dr. K. Adisesha

skeleton System.pdf (skeleton system wow)

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إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
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تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
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THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...

The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.

INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION

The document discuss about the hospitals and it's organization .

CapTechTalks Webinar Slides June 2024 Donovan Wright.pptx

Slides from a Capitol Technology University webinar held June 20, 2024. The webinar featured Dr. Donovan Wright, presenting on the Department of Defense Digital Transformation.

Educational Technology in the Health Sciences

Plenary presentation at the NTTC Inter-university Workshop, 18 June 2024, Manila Prince Hotel.

CHUYÊN ĐỀ ÔN TẬP VÀ PHÁT TRIỂN CÂU HỎI TRONG ĐỀ MINH HỌA THI TỐT NGHIỆP THPT ...

CHUYÊN ĐỀ ÔN TẬP VÀ PHÁT TRIỂN CÂU HỎI TRONG ĐỀ MINH HỌA THI TỐT NGHIỆP THPT ...Nguyen Thanh Tu Collection

https://app.box.com/s/qspvswamcohjtbvbbhjad04lg65waylfBrand Guideline of Bashundhara A4 Paper - 2024

It outlines the basic identity elements such as symbol, logotype, colors, and typefaces. It provides examples of applying the identity to materials like letterhead, business cards, reports, folders, and websites.

A Free 200-Page eBook ~ Brain and Mind Exercise.pptx

(A Free eBook comprising 3 Sets of Presentation of a selection of Puzzles, Brain Teasers and Thinking Problems to exercise both the mind and the Right and Left Brain. To help keep the mind and brain fit and healthy. Good for both the young and old alike.
Answers are given for all the puzzles and problems.)
With Metta,
Bro. Oh Teik Bin 🙏🤓🤔🥰

Diversity Quiz Prelims by Quiz Club, IIT Kanpur

Prelims of Diversity Quiz, conducted by Quiz Club, IIT Kanpur in collaboration with Unmukt on 22 June 2024.

220711130100 udita Chakraborty Aims and objectives of national policy on inf...

220711130100 udita Chakraborty Aims and objectives of national policy on inf...

Bonku-Babus-Friend by Sathyajith Ray (9)

Bonku-Babus-Friend by Sathyajith Ray (9)

Accounting for Restricted Grants When and How To Record Properly

Accounting for Restricted Grants When and How To Record Properly

Pharmaceutics Pharmaceuticals best of brub

Pharmaceutics Pharmaceuticals best of brub

Standardized tool for Intelligence test.

Standardized tool for Intelligence test.

How to Download & Install Module From the Odoo App Store in Odoo 17

How to Download & Install Module From the Odoo App Store in Odoo 17

HYPERTENSION - SLIDE SHARE PRESENTATION.

HYPERTENSION - SLIDE SHARE PRESENTATION.

Ch-4 Forest Society and colonialism 2.pdf

Ch-4 Forest Society and colonialism 2.pdf

How to Fix [Errno 98] address already in use

How to Fix [Errno 98] address already in use

Observational Learning

Observational Learning

Data Structure using C by Dr. K Adisesha .ppsx

Data Structure using C by Dr. K Adisesha .ppsx

skeleton System.pdf (skeleton system wow)

skeleton System.pdf (skeleton system wow)

THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...

THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...

INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION

INTRODUCTION TO HOSPITALS & AND ITS ORGANIZATION

CapTechTalks Webinar Slides June 2024 Donovan Wright.pptx

CapTechTalks Webinar Slides June 2024 Donovan Wright.pptx

Educational Technology in the Health Sciences

Educational Technology in the Health Sciences

CHUYÊN ĐỀ ÔN TẬP VÀ PHÁT TRIỂN CÂU HỎI TRONG ĐỀ MINH HỌA THI TỐT NGHIỆP THPT ...

CHUYÊN ĐỀ ÔN TẬP VÀ PHÁT TRIỂN CÂU HỎI TRONG ĐỀ MINH HỌA THI TỐT NGHIỆP THPT ...

Brand Guideline of Bashundhara A4 Paper - 2024

Brand Guideline of Bashundhara A4 Paper - 2024

A Free 200-Page eBook ~ Brain and Mind Exercise.pptx

A Free 200-Page eBook ~ Brain and Mind Exercise.pptx

Diversity Quiz Prelims by Quiz Club, IIT Kanpur

Diversity Quiz Prelims by Quiz Club, IIT Kanpur

- 1. Rectangular Coordinate Sys tem Relations and Functions Representations of Relations and Functions Ordered Pairs Equations/ Formulas Domain and Dependent Independent Variables Slope and Intercepts 6 5 4 3 2 1 7 8 0 −1 2 0 0 2 9 1 2 4 1 Paper #3a CARYL MAE S. PUERTOLLANO Subject: ICT in Mathematics Education Professor: Ms. Rosemarie Galvez * Create a Handout using MS Word 1. SmartArt Here is a sample map of the lessons that will be covered in this module: 2a. Linear Algebra Mapping Diagram Table Graphs Range and Appl ications PARTITION OF MATRICES; BLOCK MULTIPLICATION Linear Functions We now consider the following partitioned matrices of the same size to see how the operations work in practice: 퐴 = [ 1 2 3 4 6 0 7 8 0 −3 1 6 0 2 9 1 2 4 ], 퐵 = [ 0 2 −1 0 −3 5 1 −2 4 −2 2 −5 4 −4 6 1 0 −4 ], 퐶 = [ ] Now, A + B can be computed blockwise as A and B are partitioned the same way with corresponding blocks or submatrices having the same sizes. However, even though A and C are
- 2. matrices of the same size (so A + C can be computed adding elements entrywise) A + C cannot be computed blockwise as the blocks of A and C are not comparable. 2b. Number Theory Properties of Summation 1. Σ 푐 푛푖 =1 = 푐 + 푐 + ⋯ + 푐 = 푐푛. 2. Σ 푐푥푖 = 푐Σ 푥푖 . 푛푖 =1 2 푛푖 =1 3. Σ (푥푖 + 푦푖 ) = Σ 푥푖 + Σ 푦푖 . 푛푖 =1 푛푖 = 1 푛푖 =푛 2c. Algebra Simplifying Rational Algebraic Expression 푥2+3푥+2 푥2−1 Solution: 푥2+3푥+2 푥2−1 = (푥+1)(푥+2) (푥+1)(푥−1) = 푥+2 푥−1 2d. Calculus Evaluate ∫ 푥2 푥3 +1 푑푥 1 0 Solution: 푢 = 푥 3 + 1 푑푢 = 3푥 2푑푥 푑푢 3푥2 = 푑푥 ∫ 푥 2 푥 3 + 1 푑푥 = ∫ 푥 2 푢 ∙ 푑푢 3푥 2 = 1 3 ∫ 1 푢 푑푢 = What factoring method is used in this step? 1 3 푥=1 푥=0 푙푛|푢| = 1 3 1 푙푛|푥 3 + 1|| 0 푥=1 푥=0 1 0 = 1 3 푙푛|1 + 1| − 1 3 푙푛|0 + 1| = 1 3 푙푛|2| − 1 3 푙푛|1| = 1 3 푙푛|2| − 0 = 1 3 푙푛|2|
- 3. 3 2e. Trigonometry The graph of 푦 = 푠푖푛푥 is shown below. 3. Geometric figures with labels. 1. Name the parts of the circle. F G H D K L M A B C E 1 -1 휋 2 3휋 2휋 3휋 2 ● ● ●