Linear equation in one variable for class VIII by G R Ahmed MD. G R Ahmed
This document discusses linear equations in one variable. It defines linear equations as those where the highest power of the variable is 1. It provides examples of linear and non-linear equations. It also discusses how to determine if a value is a solution to a linear equation and how to solve linear equations using the addition, subtraction, and multiplication properties of equality. It provides examples of solving linear equations with fractions and with consecutive integers.
The document discusses using definite integrals to find the area under a curve. It provides examples of finding the area of various shaded regions bounded by curves and the x-axis using the formula: area = ∫f(x)dx from x=a to x=b. It also shows how to find the coordinates of points where two curves intersect in order to divide the region into multiple areas and calculate them separately.
Este documento presenta 20 trinomios para ser factorizados por estudiantes de octavo grado siguiendo casos específicos. La maestra Marcela Torres guiará a los estudiantes en factorizar cada trinomio utilizando las técnicas apropiadas como la suma y diferencia de cuadrados o cubos.
The document provides instructions and examples for factoring quadratic expressions using the box method. It explains the steps as:
1) Draw a box and write the coefficients of the quadratic term, linear term, and constant term.
2) Find two numbers whose sum is the linear coefficient and whose product is the constant term.
3) Write these two numbers above and below the box to factor the expression.
Several examples are worked out step-by-step and the reader is prompted to try additional problems and check their own work.
El documento describe los sistemas numéricos binario y decimal, y cómo realizar operaciones como convertir entre ellos, sumar, restar, multiplicar y dividir números binarios. Explica los pasos para cada operación y provee ejemplos ilustrativos.
The documents show tables for multiplying numbers 2-8 by 1-10. Each table is presented as a grid with dots representing the products. The student is asked to fill in the missing products below each table. This allows the student to practice and study their multiplication tables.
- Two functions, f(x) and g(x), are combined in similar ways to addition, subtraction, multiplication, and division of real numbers. This allows the formation of new functions.
- The document demonstrates examples of combining f(x)=x+1 and g(x)=2x using addition, subtraction, multiplication, and division. It shows the algebraic steps to derive each new function h(x) and evaluates h(x) for various x values.
- Graphs of the functions h(x) are provided to illustrate the relationships between the combined functions.
This document lists and summarizes various tools that are useful for competitive programming on AtCoder. It includes websites for problem archives, rating simulators, online databases for integer sequences, tools for replaying past contests, comparing ratings, and more. It also lists several Chrome extensions that provide additional features for AtCoder, such as coloring solved problems or displaying statistics.
This document defines and provides examples of quadratic functions. A quadratic function is a polynomial function of degree two in the form f(x) = ax^2 + bx + c, where a ≠ 0. It must have the highest degree of 2, a non-zero coefficient for x^2, and no negative or rational exponents. Examples are provided to identify quadratic functions from sets of equations. The graph of a quadratic function is a parabola that can open upward or downward depending on the sign of a. Additional examples demonstrate how changing the coefficients affects the graph shape. The vertex form of a quadratic function is given as f(x) = a(x-h)^2 + k, where the vertex is (
1. This document provides explanations for the AtCoder Regular Contest 038 programming problems in Japanese.
2. It describes the logic and pseudocode for solving a problem about determining if a knight piece on a chessboard will win or lose from its current position. Dynamic programming is used to store and lookup previously computed results.
3. It then provides sample test cases and outputs for the knight problem.
Este documento descreve uma técnica para apostar na loteria Lotofácil utilizando 14 dezenas fixas em 11 apostas diferentes. Ao usar esta estratégia, aumenta-se a chance de acertar de 10 a 14 dezenas e ganhar prêmios que podem chegar a R$3.800. A técnica envolve escolher 14 dezenas fixas e completar cada um dos 11 bilhetes com uma das 11 dezenas restantes.
Linear equation in one variable for class VIII by G R Ahmed MD. G R Ahmed
This document discusses linear equations in one variable. It defines linear equations as those where the highest power of the variable is 1. It provides examples of linear and non-linear equations. It also discusses how to determine if a value is a solution to a linear equation and how to solve linear equations using the addition, subtraction, and multiplication properties of equality. It provides examples of solving linear equations with fractions and with consecutive integers.
The document discusses using definite integrals to find the area under a curve. It provides examples of finding the area of various shaded regions bounded by curves and the x-axis using the formula: area = ∫f(x)dx from x=a to x=b. It also shows how to find the coordinates of points where two curves intersect in order to divide the region into multiple areas and calculate them separately.
Este documento presenta 20 trinomios para ser factorizados por estudiantes de octavo grado siguiendo casos específicos. La maestra Marcela Torres guiará a los estudiantes en factorizar cada trinomio utilizando las técnicas apropiadas como la suma y diferencia de cuadrados o cubos.
The document provides instructions and examples for factoring quadratic expressions using the box method. It explains the steps as:
1) Draw a box and write the coefficients of the quadratic term, linear term, and constant term.
2) Find two numbers whose sum is the linear coefficient and whose product is the constant term.
3) Write these two numbers above and below the box to factor the expression.
Several examples are worked out step-by-step and the reader is prompted to try additional problems and check their own work.
El documento describe los sistemas numéricos binario y decimal, y cómo realizar operaciones como convertir entre ellos, sumar, restar, multiplicar y dividir números binarios. Explica los pasos para cada operación y provee ejemplos ilustrativos.
The documents show tables for multiplying numbers 2-8 by 1-10. Each table is presented as a grid with dots representing the products. The student is asked to fill in the missing products below each table. This allows the student to practice and study their multiplication tables.
- Two functions, f(x) and g(x), are combined in similar ways to addition, subtraction, multiplication, and division of real numbers. This allows the formation of new functions.
- The document demonstrates examples of combining f(x)=x+1 and g(x)=2x using addition, subtraction, multiplication, and division. It shows the algebraic steps to derive each new function h(x) and evaluates h(x) for various x values.
- Graphs of the functions h(x) are provided to illustrate the relationships between the combined functions.
This document lists and summarizes various tools that are useful for competitive programming on AtCoder. It includes websites for problem archives, rating simulators, online databases for integer sequences, tools for replaying past contests, comparing ratings, and more. It also lists several Chrome extensions that provide additional features for AtCoder, such as coloring solved problems or displaying statistics.
This document defines and provides examples of quadratic functions. A quadratic function is a polynomial function of degree two in the form f(x) = ax^2 + bx + c, where a ≠ 0. It must have the highest degree of 2, a non-zero coefficient for x^2, and no negative or rational exponents. Examples are provided to identify quadratic functions from sets of equations. The graph of a quadratic function is a parabola that can open upward or downward depending on the sign of a. Additional examples demonstrate how changing the coefficients affects the graph shape. The vertex form of a quadratic function is given as f(x) = a(x-h)^2 + k, where the vertex is (
1. This document provides explanations for the AtCoder Regular Contest 038 programming problems in Japanese.
2. It describes the logic and pseudocode for solving a problem about determining if a knight piece on a chessboard will win or lose from its current position. Dynamic programming is used to store and lookup previously computed results.
3. It then provides sample test cases and outputs for the knight problem.
Este documento descreve uma técnica para apostar na loteria Lotofácil utilizando 14 dezenas fixas em 11 apostas diferentes. Ao usar esta estratégia, aumenta-se a chance de acertar de 10 a 14 dezenas e ganhar prêmios que podem chegar a R$3.800. A técnica envolve escolher 14 dezenas fixas e completar cada um dos 11 bilhetes com uma das 11 dezenas restantes.