This document provides an overview of key concepts in real numbers and geometry. It defines sets and set operations like union, intersection, and difference. It describes the properties of real numbers, including natural numbers, integers, fractions, algebraic numbers, and transcendental numbers. The document also covers inequalities, absolute value, the number line, distance and midpoint formulas, and representations of conic sections like circles, parabolas, ellipses, and hyperbolas. Examples are provided to illustrate solving inequalities with absolute value, finding distance and midpoint, and the standard forms of conic sections. In conclusion, it references two textbooks on calculus and geometry.
This document is from IFET College of Engineering and presents information on solving second order linear differential equations with constant coefficients. It defines such an equation as one where the highest order derivative is of order 2 and all coefficients are constants. The general solution is described as the sum of the complementary function and particular integral. Various cases are discussed for the complementary function depending on whether the roots are real/complex and distinct or repeated. Methods like variation of parameters and Cauchy's and Legendre's equations are also mentioned for solving related problems.
The document defines the determinant of a square matrix. For a 1x1 matrix with value k, the determinant is defined to be k. For a 2x2 matrix with values a, b, c, d, the determinant is defined as ad - bc. This definition is motivated geometrically as representing the signed area of the parallelogram formed by the vector points (a,b) and (c,d). It is also motivated algebraically in that a system of equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero. Cramer's rule is presented for solving systems of linear equations.
5 4 equations that may be reduced to quadraticsmath123b
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting (x/(x-1)) with y, then solving the resulting quadratic equation for y and back substituting to find values of x. This reduction technique converts difficult equations into two easier equations to solve: the quadratic after substitution and then solving for the original variable.
Mcq differential and ordinary differential equationSayyad Shafi
This document contains multiple choice questions about differentiation, ordinary differential equations, and partial differential equations. Some key points covered are:
- The order of a differential equation is the highest derivative present. The degree is the exponent of the highest derivative.
- A partial differential equation has two or more independent variables.
- The steps to obtain a differential equation from a given function are to differentiate with respect to the independent variable, and continue differentiating until the number of arbitrary constants is reached.
- The solution of a second-order differential equation contains two arbitrary constants.
- Linear differential equations have dependent variables and derivatives that are of first degree only, with no product or transcendental terms.
The document provides properties of determinants and examples of their applications. It also gives tips for solving problems based on properties of determinants. Finally, it lists 20 assignment questions related to matrices and determinants, covering topics like solving systems of equations using matrices, finding the inverse of a matrix, and applying properties of determinants.
The question asks to solve a system of linear inequalities graphically. The system is: x + y ≤ 9, y > x, x ≥ 0. The solution region is the shaded area where all the individual inequality regions overlap, which is the triangular region bounded by the lines y = x, x = 0, and x + y = 9.
The document describes three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It can only be used if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document is from IFET College of Engineering and presents information on solving second order linear differential equations with constant coefficients. It defines such an equation as one where the highest order derivative is of order 2 and all coefficients are constants. The general solution is described as the sum of the complementary function and particular integral. Various cases are discussed for the complementary function depending on whether the roots are real/complex and distinct or repeated. Methods like variation of parameters and Cauchy's and Legendre's equations are also mentioned for solving related problems.
The document defines the determinant of a square matrix. For a 1x1 matrix with value k, the determinant is defined to be k. For a 2x2 matrix with values a, b, c, d, the determinant is defined as ad - bc. This definition is motivated geometrically as representing the signed area of the parallelogram formed by the vector points (a,b) and (c,d). It is also motivated algebraically in that a system of equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero. Cramer's rule is presented for solving systems of linear equations.
5 4 equations that may be reduced to quadraticsmath123b
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting (x/(x-1)) with y, then solving the resulting quadratic equation for y and back substituting to find values of x. This reduction technique converts difficult equations into two easier equations to solve: the quadratic after substitution and then solving for the original variable.
Mcq differential and ordinary differential equationSayyad Shafi
This document contains multiple choice questions about differentiation, ordinary differential equations, and partial differential equations. Some key points covered are:
- The order of a differential equation is the highest derivative present. The degree is the exponent of the highest derivative.
- A partial differential equation has two or more independent variables.
- The steps to obtain a differential equation from a given function are to differentiate with respect to the independent variable, and continue differentiating until the number of arbitrary constants is reached.
- The solution of a second-order differential equation contains two arbitrary constants.
- Linear differential equations have dependent variables and derivatives that are of first degree only, with no product or transcendental terms.
The document provides properties of determinants and examples of their applications. It also gives tips for solving problems based on properties of determinants. Finally, it lists 20 assignment questions related to matrices and determinants, covering topics like solving systems of equations using matrices, finding the inverse of a matrix, and applying properties of determinants.
The question asks to solve a system of linear inequalities graphically. The system is: x + y ≤ 9, y > x, x ≥ 0. The solution region is the shaded area where all the individual inequality regions overlap, which is the triangular region bounded by the lines y = x, x = 0, and x + y = 9.
The document describes three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It can only be used if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document contains 10 math questions related to trigonometric functions, sets, relations and functions, complex numbers, sequences and series, straight lines, conic sections. The questions range from proving identities and equations to finding specific values based on given information. They require various trigonometric, algebraic and geometric problem solving skills at a higher-order thinking level.
This document provides a summary of topics related to algebra, functions, and calculus including: linear and quadratic expressions, simultaneous equations, completing the square, trigonometric ratios, differentiation, tangents, normals, and finding stationary points through higher derivatives. It outlines key steps and methods for solving various types of problems within these topics.
- The document discusses graphs of linear and quadratic equations.
- Linear equations produce straight line graphs, while quadratic equations produce curved graphs called parabolas.
- To graph a parabola, one finds the vertex using the vertex formula, then makes a table of x and y values centered around the vertex to plot the points symmetrically.
This document provides information about various geometric concepts in Cartesian coordinates (R2). It defines R2 as the set of all ordered pairs (a,b) of real numbers. It discusses representing points in R2 using coordinates, and defines concepts like distance, midpoint, linear equations, circles, parabolas, ellipses, and hyperbolas. It provides examples of finding distances between points, finding midpoints of line segments, graphing linear equations, finding equations of circles, and identifying graphs of parabolas, ellipses and hyperbolas based on their standard equations.
The document discusses determinants of matrices and their geometric interpretations. It begins by defining a matrix as a rectangular table of numbers or formulas. It then explains that the determinant of a 2x2 matrix gives the signed area of the parallelogram defined by the row vectors of the matrix. The sign of the determinant indicates whether the parallelogram is swept clockwise or counterclockwise. Finally, it generalizes these concepts to 3x3 matrices by writing out the formula for the determinant as a sum of signed areas of parallelograms.
The document provides an overview of quadratic equations, including:
1) Quadratic equations take the form of ax^2 + bx + c = 0 and involve an unknown (x) and known coefficients (a, b, c).
2) Quadratic equations can be solved through factoring, completing the square, using the quadratic formula, or graphing.
3) The quadratic formula provides the general solution for quadratic equations as x = (-b ± √(b^2 - 4ac))/2a.
The document discusses graphs of quadratic equations. It explains that quadratic equations form parabolic graphs rather than straight lines. It provides examples of graphing quadratic functions by first finding the vertex using a formula, then making a table of x and y values centered around the vertex to plot points symmetrically. Key properties of parabolas are that they are symmetric around the vertex, which is the highest/lowest point on the center line.
The document provides an overview of terms and steps used to solve linear and quadratic equations. It defines variables, coefficients, constants, expressions, and equations. It then outlines the steps to solve linear equations which are: 1) simplify, 2) move variables, 3) isolate variables by undoing addition/subtraction and multiplication/division, and 4) check the answer. Examples are provided to demonstrate each step. The document also provides an overview of solving quadratic equations by factoring or using the quadratic formula. More practice problems are provided for the reader to solve.
The document discusses first degree (linear) functions. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential-log formulas. Linear functions of the form f(x)=mx+b are especially important, where m is the slope and b is the y-intercept. The slope-intercept form allows expressions of the form Ax+By=C to be written as functions with y as the output. Examples are given of finding the slope and form of linear equations.
The document discusses key concepts in coordinate geometry, including:
- Length, gradient, and midpoint of lines between two points
- Finding the distance and equation of a line given two points or the gradient and one point
- Parallel and perpendicular lines having related gradients
- Finding the coordinates of points of intersection between two lines or a line and curve by solving their equations simultaneously
- Using the discriminant of a quadratic equation to determine the number of intersection points between a curve and line
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMSRACSOelimu
1. A function is a relationship between inputs and outputs where each input corresponds to exactly one output. The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
2. For a function to be one-to-one, each output must correspond to only one input. This can be tested using the horizontal line test - drawing horizontal lines on the graph. Restricting the domain can make non-one-to-one functions one-to-one.
3. The inverse of a function undoes the input-output relationship by switching the domain and range. Only one-to-one functions have inverses. The graph of an inverse function passes the vertical
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
The document discusses the difference quotient formula for calculating the slope between two points (x1,y1) and (x2,y2) on a function y=f(x). It shows that the slope m is equal to (f(x+h)-f(x))/h, where h is the difference between x1 and x2. This "difference quotient" formula allows slopes to be calculated from the values of a function at two nearby points. Examples are given of simplifying the difference quotient for quadratic and rational functions.
1. This document provides a multiple choice questions from the topic of Ordinary Differential Equations of Higher Order that is part of the Engineering Mathematics-II course compiled by Dr. Deepa Chauhan.
2. It contains 43 multiple choice questions testing concepts related to differential equations including finding roots of auxiliary equations, determining orders and degrees of equations, finding particular integrals, solving differential equations, and more.
3. The questions are accompanied by options for the answers.
The document discusses three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It is only applicable if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
1. The document provides an overview of important topics covered in Form 4 and Form 5 mathematics. These include functions, quadratic equations, trigonometry, statistics, calculus, and coordinate geometry.
2. Examples of how to solve different types of problems are given for each topic, such as finding the sum and product of roots for quadratic equations or using rules of logarithms to simplify logarithmic expressions.
3. Strategies for solving problems involving concepts like differentiation, integration, progressions, and linear laws are outlined. Methods for finding volumes or areas under curves are also summarized briefly.
Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.
The document provides information on various math topics including:
1. Graph transformations including stretching and compressing graphs along the x and y axes.
2. Similarity and congruency of triangles.
3. Differentiation including differentiating polynomials and finding derivatives.
4. Integration including integrating polynomials and using integration to find areas.
5. Kinematics equations for velocity, acceleration, and displacement.
6. The binomial distribution and Pascal's triangle for expanding binomial expressions.
7. Using the discriminant of a quadratic equation to determine the nature of its roots.
This document defines sets and subsets, classifies different types of sets such as finite, infinite, empty and unit sets. It also discusses operations on sets like union and intersection. Real number sets such as natural, integer, rational and irrational numbers are defined. Inequalities, absolute value inequalities and their properties are explained. Intervals such as open, closed and infinite intervals are classified. The numeric plane and Cartesian product are defined. Graphical representations of conic sections like ellipses, circles, parabolas and hyperbolas are shown. Examples of solving inequalities and simplifying fractions are provided.
The document discusses equations and their definitions and classifications. It defines equality, equations, identities, variables, terms of an equation, numerical and literal equations, types of equations including polynomial, rational, radical, and absolute value equations. It provides examples of solving linear, rational, and word problems involving equations. Key steps in solving equations are outlined such as isolating the variable, using properties of equality, and verifying solutions.
This document contains 10 math questions related to trigonometric functions, sets, relations and functions, complex numbers, sequences and series, straight lines, conic sections. The questions range from proving identities and equations to finding specific values based on given information. They require various trigonometric, algebraic and geometric problem solving skills at a higher-order thinking level.
This document provides a summary of topics related to algebra, functions, and calculus including: linear and quadratic expressions, simultaneous equations, completing the square, trigonometric ratios, differentiation, tangents, normals, and finding stationary points through higher derivatives. It outlines key steps and methods for solving various types of problems within these topics.
- The document discusses graphs of linear and quadratic equations.
- Linear equations produce straight line graphs, while quadratic equations produce curved graphs called parabolas.
- To graph a parabola, one finds the vertex using the vertex formula, then makes a table of x and y values centered around the vertex to plot the points symmetrically.
This document provides information about various geometric concepts in Cartesian coordinates (R2). It defines R2 as the set of all ordered pairs (a,b) of real numbers. It discusses representing points in R2 using coordinates, and defines concepts like distance, midpoint, linear equations, circles, parabolas, ellipses, and hyperbolas. It provides examples of finding distances between points, finding midpoints of line segments, graphing linear equations, finding equations of circles, and identifying graphs of parabolas, ellipses and hyperbolas based on their standard equations.
The document discusses determinants of matrices and their geometric interpretations. It begins by defining a matrix as a rectangular table of numbers or formulas. It then explains that the determinant of a 2x2 matrix gives the signed area of the parallelogram defined by the row vectors of the matrix. The sign of the determinant indicates whether the parallelogram is swept clockwise or counterclockwise. Finally, it generalizes these concepts to 3x3 matrices by writing out the formula for the determinant as a sum of signed areas of parallelograms.
The document provides an overview of quadratic equations, including:
1) Quadratic equations take the form of ax^2 + bx + c = 0 and involve an unknown (x) and known coefficients (a, b, c).
2) Quadratic equations can be solved through factoring, completing the square, using the quadratic formula, or graphing.
3) The quadratic formula provides the general solution for quadratic equations as x = (-b ± √(b^2 - 4ac))/2a.
The document discusses graphs of quadratic equations. It explains that quadratic equations form parabolic graphs rather than straight lines. It provides examples of graphing quadratic functions by first finding the vertex using a formula, then making a table of x and y values centered around the vertex to plot points symmetrically. Key properties of parabolas are that they are symmetric around the vertex, which is the highest/lowest point on the center line.
The document provides an overview of terms and steps used to solve linear and quadratic equations. It defines variables, coefficients, constants, expressions, and equations. It then outlines the steps to solve linear equations which are: 1) simplify, 2) move variables, 3) isolate variables by undoing addition/subtraction and multiplication/division, and 4) check the answer. Examples are provided to demonstrate each step. The document also provides an overview of solving quadratic equations by factoring or using the quadratic formula. More practice problems are provided for the reader to solve.
The document discusses first degree (linear) functions. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential-log formulas. Linear functions of the form f(x)=mx+b are especially important, where m is the slope and b is the y-intercept. The slope-intercept form allows expressions of the form Ax+By=C to be written as functions with y as the output. Examples are given of finding the slope and form of linear equations.
The document discusses key concepts in coordinate geometry, including:
- Length, gradient, and midpoint of lines between two points
- Finding the distance and equation of a line given two points or the gradient and one point
- Parallel and perpendicular lines having related gradients
- Finding the coordinates of points of intersection between two lines or a line and curve by solving their equations simultaneously
- Using the discriminant of a quadratic equation to determine the number of intersection points between a curve and line
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMSRACSOelimu
1. A function is a relationship between inputs and outputs where each input corresponds to exactly one output. The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
2. For a function to be one-to-one, each output must correspond to only one input. This can be tested using the horizontal line test - drawing horizontal lines on the graph. Restricting the domain can make non-one-to-one functions one-to-one.
3. The inverse of a function undoes the input-output relationship by switching the domain and range. Only one-to-one functions have inverses. The graph of an inverse function passes the vertical
The document defines matrices and their properties, including symmetric, skew-symmetric, and determinant. It provides examples of solving systems of equations using matrices and their inverses. It also discusses properties of determinants, including properties related to symmetric and skew-symmetric matrices. Inverse trigonometric functions are defined, including their domains, ranges, and relationships between inverse functions using addition and subtraction formulas. Sample problems are provided to solve systems of equations and evaluate determinants.
The document discusses the difference quotient formula for calculating the slope between two points (x1,y1) and (x2,y2) on a function y=f(x). It shows that the slope m is equal to (f(x+h)-f(x))/h, where h is the difference between x1 and x2. This "difference quotient" formula allows slopes to be calculated from the values of a function at two nearby points. Examples are given of simplifying the difference quotient for quadratic and rational functions.
1. This document provides a multiple choice questions from the topic of Ordinary Differential Equations of Higher Order that is part of the Engineering Mathematics-II course compiled by Dr. Deepa Chauhan.
2. It contains 43 multiple choice questions testing concepts related to differential equations including finding roots of auxiliary equations, determining orders and degrees of equations, finding particular integrals, solving differential equations, and more.
3. The questions are accompanied by options for the answers.
The document discusses three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It is only applicable if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
1. The document provides an overview of important topics covered in Form 4 and Form 5 mathematics. These include functions, quadratic equations, trigonometry, statistics, calculus, and coordinate geometry.
2. Examples of how to solve different types of problems are given for each topic, such as finding the sum and product of roots for quadratic equations or using rules of logarithms to simplify logarithmic expressions.
3. Strategies for solving problems involving concepts like differentiation, integration, progressions, and linear laws are outlined. Methods for finding volumes or areas under curves are also summarized briefly.
Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.
The document provides information on various math topics including:
1. Graph transformations including stretching and compressing graphs along the x and y axes.
2. Similarity and congruency of triangles.
3. Differentiation including differentiating polynomials and finding derivatives.
4. Integration including integrating polynomials and using integration to find areas.
5. Kinematics equations for velocity, acceleration, and displacement.
6. The binomial distribution and Pascal's triangle for expanding binomial expressions.
7. Using the discriminant of a quadratic equation to determine the nature of its roots.
This document defines sets and subsets, classifies different types of sets such as finite, infinite, empty and unit sets. It also discusses operations on sets like union and intersection. Real number sets such as natural, integer, rational and irrational numbers are defined. Inequalities, absolute value inequalities and their properties are explained. Intervals such as open, closed and infinite intervals are classified. The numeric plane and Cartesian product are defined. Graphical representations of conic sections like ellipses, circles, parabolas and hyperbolas are shown. Examples of solving inequalities and simplifying fractions are provided.
The document discusses equations and their definitions and classifications. It defines equality, equations, identities, variables, terms of an equation, numerical and literal equations, types of equations including polynomial, rational, radical, and absolute value equations. It provides examples of solving linear, rational, and word problems involving equations. Key steps in solving equations are outlined such as isolating the variable, using properties of equality, and verifying solutions.
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
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.
The document provides an overview of various topics in analytic geometry, including circle equations, distance equations, systems of two and three variable equations, linear inequalities, rational inequalities, and intersections of inequalities. It defines key concepts, provides examples of how to solve different types of problems, and notes things to remember when working with inequalities.
The document defines sets and set operations like union, intersection, and difference. It discusses the properties of real numbers including rational and irrational numbers. It also covers topics like absolute value, inequalities, distance between points, and the midpoint formula. Examples are provided to illustrate definitions of circles, parabolas, and graphing conic sections cut from a cone.
This document introduces the Method of Least Squares (or Minimum Squares) for fitting curves to data points. It explains that this method finds the coefficients of a function that best approximates the relationship between x- and y-values in a dataset by minimizing the sum of squared residuals between the actual and predicted y-values. The document provides an example of using a linear and quadratic function to fit a dataset, showing how to set up and solve the normal equations to determine the coefficients. It also discusses evaluating the quality of fit using the R-squared value.
This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.
1) The document discusses various geometric concepts in multi-variable calculus including the Cartesian plane R2, distance between points, midpoint of a line segment, circles, parabolas, ellipses, and hyperbolas.
2) It provides examples of solving problems related to these concepts, such as proving points are collinear, finding midpoints of diagonals of a quadrilateral, and graphing various equations.
3) The document concludes by listing two references used in teaching these multi-variable calculus topics.
This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.
This document discusses equations and their solutions. It defines key terms like constants, variables, and types of equations like linear equations with one or two variables. It describes methods for solving different types of equations, including using properties of operations and graphical methods to solve systems of linear equations. Sample problems are worked through applying these methods. The document provides context on the importance of equations throughout history and in various sciences.
The document discusses various algebraic expressions and operations, including:
- Adding and subtracting algebraic expressions by combining like terms
- Multiplying algebraic expressions using the distributive property
- Dividing algebraic expressions using long division or factoring the divisor out
- Evaluating algebraic expressions for given numeric values of variables
- Factoring expressions using special product formulas like difference of squares
Here are the steps to solve this problem numerically in MATLAB:
1. Define the 2nd order ODE for the pendulum as two first order equations:
y1' = y2
y2' = -sin(y1)
2. Create an M-file function pendulum.m that returns the right hand side:
function dy = pendulum(t,y)
dy = [y(2); -sin(y(1))];
end
3. Use an ODE solver like ode45 to integrate from t=0 to t=6pi with initial conditions y1(0)=pi, y2(0)=0:
[t,y] = ode45
Quadratic equations are polynomial equations of the second degree that can be written in the general form of ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. There are three main ways to solve quadratic equations: using the quadratic formula, factoring, or completing the square. The quadratic formula provides the exact solutions and can be used to solve any quadratic equation. Factoring and completing the square involve rewriting the equation in an equivalent form to reveal the solutions.
The document provides steps and examples for solving various types of word problems in algebra, including number, mixture, rate/time/distance, work, coin, and geometric problems. It also covers solving quadratic equations using methods like the square root property, completing the square, quadratic formula, factoring, and using the discriminant. Finally, it discusses linear inequalities, including properties related to addition, multiplication, division, and subtraction of inequalities.
Paso 2 contextualizar y profundizar el conocimiento sobre expresiones algebr...Trigogeogebraunad
This document provides instructions and solutions for 7 math exercises involving algebraic expressions and polynomials. It explains how to factor expressions using difference of squares formulas, perform polynomial division using synthetic division, find domains of rational functions, and simplify algebraic fractions. Step-by-step workings are shown for each exercise. Geogebra is used to check solutions for graphing and domains of functions.
1. The document discusses matrices, including definitions, types of matrices, operations on matrices such as addition, subtraction, multiplication, and properties of these operations.
2. It also covers solving systems of linear equations using matrices, including finding the inverse and using Grammer's Rule.
3. The document concludes with an introduction to vectors, including definitions, operations like addition and subtraction, unit vectors, and applications of vectors such as the dot product and cross product.
The document defines and provides examples of quadratic equations. It begins by stating that a quadratic equation is a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Roots of a quadratic equation are the values that make the equation equal to 0. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula. The discriminant can be used to determine the nature of the roots.
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Numeros reales y_plano_numerico1.1_compressed
1. República Bolivariana de Venezuela
Ministerio del Poder Popular para la Educación
Universidad Politécnica Territorial Andrés Eloy Blanco
Barquisimeto, Lara
NÚMEROS REALES
PLANO NUMÉRICO
ESTUDIANTE:
Antonela Santana
Matera: Matemáticas
Sección: #0201
Grupo “B”
2. 1. Conjuntos:
Son elementos diferenciados entre sí pero que poseen en común ciertas propiedades o
características, y que pueden tener entre ellos, o con los elementos de otros conjuntos,
ciertas relaciones.
Un conjunto puede tener un número finito o infinito de elementos, en matemáticas
es común denotar a los elementos mediante letras minúsculas y a los conjuntos por
letras mayúsculas, así por ejemplo: C = {a, b, c, d, e, f, g, h}
En ocasiones un conjunto viene expresado por la propiedad (o propiedades) que
cumplen sus elementos, por ejemplo: C = { 𝑥 ∈ 𝑅, 1 ≤ 𝑥 ≤ 2}.
2. Operaciones con conjuntos:
Estas nos permiten realizar operaciones sobre los conjuntos para obtener un
nuevo conjunto. Tenemos 3 tipos:
a) Unión: nos permite unir dos o más conjuntos para formar otro conjunto
que contendrá a todos los elementos que queremos unir pero sin que se
repitan. Ejemplo:
Dados los conjuntos A {1,2,5,8} y B {3,4,7,9} la unión entre ellos será:
𝐴 𝖴 𝐵 {1,2,3,4,5,7,8,9}
Dados los conjuntos A{-1,5,7,9} y B{-3,0,2,6}, la unión entre ellos
será:
𝐴 𝖴 𝐵 {−3, −1, 0,2,5,6,7,9}
b) Intersección: Es la operación que nos permite formar un conjunto, sólo
con los elementos comunes involucrados en la operación. Ejemplo:
3. Dados los conjuntos A{1,6,8,5,9} y B {2,5,7,8,10}, entonces su
intersección será:
𝐴 ∩ 𝐵 {5,8}
Dados los conjuntos A{-2,-1,1,3} y B= {-1,1,4,5} su resultado será:
𝐴 𝖴 𝐵 {−1,1}
c) Diferencia de conjuntos: Es la operación que nos permite formar un
conjunto, en donde de dos conjuntos el conjunto resultante es el que
tendrá todos los elementos que pertenecen al primero pero no al
segundo. Ejemplo:
Dados los conjuntos A{-5,-3,-1,5} y B{1,2,3,4}, entonces su resultado es:
𝐴 − 𝐵{−5, −3, −1,5}
Dados los conjuntos A{5,7,9,10} y B{1,2,4,6}, tendrá como diferencia
𝐵 − 𝐴{1,2,4,6}
d) Diferencia simétricas de conjuntos: Es la operación que nos permite
formar un conjunto, en donde de dos conjuntos el conjunto resultante
es el que tendrá todos los elementos que no sean comunes a ambos
conjuntos. Ejemplo:
Dados los conjuntos A{1.,2,3,5} y B{8,6,1,3}; su resultado es:
𝐴 ∆ 𝐵{2,5,6,8}
Dados los conjuntos A{6,7,1,2} y B{1,2,6,8}; su diferencia asimétrica
será:
𝐴 ∆ 𝐵 {7,8}
4. 3. Números Reales:
Es la operación que nos permite formar un conjunto, en donde de dos conjuntos el
conjunto resultante es el que tendrá todos los elementos que no sean comunes a
ambos conjuntos.
a) Números naturales ( Ν ) : son todos los números (1,2,3,…)
b) Números Enteros ( Ζ ) : son los naturales, más el 0 y todos los negativos
(-3,-2,-1,0,1,2,3).
c) Números Fraccionarios: Se expresan mediante fracción 𝑎 con a y b
𝑏
enteros. B ≠0
d) Números algebraicos: Vienen de la solución de alguna ecuación
algebraica. Ej: √3
e) Números trascendentales: No se representan mediante un número
finito. Vienen de las funciones trascendente como las logarítmicas,
trigonométricas y exponenciales. Ej: “ ℇ “ (Euler)
Los números reales tienen propiedades como por ejemplo:
Propiedad Conmutativa
Propiedad Asociativa
Propiedad Identidad
5. Propiedad Inversa
Propiedad Distributiva
Propiedad Reflexiva
Propiedad Simétrica
Propiedad Transitiva
Propiedad Uniforme
Propiedad Cancelativa
4. Desigualdades:
Es encontrar el conjunto de todos los números reales que la hacen verdadera. En
contraste con una ecuación, cuyo conjunto solución, en general, consta de un número
o quizás un conjunto finito de números, el conjunto solución de una desigualdad
comúnmente consta de un intervalo completo de número o, en algunos casos, la unión
de tales intervalos.
Tenemos intervalos abiertos tales como (a,b) que son todos los incluidos entre ellos
pero a su vez sin incluirlos; mientras que los intervalos cerrados como [a,b] que son
aquellos que tienes los incluido entre ellos incluyendo los extremos.
Para resolver desigualdades hay que transformar la desigualdad un paso cada vez
hasta que el conjunto sea obvio. Las herramientas para este caso son las propiedades
y estas se deben realizar sin cambiar el conjunto solución, algunas de estas opciones
son:
a) Se puede ajadir el mismo número a ambos miembros de una
desigualdad
b) Se pueden multiplicar ambos miembros de una desigualdad por un
número positivo.
c) Se pueden multiplicar ambos miembros de una desigualdad por un
número negativo, pero aquí se debe invertir el signo de desigualdad.
6. 4.1) Propiedades de las desigualdades
Ley de tricotomía: a = b ; a < b ; a > b
Ley de transitividad: 𝑎 < 𝑏 𝑦 𝑏 < 𝑐 ⟹ 𝑎 < 𝑐
Ley aditiva: 𝑎 < 𝑏 ⟹ 𝑎 + 𝑐 < 𝑏 + 𝑐 , ∀ 𝑐 ∈ ℝ
Ley multiplicativa: 𝑎 < 𝑏 ⟺ 𝑎𝑐 < 𝑏𝑐, ∀ 𝑐 > 0
𝑎 < 𝑏 ⟺ 𝑎𝑐 > 𝑏𝑐, ∀ 𝑐 < 0
0 < 𝑎 < 𝑏 ó 𝑎 < 𝑏 < 0 ⟹
1
>
1
𝑎 𝑏
5. Valor absoluto:
Es el valor real que tiene un número más allá de su signo. En otras palabras el valor
absoluto de x que se designa mediante |x| y se define como:
|x| = X; si x ≥ 0
|x| = -X; si x ≤ 0
Ejemplo #1: Si tenemos |4|, entonces
|4| = 4 o |4| = -4
Ejemplo #2: Si tenemos |x – y| , entonces
|x – y| = |x| - |y|
6. Desigualdades con valor absoluto:
Para resolver desigualdades con valor absoluto es necesario aplicar las
propiedades del valor absoluto para poder eliminar dicho valor, estás son:
|𝑥| < 𝑎 ⟺ −𝑎 < 𝑥 < 𝑎
|𝑥| > 𝑎 ⟺ 𝑥 < −𝑎 𝑜 𝑥 > 𝑎
Ejemplo #1:
Resolver la inecuación | 2x – 3 | < 5
7. -5 < 2x -3 < 5
-5 + 3 < 2x < 5 + 3
-2 < 2x < 8
−2
< x <
8 -1 4
2 2
−1 < x < 4
7. Plano numérico:
Es la representación gráfica matemática donde dos líneas numeradas se interceptan
en un punto llamado “origen”. Estas rectas nos permiten establecer una correspondencia
entre los puntos P del plano y los pares ordenados (x, y). Al eje (x) se le llama eje de
las abscisas mientras al eje (y) se le conoce como el eje de las ordenadas.
A partir de acá podemos medir distintos puntos en la recta entre ellos:
7.1) Distancia: Para buscar la distancia entre dos puntos del plano
debemos seguir la siguiente fórmula:
8. 𝑑(𝑃1, 𝑃2) = √( 𝑥2 − 𝑥1)2 + (𝑦2 −𝑦1)2
Ejemplo: Encontrar la distancia y probar que son vértices de un triángulo
rectángulo:
A = (1, 1) ; B = (3, 0) ; C = (4, 7)
Solución: Se calculará la solución de los lados del triángulo mediante la fórmula de
distancia
d (A,B) = √(3 − 1)2
+ (0 − 1)2
= √22
+ 12
= √5
d (A,C) = √(4 − 1)2
+ (7 − 1)2
= √32
+ 62
= √45
d (B, C) = √(4 − 3)2
+ (7 − 0)2
= √12
+ 72
= √50
Como cumple que:
: . El triángulo debe ser rectángulo según el teorema de Pitágoras.
d ( A,B) 2 + d (A,C)2 = 5 + 45 = 50 = d (B,C)2
9. 7.2) Punto Medio: Es aquel segmento de la recta que va desde P( x1 , y1 )
hasta Q( x2 – y2 ) . La fórmula utilizada para este caso es:
M =
𝑥1+ 𝑥2
2
,
𝑦1+ 𝑦2
2
Ejemplo: Hallar el punto medio del segmento de la recta de extremos: (-3, 0) y
(1, 2)
Solución:
−3+1
M =
2
,
0+2
2
=
−2
,
2
=
2 2
8. Representación de cónicas:
Se denomina sección cónica (o simplemente cónica) a todas las curvas resultantes de
las diferentes intersecciones entre un cono y un plano; si dicho plano no pasa por el
vértice, se obtienen las cónicas propiamente dichas. Se clasifican en cuatro tipos:
elipse, parábola, hipérbola y circunferencia.
8.1) Circunferencia: es el lugar geométrico de los puntos de un plano que
equidistan de otro punto fijo y coplanario llamado centro en una cantidad
constante que se denomina radio.
Su ecuación general es: Ax 2 + By 2 + Cx + Dy + E = 0
La circunferencia de centro C = (h, k) y radio ( r ). Esta se utilizará cuando sea
trasladada y tiene por ecuación
( x – h ) 2 + ( y – k ) 2 = r 2
( -1 , 1 )
10. En particular si el centro es el origen:
x 2 + y 2 = r 2
8.2) Parábolas: se le llama así al gráfico de cualquiera de las dos ecuaciones
siguientes, donde a, b y c son constantes con a ≠ 0.
Ecuación general: Ax 2 + Bx + Cy + D = 0
Su ecuación canónica:
Abre hacia arriba: ( 𝑥 − ℎ) 2
= 4𝑝 (𝑦 − 𝑘 ) Abre
hacia abajo: ( 𝑥 − ℎ) 2
= −4𝑝 (𝑦 − 𝑘 ) Abre hacia
la derecha: ( 𝑦 − 𝑘) 2
= 4𝑝 (𝑥 − ℎ )
Abre hacia la izquierda: ( 𝑦 − 𝑘) 2
= −4𝑝 (𝑥 − ℎ)
11. 2 2
8.3) Elipse: Se trata de una circunferencia achatada que se caracteriza porque la
suma de las distancias desde cualquiera de sus puntos P hasta otros dos puntos
denominados focos (F y F ' ) es siempre la misma.
Su ecuación general es: Ax 2 +By 2 + Cx + Dx + C = 0
A x B > 0
Ecuación canónica: 𝑋
2
+
𝑌2
𝑎 𝑏
8.4) Hipérbola: Es el lugar geométrico de los puntos del plano cuya diferencia de
distancias a dos puntos fijos llamados focos es constante.
Su ecuación general es: Ax 2 +By 2 + Cx + Dx + C = 0
A x B < 0
Ecuación canónica: 𝑥
2
𝑎
−
𝑦2
= 1 ( si abre en x )
𝑏
𝑦2
𝑎2 −
𝑥2
𝑏2
= 1 ( si abre en y )
12. Valores Y
6
5
4
3
2
1
0
0 1 2 3 4
E J E R C I C I O S:
1) Ejercicio de desigualdad con valor absoluto:
| 3𝑥 + 1 | < 15
⟹ −15 < 3𝑥 + 1 < 15
⟹ −15 − 1 < 3𝑥 < 15 − 1
⟹ −16 < 3𝑥 < 14
2) Ejercicio de distancia y punto medio:
Hallar la distancia entre los pares de puntos P y Q , y encontrar el punto
medio:
P= ( 1 , 3)
Q= (3 , 5)
Para encontrar la distancia usaremos la fórmula:
(𝑃1, 𝑃2) = √( 𝑥2 − 𝑥1)2 + (𝑦2 −𝑦1)2
Sustituimos:
𝑑(𝑃𝑄) = √( 3 − 1)2 + (5 − 3)2
𝑑(𝑃𝑄) √22
+ 22
(Q)
(P )
−16 14
⟹ < 𝑥 <
3 3
13. 𝑑(𝑃𝑄) = √4 + 4
𝑑(𝑃𝑄) = 2√2
Una vez hallada la distancia, procedemos a encontrar el punto medio:
M =
𝑥1+ 𝑥2
2
,
𝑦1+ 𝑦2
2
M =
1+ 3
2
M = 4
2
,
3+ 5
2
,
8
2
M = (2 , 4)
: . El resultado para éste ejercicio es: 2√2 , (2,4)
14. Referencias Bibliográficas
Purcell E. y Varberg D (1993) Calculo con geometría analítica 6ta edición.
Ayres, F. (1971) Calculo diferencial e integral. Teoría y 1175 ejercicios
Saenz J. (2005) Calculo diferencial para ingeniería. 2da edición
Bricejo Y. (S/F) Números Reales
Bricejo Y. (S/F) Propiedades de Los Números Reales
Bricejo Y. (S/F) Inecuaciones y Desigualdades