The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example problem demonstrating these steps, eliminating variables through multiplication and addition of the equations until a single variable remains that can be solved for. The document notes that simultaneous equations with the same number of variables and equations can always be solved using this method.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
The document describes putting a system of linear equations into triangular form. It contains a 3x3 system of linear equations. The summary explains that through Gaussian elimination by eliminating variables from equations, the system can be transformed into an upper triangular matrix with equations arranged from top to bottom to easily solve for the variables through back substitution.
The document discusses solving quadratic equations by completing the square. It defines completing the square as turning a quadratic into a perfect square trinomial that can be factored into a binomial squared. The steps for completing the square are provided. Examples of solving quadratic equations by both using the square root property and completing the square are shown and worked through step-by-step.
Here are 3 practice problems from the problem set with solutions:
1) Simplify: 8x + 12x
20x
2) Evaluate the expression 5x + 2x when x = 3:
7x
21
3) Simplify and combine like terms: 4y - 2y + 7y - y
8y
Work through the rest of the assigned problems carefully and check your work. Ask for help if you get stuck on any part of the process. Tackling a full problem set is an excellent way to reinforce the concepts and build skills in working with variable expressions.
The document discusses the power of powers property for exponents. It states that when a number is raised to a power that is then raised to another power, you multiply the exponents. Several examples are provided to demonstrate this, such as (2^3)^2 = 2^6 and (3^3)^4 = 3^12. It also discusses how to simplify expressions with monomials raised to powers using this same property, such as (xy)^2 = x^2y^2.
This document discusses the derivation of a Quotient Rule Integration by Parts formula. It shows how the student Victor Reynolds asked if a similar formula could be derived from the Quotient Rule as the standard Integration by Parts formula is derived from the Product Rule. The author proceeds to derive such a Quotient Rule Integration by Parts formula. An example application of the new formula is also shown. However, the formula does not appear in calculus texts because it provides only a slight technical advantage over the standard formula and requires the same integral computations.
This document contains 26 math problems involving derivatives of logarithmic, exponential, and inverse trigonometric functions. The problems include finding derivatives of expressions, setting up and solving differential equations, and determining relationships between derivatives.
1. This section introduces substitution methods for exact differential equations. It provides examples of homogeneous differential equations and their solutions obtained through substitutions that transform the equations into separable form.
2. Fifteen problems walk through specific substitution methods and solutions for homogeneous differential equations. Additional examples demonstrate substitutions that transform Bernoulli equations into linear equations.
3. Examples of optional material on airplane flight trajectories are included, as well as substitution methods and solutions for additional differential equation problems.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
The document describes putting a system of linear equations into triangular form. It contains a 3x3 system of linear equations. The summary explains that through Gaussian elimination by eliminating variables from equations, the system can be transformed into an upper triangular matrix with equations arranged from top to bottom to easily solve for the variables through back substitution.
The document discusses solving quadratic equations by completing the square. It defines completing the square as turning a quadratic into a perfect square trinomial that can be factored into a binomial squared. The steps for completing the square are provided. Examples of solving quadratic equations by both using the square root property and completing the square are shown and worked through step-by-step.
Here are 3 practice problems from the problem set with solutions:
1) Simplify: 8x + 12x
20x
2) Evaluate the expression 5x + 2x when x = 3:
7x
21
3) Simplify and combine like terms: 4y - 2y + 7y - y
8y
Work through the rest of the assigned problems carefully and check your work. Ask for help if you get stuck on any part of the process. Tackling a full problem set is an excellent way to reinforce the concepts and build skills in working with variable expressions.
The document discusses the power of powers property for exponents. It states that when a number is raised to a power that is then raised to another power, you multiply the exponents. Several examples are provided to demonstrate this, such as (2^3)^2 = 2^6 and (3^3)^4 = 3^12. It also discusses how to simplify expressions with monomials raised to powers using this same property, such as (xy)^2 = x^2y^2.
This document discusses the derivation of a Quotient Rule Integration by Parts formula. It shows how the student Victor Reynolds asked if a similar formula could be derived from the Quotient Rule as the standard Integration by Parts formula is derived from the Product Rule. The author proceeds to derive such a Quotient Rule Integration by Parts formula. An example application of the new formula is also shown. However, the formula does not appear in calculus texts because it provides only a slight technical advantage over the standard formula and requires the same integral computations.
This document contains 26 math problems involving derivatives of logarithmic, exponential, and inverse trigonometric functions. The problems include finding derivatives of expressions, setting up and solving differential equations, and determining relationships between derivatives.
1. This section introduces substitution methods for exact differential equations. It provides examples of homogeneous differential equations and their solutions obtained through substitutions that transform the equations into separable form.
2. Fifteen problems walk through specific substitution methods and solutions for homogeneous differential equations. Additional examples demonstrate substitutions that transform Bernoulli equations into linear equations.
3. Examples of optional material on airplane flight trajectories are included, as well as substitution methods and solutions for additional differential equation problems.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
The document discusses using the remainder theorem to find the remainder of a polynomial function divided by a binomial function without fully performing the division. It provides two examples:
1) When f(x) = x^3 - 3x^2 + 7 is divided by g(x) = x + 2, the remainder is -13.
2) When f(x) = x^3 - 3x^2 + 5x - 1 is divided by g(x) = x - 1, the remainder is 2.
The key steps are to set the binomial factor equal to 0 to find the value to substitute into the polynomial, then evaluate the polynomial at that value to determine the remainder.
The document provides instructions on different methods for factoring polynomials, including:
1) Pulling out common factors.
2) Looking for special forms like the difference of squares or perfect square trinomial.
3) Factoring trinomials by finding two binomial factors whose product is the trinomial using FOIL.
The document provides a review of the three methods to solve systems of equations: graphing, substitution, and elimination. It includes examples of systems of equations to solve using each method. Checkpoint questions are provided to have the student practice solving systems of equations by graphing, substitution, and elimination.
Multiple Choice Questions of Successive Differentiation (Calculus) for B.Sc. 1st Semester (Panjab University ) Mathematics students.There are 2o questions with answer keys.
The student will learn to solve systems of equations using elimination with addition and subtraction. This involves putting the equations in standard form, eliminating one variable by adding or subtracting the equations, solving for the eliminated variable, plugging back into one equation to solve for the other variable, and checking the solution. Two examples are shown of solving systems of two equations with two variables using this elimination method.
The student will be able to solve systems of equations using elimination with addition and subtraction. There are 5 steps to solving a system by elimination: 1) put the equations in standard form, 2) determine which variable to eliminate, 3) add or subtract the equations to eliminate the variable, 4) plug back into one equation to find the other variable, and 5) check the solution by substituting into both original equations. Two examples are provided to demonstrate the process.
11X1 T05 06 line through point of intersection (2010)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
The document discusses multiplying and dividing variable expressions. It provides examples of simplifying expressions using the distributive property and the property of the opposite of a sum. It also demonstrates dividing variable expressions by writing the division as a fraction and simplifying. Key steps include distributing terms, dividing each term in the numerator by the denominator, and evaluating expressions for given variable values.
This document contains 32 systems of equations and their solutions. The systems include linear equations, quadratic equations, and equations containing variables multiplied together. Solving the systems requires skills like adding or subtracting equations, substituting values, and solving quadratics. The solutions are provided in fractional or decimal form depending on the system.
This document provides examples of solving systems of nonlinear equations in two variables. It begins with definitions, including that a nonlinear system contains at least one equation that is not of the form Ax + By = C. Methods for solving nonlinear systems include substitution and addition. Examples walk through both methods step-by-step for various systems. Key steps are rewriting equations in terms of variables, substituting values, solving resulting equations, back-substituting, and checking solutions satisfy both original equations.
This module introduces linear functions of the form f(x) = mx + b. Learners will develop skills to determine the slope, trend, intercepts, and points of linear functions. The module is designed to help learners:
1) Determine slope, trend, intercepts, and points of a linear function given f(x) = mx + b
2) Determine f(x) = mx + b given various conditions like slope and intercepts, two points, etc.
The document provides lessons that explain how to find these properties of linear functions through examples. Practice problems are also included for learners to test their understanding.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
The document provides a multi-part algebra review covering topics such as simplifying expressions, combining like terms, factoring, operations with exponents, and rationalizing denominators. It contains over 20 practice problems testing these skills. The problems range in complexity from combining simple terms to factoring polynomials and performing multiple operations with exponents.
The document discusses trigonometry concepts related to 3D shapes and solving problems involving angles of elevation. Specifically:
- When doing 3D trigonometry, it is often useful to redraw shapes in 2D to analyze them.
- A worked example problem is shown to find the distance and bearing between a life raft (David's position) and a search vessel (Anna's position) based on angles of elevation they each observe of a mountain peak.
- Applying trigonometric relationships involving angles and the mountain's known height, the distance between David and Anna is calculated to be 2799 meters, and the bearing of David from Anna is calculated to be 249°51'.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document defines and provides examples of absolute value. Absolute value is the distance of a number from 0, regardless of its direction. It is solved as the number if it is positive, and the opposite of the number if it is negative. Examples of evaluating absolute value expressions are provided. The document also discusses solving absolute value equations and inequalities, providing examples of setting up and solving different absolute value equations for the variable.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
The document discusses using the remainder theorem to find the remainder of a polynomial function divided by a binomial function without fully performing the division. It provides two examples:
1) When f(x) = x^3 - 3x^2 + 7 is divided by g(x) = x + 2, the remainder is -13.
2) When f(x) = x^3 - 3x^2 + 5x - 1 is divided by g(x) = x - 1, the remainder is 2.
The key steps are to set the binomial factor equal to 0 to find the value to substitute into the polynomial, then evaluate the polynomial at that value to determine the remainder.
The document provides instructions on different methods for factoring polynomials, including:
1) Pulling out common factors.
2) Looking for special forms like the difference of squares or perfect square trinomial.
3) Factoring trinomials by finding two binomial factors whose product is the trinomial using FOIL.
The document provides a review of the three methods to solve systems of equations: graphing, substitution, and elimination. It includes examples of systems of equations to solve using each method. Checkpoint questions are provided to have the student practice solving systems of equations by graphing, substitution, and elimination.
Multiple Choice Questions of Successive Differentiation (Calculus) for B.Sc. 1st Semester (Panjab University ) Mathematics students.There are 2o questions with answer keys.
The student will learn to solve systems of equations using elimination with addition and subtraction. This involves putting the equations in standard form, eliminating one variable by adding or subtracting the equations, solving for the eliminated variable, plugging back into one equation to solve for the other variable, and checking the solution. Two examples are shown of solving systems of two equations with two variables using this elimination method.
The student will be able to solve systems of equations using elimination with addition and subtraction. There are 5 steps to solving a system by elimination: 1) put the equations in standard form, 2) determine which variable to eliminate, 3) add or subtract the equations to eliminate the variable, 4) plug back into one equation to find the other variable, and 5) check the solution by substituting into both original equations. Two examples are provided to demonstrate the process.
11X1 T05 06 line through point of intersection (2010)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
1. The document discusses functions and relations through examples and questions.
2. It covers finding the value of functions, solving equations involving functions, and evaluating composite functions.
3. Key concepts covered include domain, codomain, range, one-to-one, many-to-one, one-to-many and many-to-many relations.
The document discusses multiplying and dividing variable expressions. It provides examples of simplifying expressions using the distributive property and the property of the opposite of a sum. It also demonstrates dividing variable expressions by writing the division as a fraction and simplifying. Key steps include distributing terms, dividing each term in the numerator by the denominator, and evaluating expressions for given variable values.
This document contains 32 systems of equations and their solutions. The systems include linear equations, quadratic equations, and equations containing variables multiplied together. Solving the systems requires skills like adding or subtracting equations, substituting values, and solving quadratics. The solutions are provided in fractional or decimal form depending on the system.
This document provides examples of solving systems of nonlinear equations in two variables. It begins with definitions, including that a nonlinear system contains at least one equation that is not of the form Ax + By = C. Methods for solving nonlinear systems include substitution and addition. Examples walk through both methods step-by-step for various systems. Key steps are rewriting equations in terms of variables, substituting values, solving resulting equations, back-substituting, and checking solutions satisfy both original equations.
This module introduces linear functions of the form f(x) = mx + b. Learners will develop skills to determine the slope, trend, intercepts, and points of linear functions. The module is designed to help learners:
1) Determine slope, trend, intercepts, and points of a linear function given f(x) = mx + b
2) Determine f(x) = mx + b given various conditions like slope and intercepts, two points, etc.
The document provides lessons that explain how to find these properties of linear functions through examples. Practice problems are also included for learners to test their understanding.
This document contains an exercise set from a chapter on functions. It includes 35 multi-part math problems testing concepts like domains and ranges of functions, rates of change, and word problems involving temperature, speed, and geometric shapes. The problems cover skills like determining maximum/minimum values, solving equations, and sketching graphs of functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
The document provides a multi-part algebra review covering topics such as simplifying expressions, combining like terms, factoring, operations with exponents, and rationalizing denominators. It contains over 20 practice problems testing these skills. The problems range in complexity from combining simple terms to factoring polynomials and performing multiple operations with exponents.
The document discusses trigonometry concepts related to 3D shapes and solving problems involving angles of elevation. Specifically:
- When doing 3D trigonometry, it is often useful to redraw shapes in 2D to analyze them.
- A worked example problem is shown to find the distance and bearing between a life raft (David's position) and a search vessel (Anna's position) based on angles of elevation they each observe of a mountain peak.
- Applying trigonometric relationships involving angles and the mountain's known height, the distance between David and Anna is calculated to be 2799 meters, and the bearing of David from Anna is calculated to be 249°51'.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document defines and provides examples of absolute value. Absolute value is the distance of a number from 0, regardless of its direction. It is solved as the number if it is positive, and the opposite of the number if it is negative. Examples of evaluating absolute value expressions are provided. The document also discusses solving absolute value equations and inequalities, providing examples of setting up and solving different absolute value equations for the variable.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The document discusses coordinate geometry concepts including the distance formula and midpoint formula. It explains that the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint. It also discusses dividing intervals, noting that for a level 2 math exam it is restricted to midpoint divisions in a 1:1 ratio, while an extension 1 exam can involve any ratio for internal or external divisions. Examples are provided to illustrate the concepts.
The document discusses trigonometric functions and radian measure. It states that 360 degrees equals 2 pi radians. It then provides a table with common conversions between degrees and radians for angles from 30 to 360 degrees. Examples are also given of converting degrees to radians and radians to degrees.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document discusses the fundamental theorem of algebra and properties of complex zeros of polynomials. It provides examples of finding all zeros of polynomials by factoring, using the quadratic formula, and the difference of squares/cubes formulas. It also demonstrates using the "sum and product method" to find the polynomial of lowest degree with given complex zeros, which involves taking the sum and product of the zeros.
This document provides a study guide for Chapter 6 of Algebra 2 covering evaluating expressions using laws of exponents, determining if functions are polynomials, describing polynomial end behavior, factoring polynomials, solving polynomial equations, dividing polynomials using long and synthetic division, finding zeros of polynomials, writing polynomials given their zeros, and graphing polynomials. It includes 79 problems to work through involving these topics.
This document provides a study guide for Chapter 6 of Algebra 2 covering evaluating expressions using laws of exponents, determining if functions are polynomials, describing polynomial end behavior, factoring polynomials, solving polynomial equations, dividing polynomials using long and synthetic division, finding zeros of polynomials, writing polynomials given their zeros, and graphing polynomials. It includes 79 problems to work through involving these topics.
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
This document contains solutions to mathematics questions from the 2010 HSC exam in Australia. Question 1 involves solving equations, inequalities and finding derivatives. Question 2 involves finding derivatives of trigonometric functions. Question 3 involves vectors, gradients and parallel lines. Question 4 involves arithmetic progressions, integrals and area under curves. Question 5 involves volumes, surface areas, maxima and minima. Question 6 involves factorizing polynomials, discriminants and finding angles and areas of figures.
1) This document contains a review of various algebra 2 concepts across 9 standards, including solving linear equations, quadratic equations, systems of equations, exponents, functions, and probability.
2) Several problems provide examples of solving systems of equations, factoring quadratic expressions, graphing quadratic and exponential functions, simplifying expressions with exponents, and calculating probabilities of independent and dependent events.
3) The review covers a wide range of algebra 2 topics to help students prepare for an upcoming benchmark exam.
4.2 derivatives of logarithmic functionsdicosmo178
This document discusses implicit and explicit differentiation.
It provides examples of taking the derivative of equations in both implicit and explicit form. It also shows how to find the derivative at a point, such as finding the slope of an implicitly defined equation at the point (1,1).
This document contains solutions to exercises from a pre-calculus textbook involving inverse functions and relations. Some of the key questions answered include:
- Sketching the graphs of functions and their inverses after transformations like reflections
- Finding equations that represent the inverse of various given functions
- Determining whether pairs of functions are inverses of each other by comparing their equations
- Restricting domains of functions to make their inverses functions as well
- Finding coordinates of points on inverse relations after translations
- Sketching graphs of inverses based on restrictions of the domain of the original relation
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
Integrating factors found by inspectionShin Kaname
1. The document discusses using exact differentials to solve integration problems.
2. It provides examples of using exact differentials and integrating terms to find solutions.
3. The solutions found are particular solutions for the given values of x and y in each problem.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
The document provides information on determining limits of algebraic functions. It discusses different methods for calculating limits, including dividing the numerator and denominator by the highest power term, and multiplying by the conjugate of the numerator and denominator. Examples are provided to illustrate each method and determine limits as the variable approaches a value.
The document discusses factoring binomials into two binomials by factoring the difference of squares. It provides examples of factoring expressions like x^2 - 25, y^2 - 144, 4x^2 - 16 into the form (x-a)(x+a), showing they can be factored when the second term is a perfect square. It notes the common pattern and explains the rules for factoring binomials into two binomials, that the first term must be a square and the second term must be negative.
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This document discusses special polynomial products and factoring polynomials. It begins by covering special product rules for multiplying binomials, squaring binomials, taking the cube of a binomial, and multiplying a binomial and trinomial. It then discusses different factoring techniques, including factoring a common monomial, difference of squares, factoring trinomials, factoring perfect square trinomials, sum and difference of cubes, and factoring by grouping. The document provides examples of each type of special product and factoring technique. It aims to teach how to find special products and completely factor polynomials into prime factors.
G8 Math Q1- Week 1-2 Special Products and Factors (1).pptxCatherineGanLabaro
This document discusses special polynomial products and factoring polynomials. It begins by introducing special products such as the product of two binomials, the square of a binomial, and the product of the sum and difference of two terms. It then covers factoring techniques like factoring a common monomial, difference of squares, factoring trinomials, and factoring by grouping. The document provides examples for each type of special product and factoring method. It aims to teach how to find special products in a faster way than long multiplication and how to completely factor polynomials into prime factors.
This document provides a review of key algebra 1 concepts including equations of lines, solving various types of equations, factoring polynomials, expressions and equations involving perimeter, area, and geometry. Students are given examples to solve of each concept, including solving systems of equations, simplifying expressions, graphing lines, and determining equations of lines given points or other criteria. The review covers standard form, slope-intercept form, point-slope form of a line, solving linear and quadratic equations, factoring polynomials, perimeter, area, geometry relationships, graphing, and determining equations of lines from information provided.
This document provides a tutorial on calculus that includes 4 problems:
1) Converting improper rational functions to proper rational functions.
2) Simplifying expressions with exponents, logarithms, and natural logarithms.
3) Simplifying expressions with common and natural logarithms.
4) Solving equations with exponents, logarithms, and natural logarithms.
Similar to 11 x1 t01 09 simultaneous equations (2012) (20)
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
4. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
5. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
e.g. (i ) 2 x 3 y 21 (1)
5 x 2 y 3 (2)
6. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
e.g. (i ) 2 x 3 y 21 (1)
5 x 2 y 3 (2)
Multiply (1) by 2 and (2) by 3
7. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
e.g. (i ) 2 x 3 y 21 (1)
5 x 2 y 3 (2)
Multiply (1) by 2 and (2) by 3
4 x 6 y 42
15 x 6 y 9
8. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
e.g. (i ) 2 x 3 y 21 (1)
5 x 2 y 3 (2)
Multiply (1) by 2 and (2) by 3
4 x 6 y 42
15 x 6 y 9
11x = 33
9. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
e.g. (i ) 2 x 3 y 21 (1)
5 x 2 y 3 (2)
Multiply (1) by 2 and (2) by 3
4 x 6 y 42
15 x 6 y 9
11x = 33
x = 3
10. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
e.g. (i ) 2 x 3 y 21 (1)
5 x 2 y 3 (2)
Multiply (1) by 2 and (2) by 3
4 x 6 y 42
15 x 6 y 9
11x = 33 2 3 3 y 21
x = 3 3 y 27
y9
11. Simultaneous Equations
1) Eliminate a variable
2) Solve for the other variable
3) Substitute to find the eliminated variable
e.g. (i ) 2 x 3 y 21 (1)
5 x 2 y 3 (2)
Multiply (1) by 2 and (2) by 3
4 x 6 y 42
15 x 6 y 9
11x = 33 2 3 3 y 21
x = 3 3 y 27
y9
x 3, y 9
13. (ii ) 2 x y 14 (1)
x 2 y 2 9 (2)
Make y the subject in (1)
y 14 2 x
14. (ii ) 2 x y 14 (1)
x 2 y 2 9 (2)
Make y the subject in (1)
y 14 2 x
Substitute into (2)
x 2 14 2 x 9
2
15. (ii ) 2 x y 14 (1)
x 2 y 2 9 (2)
Make y the subject in (1)
y 14 2 x
Substitute into (2)
x 2 14 2 x 9
2
x 2 196 56 x 4 x 2 9
3 x 2 56 x 205 0
16. (ii ) 2 x y 14 (1)
x 2 y 2 9 (2)
Make y the subject in (1)
y 14 2 x
Substitute into (2)
x 2 14 2 x 9
2
x 2 196 56 x 4 x 2 9
3 x 2 56 x 205 0
3x 41 x 5 0
41
x 5 or x
3
17. (ii ) 2 x y 14 (1)
x 2 y 2 9 (2)
Make y the subject in (1)
y 14 2 x
Substitute into (2)
x 2 14 2 x 9
2
x 2 196 56 x 4 x 2 9
3 x 2 56 x 205 0
3x 41 x 5 0
41
x 5 or x
3
41 14
2 5 y 14 or 2 y
3
40
y 4 or y
3
18. (ii ) 2 x y 14 (1)
x 2 y 2 9 (2)
Make y the subject in (1)
y 14 2 x
Substitute into (2)
x 2 14 2 x 9
2
x 2 196 56 x 4 x 2 9
3 x 2 56 x 205 0
3x 41 x 5 0
41
x 5 or x
3
41 14
2 5 y 14 or 2 y
3
41 40
40 x 5, y 4 or x , y
y 4 or y 3 3
3
19. (iii ) x 2 y z 5 (1)
2 x 3 y 4 z 28 (2) any time you have the same number of
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
20. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
21. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2
2 x 4 y 2 z 10
2 x 3 y 4 z 28
22. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2
2 x 4 y 2 z 10
2 x 3 y 4 z 28
7 y 6 z 38
23. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2 Multiply (1) by 4
2 x 4 y 2 z 10 4 x 8 y 4 z 20
2 x 3 y 4 z 28 4 x 5 y 3 z 10
7 y 6 z 38
24. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2 Multiply (1) by 4
2 x 4 y 2 z 10 4 x 8 y 4 z 20
2 x 3 y 4 z 28 4 x 5 y 3 z 10
7 y 6 z 38 3 y z 10
25. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2 Multiply (1) by 4
2 x 4 y 2 z 10 4 x 8 y 4 z 20
2 x 3 y 4 z 28 4 x 5 y 3 z 10
7 y 6 z 38 3 y z 10
Solve these new equations simultaneously
26. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2 Multiply (1) by 4
2 x 4 y 2 z 10 4 x 8 y 4 z 20
2 x 3 y 4 z 28 4 x 5 y 3 z 10
7 y 6 z 38 3 y z 10
Solve these new equations simultaneously
7 y 6 z 38
18 y 6 z 60
27. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2 Multiply (1) by 4
2 x 4 y 2 z 10 4 x 8 y 4 z 20
2 x 3 y 4 z 28 4 x 5 y 3 z 10
7 y 6 z 38 3 y z 10
Solve these new equations simultaneously
7 y 6 z 38
18 y 6 z 60
28. (iii ) x 2 y z 5 (1)
any time you have the same number of
2 x 3 y 4 z 28 (2)
pronumerals as equations it should
4 x 5 y 3 z 10 (3) be possible to find their values
Create two pairs of two equations and
eliminate the same variable from both
Multiply (1) by 2 Multiply (1) by 4
2 x 4 y 2 z 10 4 x 8 y 4 z 20
2 x 3 y 4 z 28 4 x 5 y 3 z 10
7 y 6 z 38 3 y z 10
Solve these new equations simultaneously
7 y 6 z 38
18 y 6 z 60
11 y 22
y 2