This document discusses operations on rational expressions, including:
1) Reducing rational expressions to lowest terms by dividing the numerator and denominator by their greatest common factor.
2) Multiplying rational expressions follows the same rules as multiplying fractions - the numerators are multiplied and the denominators are multiplied.
3) Dividing rational expressions follows the same rules as dividing fractions - the expression is written as the first rational expression over the second.
This document provides an overview of solving nonlinear equations. It discusses solving single nonlinear equations and systems of nonlinear equations. It covers existence and uniqueness of solutions, multiple roots, sensitivity and conditioning. Iterative methods are used to solve nonlinear equations numerically. The bisection method is introduced, which successively halves the interval containing the solution until the desired accuracy is reached. The convergence rate of the bisection method is linear, meaning the error bound is reduced by half each iteration.
This document discusses solving rational equations and eliminating extraneous solutions. It begins by explaining that rational functions are often used as models that require solving equations involving fractions. When multiplying or dividing terms, extra solutions can arise that do not satisfy the original equation. Examples are provided to demonstrate solving rational equations by clearing fractions and identifying extraneous solutions. One example solves an equation to find the minimum perimeter of a rectangle with a given area.
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
This document provides instructions for Homework 1 for the course 6.867. It is due on September 28 and will be 10% off for each day late. The homework involves exploring bias-variance tradeoffs in estimating the mean of different distributions from sample data. It provides questions to answer about maximum likelihood estimators for the mean of uniform distributions on intervals of different lengths. It also covers Bayesian estimation of probabilities for a "thick coin" that can land on heads, tails, or edge. Finally, it includes questions on Gaussian distributions, analyzing a presidential debate poll, and decision theory concepts.
The document discusses local and global maxima and minima of functions. It introduces two tests for determining whether a stationary point is a local maximum, minimum, or inflection point:
1) The Second Derivative Test examines the sign of the second derivative at the stationary point.
2) The First Derivative Test examines the signs of the first derivative immediately to the left and right of the stationary point.
The First Derivative Test is preferable when the second derivative is difficult to take or inconclusive. Both tests are demonstrated on examples to determine the nature of stationary points.
This document discusses two major theorems - the Fundamental Theorem of Algebra and the Linear Factorization Theorem. The Fundamental Theorem of Algebra states that a polynomial of degree n has n complex zeros, some of which may be repeated. The Linear Factorization Theorem states that a polynomial of degree n can be written as a product of n linear factors. The document also discusses how complex conjugate zeros, factors of polynomials with real coefficients, and finding polynomials from given zeros relate to these theorems.
Linearprog, Reading Materials for Operational Research Derbew Tesfa
The document discusses linear programming (LP), which involves optimizing a linear objective function subject to linear constraints. It provides examples of LP problems, such as production planning and transportation problems. It defines key LP concepts like the feasible region, basic solutions, basic variables, and degenerate basic feasible solutions. It also describes how to transform any LP problem into standard form and discusses properties of optimal solutions.
This document provides an overview of solving nonlinear equations. It discusses solving single nonlinear equations and systems of nonlinear equations. It covers existence and uniqueness of solutions, multiple roots, sensitivity and conditioning. Iterative methods are used to solve nonlinear equations numerically. The bisection method is introduced, which successively halves the interval containing the solution until the desired accuracy is reached. The convergence rate of the bisection method is linear, meaning the error bound is reduced by half each iteration.
This document discusses solving rational equations and eliminating extraneous solutions. It begins by explaining that rational functions are often used as models that require solving equations involving fractions. When multiplying or dividing terms, extra solutions can arise that do not satisfy the original equation. Examples are provided to demonstrate solving rational equations by clearing fractions and identifying extraneous solutions. One example solves an equation to find the minimum perimeter of a rectangle with a given area.
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
This document provides instructions for Homework 1 for the course 6.867. It is due on September 28 and will be 10% off for each day late. The homework involves exploring bias-variance tradeoffs in estimating the mean of different distributions from sample data. It provides questions to answer about maximum likelihood estimators for the mean of uniform distributions on intervals of different lengths. It also covers Bayesian estimation of probabilities for a "thick coin" that can land on heads, tails, or edge. Finally, it includes questions on Gaussian distributions, analyzing a presidential debate poll, and decision theory concepts.
The document discusses local and global maxima and minima of functions. It introduces two tests for determining whether a stationary point is a local maximum, minimum, or inflection point:
1) The Second Derivative Test examines the sign of the second derivative at the stationary point.
2) The First Derivative Test examines the signs of the first derivative immediately to the left and right of the stationary point.
The First Derivative Test is preferable when the second derivative is difficult to take or inconclusive. Both tests are demonstrated on examples to determine the nature of stationary points.
This document discusses two major theorems - the Fundamental Theorem of Algebra and the Linear Factorization Theorem. The Fundamental Theorem of Algebra states that a polynomial of degree n has n complex zeros, some of which may be repeated. The Linear Factorization Theorem states that a polynomial of degree n can be written as a product of n linear factors. The document also discusses how complex conjugate zeros, factors of polynomials with real coefficients, and finding polynomials from given zeros relate to these theorems.
Linearprog, Reading Materials for Operational Research Derbew Tesfa
The document discusses linear programming (LP), which involves optimizing a linear objective function subject to linear constraints. It provides examples of LP problems, such as production planning and transportation problems. It defines key LP concepts like the feasible region, basic solutions, basic variables, and degenerate basic feasible solutions. It also describes how to transform any LP problem into standard form and discusses properties of optimal solutions.
This document discusses functions and modeling with functions. It provides examples of defining functions from formulas, graphs, verbal descriptions, and data. One example shows defining the volume of a box as a function of the side length of squares cut out of the corners, finding the domain and maximum volume. Another example models the growth of a grain pile using a function for the volume of a cone. The document also covers constructing functions from scatter plot data and using technology for curve fitting.
This document provides an overview of the key concepts covered in Chapter 6 of a mathematics textbook, which includes:
1) Defining rational expressions as ratios of two polynomials and focusing on adding, subtracting, multiplying, and dividing rational expressions.
2) Concluding with solving rational equations and applications of rational equations.
3) Summarizing the chapter sections, which cover topics like rational expressions and functions, operations on rational expressions, and applications of rational equations.
This document discusses integration, which is the inverse operation of differentiation. It begins by explaining that integration finds the original function given its derivative, with the addition of a constant of integration. It then provides examples of basic integration techniques using a table of integrals. The document also outlines some rules for integrating sums and constant multiples of functions. Finally, it gives an example of using integration to solve an engineering problem involving the electric potential of a charged sphere.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
This document provides an overview of key concepts related to functions and their properties. It discusses function definition and notation, domain and range, continuity, increasing and decreasing functions, boundedness, local and absolute extrema, symmetry, asymptotes, and end behavior. Examples are provided to illustrate key concepts such as finding the domain and range of functions, identifying points of discontinuity, determining if a function is increasing or decreasing, and identifying local extrema. The document is intended to teach readers about important properties of functions.
This document provides an overview of solving inequalities in one variable. It discusses polynomial inequalities, rational inequalities, and other types of inequalities like those involving radicals. Examples are provided of finding the values of a variable that make a polynomial positive, negative, zero, or undefined. The document also gives examples of solving polynomial and rational inequalities algebraically and graphically. It concludes with a sample test covering skills like finding zeros of functions, solving equations and inequalities, and modeling applications like projectile motion.
Decreasing and increasing functions by arun umraossuserd6b1fd
Function analysis - characteristics of increasing and decreasing functions. How "sign" either positive or negative tells about the nature of the function, i.e. where it is increasing and where it is decreasing.
This document discusses finding the extrema (maximum and minimum values) of functions on intervals using calculus. It defines extrema and relative extrema of functions, and explains how derivatives can be used to find them. Critical numbers are points where the derivative is zero or undefined, and may indicate relative extrema. The document provides examples of finding the critical numbers of functions and using them along with endpoint values to determine the absolute maximum and minimum values over a closed interval. While critical numbers sometimes identify relative extrema, the converse is not always true - not all critical numbers yield an extremum.
Partial fraction decomposition involves writing rational functions as the sum of simpler fractional components. It is useful for integration in calculus. A rational function is broken into partial fractions by factoring the denominator and grouping terms with each linear or irreducible quadratic factor. Examples demonstrate decomposing fractions with linear factors and repeated or quadratic factors. Partial fraction decomposition allows rational functions to be integrated and graphs of rational functions to be sketched.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
The document discusses the upper and lower bound theorem for real roots of polynomial functions. The theorem states that if the remainder of synthetic division of a polynomial f(x) by x-b has only non-negative numbers, b is an upper bound for the real roots of f(x)=0. If the remainder alternates in sign when dividing f(x) by x-a, a is a lower bound. An example demonstrates finding the bounds between -3 and 2 for a 4th degree polynomial. The intermediate value theorem and fundamental theorem of algebra are also summarized.
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...ssuserd6b1fd
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
The document discusses antiderivatives and indefinite integration. It defines antiderivatives as functions whose derivatives are a given function. The general form of the antiderivative includes adding an arbitrary constant C, known as the constant of integration. It provides examples of using basic integration rules to find antiderivatives and solving differential equations. Initial conditions are introduced as a way to determine a particular solution when the general solution involves an arbitrary constant.
This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.
The document discusses methods for solving systems of two equations, including substitution and elimination. It provides examples of using each method to solve both linear and nonlinear systems. It also demonstrates how to determine if a system has no solution or infinitely many solutions. Key methods covered are the substitution method, elimination method, solving systems graphically, and applications of solving systems.
The document defines algebraic expressions and discusses various algebraic operations such as addition, subtraction, multiplication, division, and factorization of algebraic expressions. It provides examples to illustrate each operation. Factorization is described as expressing a complicated polynomial as the product of simpler polynomial factors. Common factoring techniques are mentioned, including factoring the difference of squares and factoring trinomials.
1) The document discusses exponent rules including the zero exponent rule, expanded power rule, and negative exponent rules.
2) It provides examples of applying these rules such as 53=1, (3c2/2d3)4={81c8/16d12}, and (3/r3)-2=r6/9.
3) It lists group work problems assigned from sections 4.1 and 4.2 that are due on Wednesday.
Limit, Continuity and Differentiability for JEE Main 2014Ednexa
The document discusses limits, continuity, and differentiability. It defines the limit of a function, continuity of a function at a point using three conditions, and Cauchy's definition of continuity using delta and epsilon. It also discusses left and right continuity, Heine's definition of continuity using convergent sequences, and the formal definition of continuity. Examples are provided to illustrate calculating limits and determining continuity.
This document discusses rational expressions and equations. It defines a rational expression as an algebraic expression that can be written as the quotient of two polynomials. An expression is undefined when its denominator is equal to zero, which are called the non-permissible values. The document provides examples of finding non-permissible values and simplifying rational expressions by factoring and cancelling common factors in the numerator and denominator. It concludes with an assignment to complete practice problems related to rational expressions and equations.
This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.
The document discusses properties and laws of exponents, radicals, logarithms including the definition of rational exponents, properties of logarithms such as the change of base formula, and examples of simplifying expressions using exponent laws, combining like radicals, and solving logarithmic equations by using properties of logarithms. It also provides sample problems and their step-by-step solutions for simplifying expressions and solving equations involving exponents, radicals, and logarithms.
This document discusses functions and modeling with functions. It provides examples of defining functions from formulas, graphs, verbal descriptions, and data. One example shows defining the volume of a box as a function of the side length of squares cut out of the corners, finding the domain and maximum volume. Another example models the growth of a grain pile using a function for the volume of a cone. The document also covers constructing functions from scatter plot data and using technology for curve fitting.
This document provides an overview of the key concepts covered in Chapter 6 of a mathematics textbook, which includes:
1) Defining rational expressions as ratios of two polynomials and focusing on adding, subtracting, multiplying, and dividing rational expressions.
2) Concluding with solving rational equations and applications of rational equations.
3) Summarizing the chapter sections, which cover topics like rational expressions and functions, operations on rational expressions, and applications of rational equations.
This document discusses integration, which is the inverse operation of differentiation. It begins by explaining that integration finds the original function given its derivative, with the addition of a constant of integration. It then provides examples of basic integration techniques using a table of integrals. The document also outlines some rules for integrating sums and constant multiples of functions. Finally, it gives an example of using integration to solve an engineering problem involving the electric potential of a charged sphere.
1. The document discusses calculating average values of functions using integral methods. It provides examples of finding average values over different intervals for various functions, including constants, quadratic functions, sine functions, and periodic functions.
2. Key points covered include handling zero crossings, periodicity, and whether the quantity is absolute or algebraically additive. Piecewise methods and multiplying average values over one period are discussed for periodic functions.
3. Several example problems are worked through step-by-step to demonstrate finding average values over different intervals for various functions using integral relations.
This document provides an overview of key concepts related to functions and their properties. It discusses function definition and notation, domain and range, continuity, increasing and decreasing functions, boundedness, local and absolute extrema, symmetry, asymptotes, and end behavior. Examples are provided to illustrate key concepts such as finding the domain and range of functions, identifying points of discontinuity, determining if a function is increasing or decreasing, and identifying local extrema. The document is intended to teach readers about important properties of functions.
This document provides an overview of solving inequalities in one variable. It discusses polynomial inequalities, rational inequalities, and other types of inequalities like those involving radicals. Examples are provided of finding the values of a variable that make a polynomial positive, negative, zero, or undefined. The document also gives examples of solving polynomial and rational inequalities algebraically and graphically. It concludes with a sample test covering skills like finding zeros of functions, solving equations and inequalities, and modeling applications like projectile motion.
Decreasing and increasing functions by arun umraossuserd6b1fd
Function analysis - characteristics of increasing and decreasing functions. How "sign" either positive or negative tells about the nature of the function, i.e. where it is increasing and where it is decreasing.
This document discusses finding the extrema (maximum and minimum values) of functions on intervals using calculus. It defines extrema and relative extrema of functions, and explains how derivatives can be used to find them. Critical numbers are points where the derivative is zero or undefined, and may indicate relative extrema. The document provides examples of finding the critical numbers of functions and using them along with endpoint values to determine the absolute maximum and minimum values over a closed interval. While critical numbers sometimes identify relative extrema, the converse is not always true - not all critical numbers yield an extremum.
Partial fraction decomposition involves writing rational functions as the sum of simpler fractional components. It is useful for integration in calculus. A rational function is broken into partial fractions by factoring the denominator and grouping terms with each linear or irreducible quadratic factor. Examples demonstrate decomposing fractions with linear factors and repeated or quadratic factors. Partial fraction decomposition allows rational functions to be integrated and graphs of rational functions to be sketched.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
The document discusses the upper and lower bound theorem for real roots of polynomial functions. The theorem states that if the remainder of synthetic division of a polynomial f(x) by x-b has only non-negative numbers, b is an upper bound for the real roots of f(x)=0. If the remainder alternates in sign when dividing f(x) by x-a, a is a lower bound. An example demonstrates finding the bounds between -3 and 2 for a 4th degree polynomial. The intermediate value theorem and fundamental theorem of algebra are also summarized.
Think Like Scilab and Become a Numerical Programming Expert- Notes for Beginn...ssuserd6b1fd
Notes for Scilab Programming. This notes includes the mathematics used behind scilab numerical programming. Illustrated with suitable graphics and examples. Each function is explained well with complete example. Helpful to beginners. GUI programming is also explained.
The document discusses antiderivatives and indefinite integration. It defines antiderivatives as functions whose derivatives are a given function. The general form of the antiderivative includes adding an arbitrary constant C, known as the constant of integration. It provides examples of using basic integration rules to find antiderivatives and solving differential equations. Initial conditions are introduced as a way to determine a particular solution when the general solution involves an arbitrary constant.
This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.
The document discusses methods for solving systems of two equations, including substitution and elimination. It provides examples of using each method to solve both linear and nonlinear systems. It also demonstrates how to determine if a system has no solution or infinitely many solutions. Key methods covered are the substitution method, elimination method, solving systems graphically, and applications of solving systems.
The document defines algebraic expressions and discusses various algebraic operations such as addition, subtraction, multiplication, division, and factorization of algebraic expressions. It provides examples to illustrate each operation. Factorization is described as expressing a complicated polynomial as the product of simpler polynomial factors. Common factoring techniques are mentioned, including factoring the difference of squares and factoring trinomials.
1) The document discusses exponent rules including the zero exponent rule, expanded power rule, and negative exponent rules.
2) It provides examples of applying these rules such as 53=1, (3c2/2d3)4={81c8/16d12}, and (3/r3)-2=r6/9.
3) It lists group work problems assigned from sections 4.1 and 4.2 that are due on Wednesday.
Limit, Continuity and Differentiability for JEE Main 2014Ednexa
The document discusses limits, continuity, and differentiability. It defines the limit of a function, continuity of a function at a point using three conditions, and Cauchy's definition of continuity using delta and epsilon. It also discusses left and right continuity, Heine's definition of continuity using convergent sequences, and the formal definition of continuity. Examples are provided to illustrate calculating limits and determining continuity.
This document discusses rational expressions and equations. It defines a rational expression as an algebraic expression that can be written as the quotient of two polynomials. An expression is undefined when its denominator is equal to zero, which are called the non-permissible values. The document provides examples of finding non-permissible values and simplifying rational expressions by factoring and cancelling common factors in the numerator and denominator. It concludes with an assignment to complete practice problems related to rational expressions and equations.
This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.
The document discusses properties and laws of exponents, radicals, logarithms including the definition of rational exponents, properties of logarithms such as the change of base formula, and examples of simplifying expressions using exponent laws, combining like radicals, and solving logarithmic equations by using properties of logarithms. It also provides sample problems and their step-by-step solutions for simplifying expressions and solving equations involving exponents, radicals, and logarithms.
To solve this rational equation, we first find the LCD of the fractions, which is x^2(x-5)(x-6). Then we multiply both sides of the equation by the LCD:
x^2(x-5)(x-6) = x^2(x-5)(x-6)
Canceling the common factors, we obtain:
x^2 = x^2
Since this holds for all real values of x, the solution is all real numbers. Therefore, the solution is x ∈ R.
This document provides an introduction to basic definite integration. It defines definite integration as calculating the area under a curve between two limits using antiderivatives. It demonstrates how to calculate definite integrals of simple functions and interpret the results as areas. It also discusses how the sign of the integral depends on whether the function lies above or below the x-axis. Quizzes are included to assess understanding of these concepts.
1) A radical function is of the form y = f(x) = ax + b, where changing a and b affects the graph.
2) Graphing shows that if a > 0 the graph increases, if a < 0 the graph decreases, larger a makes the graph steeper, and closer to 0 makes the graph flatter.
3) The value of b is the y-intercept, and the domain is all x ≥ 0 while the range is all y above or below b depending on if the graph increases or decreases.
This document provides an overview of differentiation formulas and concepts, including:
1) The derivative of a constant is 0. The Power Rule states that when taking the derivative of f(x)=x^n, the power is brought down and the exponent is decreased by 1.
2) Evaluation of a derivative involves taking the derivative of a function and plugging in a value. The derivative f'(x) gives the slope of the tangent line to the function f(x) at that point.
3) Leibniz notation represents the derivative of a function f(x) with respect to x as df/dx. Derivatives are used in business and economics to find marginal cost, revenue
This document discusses double integrals and their use in calculating volumes. It begins by introducing double integrals as a way to calculate the volume of a solid bounded above by a function f(x,y) over a rectangular region. It then discusses using iterated integrals to evaluate double integrals by first integrating with respect to one variable and then the other. Finally, it provides examples of using double integrals and iterated integrals to calculate volumes.
This document introduces the concept of double integrals and iterated integrals. It defines a double integral as the limit of double Riemann sums that approximate the volume under a function of two variables over a rectangular region. An iterated integral first integrates with respect to one variable, holding the other constant, resulting in a function of the remaining variable which is then integrated. This allows exact calculation of double integrals by integrating in two steps rather than approximating volume with boxes.
The document discusses relations and functions. It defines relations as mappings between a domain and range, and functions as special relations where each domain element maps to only one range element. It provides examples of different representations of relations and functions, such as numeric, graphical, symbolic and verbal. Common types of functions in algebra are described as mapping subsets of real numbers to other subsets of real numbers. The key characteristics of functions and relations are summarized.
The document provides an overview of key concepts in algebra, including:
1) Algebraic expressions involve unknown quantities represented by letters combined through operations like addition and multiplication. Examples include expressions for area and circumference.
2) To find the numerical value of an algebraic expression, substitute the given values for the unknowns and perform the operations.
3) Monomials, binomials, and polynomials are the main types of algebraic expressions. Operations like addition, multiplication, and factorization can be performed on these expressions.
4) Important algebraic identities exist, such as the difference and sum of squares, and factoring perfect square trinomials using the difference of squares formula.
Rational functions are functions of the form f(x)=polynomial/polynomial. There are six key aspects to analyze in rational functions: y-intercept, x-intercepts, vertical asymptotes, horizontal/slant asymptotes, and the graph. Vertical asymptotes occur when the denominator is 0, x-intercepts when the numerator is 0, and horizontal/slant asymptotes depend on the relative degrees of the numerator and denominator polynomials.
The document defines and provides examples of various types of functions, including:
- Polynomial functions including constant, linear, and general polynomial functions.
- Rational functions defined as the ratio of two polynomial functions.
- Trigonometric functions including sine, cosine, and their inverses.
- Other common functions like absolute value, square root, exponential, logarithmic, floor, and ceiling functions.
It also defines properties of functions like being one-to-one, even, or odd and provides examples of each.
The document discusses limits of functions. It defines one-sided limits and two-sided limits. One-sided limits indicate the limit as x approaches a from the left or right. A two-sided limit exists if both the left and right one-sided limits exist and are equal. The document also presents theorems for evaluating limits algebraically using limits of simpler functions as building blocks. Examples demonstrate applying the theorems to determine limits.
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 defines rational expressions as expressions that can be written as a ratio of two polynomials where the denominator is not 0. It provides examples of finding excluded values, which make the expression undefined, by setting the denominator equal to 0. It also discusses simplifying rational expressions by dividing out common factors in the numerator and denominator and putting them in simplest form. Examples are provided of factoring, dividing out monomials, recognizing opposites, and simplifying a rational model.
This document discusses fractions and decimals. It explains that all fractions can be written as decimals, and fractions with denominators that are powers of 2 or 5 will have terminating decimals, while other denominators will produce recurring decimals. It provides examples of converting terminating and recurring decimals to fractions. It also discusses changing fractions to decimals and vice versa. Finally, it defines rational and irrational numbers, providing examples of each, and asks students to identify numbers as rational or irrational, writing rational numbers as fractions in simplest form.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
This document summarizes exponent properties for adding, subtracting, multiplying, and dividing terms with exponents. It provides examples for each property:
1) When adding or subtracting terms with the same base and exponent, only the coefficient changes.
2) When multiplying terms, you add the exponents and multiply the coefficients.
3) When dividing terms, you subtract the bottom exponent from the top and divide the coefficients.
4) When simplifying powers of powers, you distribute the exponent and multiply the exponents.
Cosa hanno in comune un mattoncino Lego e la backdoor XZ?Speck&Tech
ABSTRACT: A prima vista, un mattoncino Lego e la backdoor XZ potrebbero avere in comune il fatto di essere entrambi blocchi di costruzione, o dipendenze di progetti creativi e software. La realtà è che un mattoncino Lego e il caso della backdoor XZ hanno molto di più di tutto ciò in comune.
Partecipate alla presentazione per immergervi in una storia di interoperabilità, standard e formati aperti, per poi discutere del ruolo importante che i contributori hanno in una comunità open source sostenibile.
BIO: Sostenitrice del software libero e dei formati standard e aperti. È stata un membro attivo dei progetti Fedora e openSUSE e ha co-fondato l'Associazione LibreItalia dove è stata coinvolta in diversi eventi, migrazioni e formazione relativi a LibreOffice. In precedenza ha lavorato a migrazioni e corsi di formazione su LibreOffice per diverse amministrazioni pubbliche e privati. Da gennaio 2020 lavora in SUSE come Software Release Engineer per Uyuni e SUSE Manager e quando non segue la sua passione per i computer e per Geeko coltiva la sua curiosità per l'astronomia (da cui deriva il suo nickname deneb_alpha).
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
AI 101: An Introduction to the Basics and Impact of Artificial IntelligenceIndexBug
Imagine a world where machines not only perform tasks but also learn, adapt, and make decisions. This is the promise of Artificial Intelligence (AI), a technology that's not just enhancing our lives but revolutionizing entire industries.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
- Tips for common problem areas, like team mailboxes, functional/test users, etc
- Practical examples and best practices to implement right away
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
1. A-32 Appendix A A BASIC ALGEBRA REVIEW
74. Construction. A rectangular open-topped box is to be (A) The area of cardboard after the corners have been
constructed out of 9- by 16-inch sheets of thin cardboard removed.
by cutting x-inch squares out of each corner and bending (B) The volume of the box.
the sides up. Express each of the following quantities as a
polynomial in both factored and expanded form.
Section A-4 Rational Expressions: Basic Operations
Reducing to Lowest Terms
Multiplication and Division
Addition and Subtraction
Compound Fractions
We now turn our attention to fractional forms. A quotient of two algebraic expres-
sions, division by 0 excluded, is called a fractional expression. If both the numer-
ator and denominator of a fractional expression are polynomials, the fractional
expression is called a rational expression. Some examples of rational expressions
are the following (recall, a nonzero constant is a polynomial of degree 0):
x 2 1 3 x2 3x 5
2 4
2x 3x 5 x 1 x 1
In this section we discuss basic operations on rational expressions, including mul-
tiplication, division, addition, and subtraction.
Since variables represent real numbers in the rational expressions we are going
to consider, the properties of real number fractions summarized in Section A-1
play a central role in much of the work that we will do.
Even though not always explicitly stated, we always assume that
variables are restricted so that division by 0 is excluded.
Reducing to Lowest Terms
We start this discussion by restating the fundamental property of fractions (from
Theorem 3 in Section A-1):
FUNDAMENTAL PROPERTY OF FRACTIONS
If a, b, and k are real numbers with b, k 0, then
ka a 2 3 3 (x 3)2 2
kb b 2 4 4 (x 3)x x
x 0, x 3
Using this property from left to right to eliminate all common factors from
the numerator and the denominator of a given fraction is referred to as reducing
2. A-4 Rational Expressions: Basic Operations A-33
a fraction to lowest terms. We are actually dividing the numerator and denom-
inator by the same nonzero common factor.
Using the property from right to left—that is, multiplying the numerator and
the denominator by the same nonzero factor—is referred to as raising a fraction
to higher terms. We will use the property in both directions in the material that
follows.
We say that a rational expression is reduced to lowest terms if the numera-
tor and denominator do not have any factors in common. Unless stated to the con-
trary, factors will be relative to the integers.
EXAMPLE Reducing Rational Expressions
1 Reduce each rational expression to the lowest terms.
x2 6x 9 (x 3)2 Factor numerator and denomina-
(A) 2 tor completely. Divide numerator
x 9 (x 3)(x 3)
and denominator by (x 3); this is
x 3 a valid operation as long as x 3
x 3 and x 3.
1
x3 1 (x 1)(x2 x 1) Dividing numerator and denomina-
(B) 2 tor by (x 1) can be indicated by
x 1 (x 1)(x 1)
1 drawing lines through both
(x 1)s and writing the resulting
quotients, 1s.
x2 x 1
x 1 and x 1
x 1
MATCHED PROBLEM Reduce each rational expression to lowest terms.
1 (A)
6x2 x 2
(B)
x4 8x
2x2 x 1 3x3 2x2 8x
EXAMPLE Reducing a Rational Expression
2 Reduce the following rational expression to lowest terms.
6x5(x2 2)2 4x3(x2 2)3 2x3(x2 2)2[3x2 2(x2 2)]
8
x x8
1
2x3(x2 2)2(x2 4)
x8
x5
2(x2 2)2(x 2)(x 2)
x5
3. A-34 Appendix A A BASIC ALGEBRA REVIEW
MATCHED PROBLEM Reduce the following rational expression to lowest terms.
2 6x4(x2 1)2 3x2(x2 1)3
x6
Remember to always factor the numerator and denominator first, then
CAUTION divide out any common factors. Do not indiscriminately eliminate terms
that appear in both the numerator and the denominator. For example,
1
2x3 y2 2x3 y2
2x3 1
y2 y2
1
Since the term y2 is not a factor of the numerator, it cannot be elimi-
nated. In fact, (2x3 y2)/y2 is already reduced to lowest terms.
Multiplication and Division
Since we are restricting variable replacements to real numbers, multiplication and
division of rational expressions follow the rules for multiplying and dividing real
number fractions (Theorem 3 in Section A-1).
MULTIPLICATION AND DIVISION
If a, b, c, and d are real numbers with b, d 0, then:
a c ac 2 x 2x
1.
b d bd 3 x 1 3(x 1)
a c a d 2 x 2 x 1
2. c 0
b d b c 3 x 1 3 x
Explore/Discuss Write a verbal description of the process of multiplying two fractions. Do
the same for the quotient of two fractions.
1
4. A-4 Rational Expressions: Basic Operations A-35
EXAMPLE Multiplying and Dividing Rational Expressions
3 Perform the indicated operations and reduce to lowest terms.
5x2 1 1 Factor numera-
10x y 3
x 2
9 10x3y (x 3)(x 3) tors and denom-
(A) inators; then
3xy 9y 4x2 12x 3y(x 3) 4x(x 3)
divide any nu-
3 1 2 1 merator and any
5x2 denominator
6 with a like com-
mon factor.
1
4 2x 2(2 x) 1 x 2
(B) (x 2) x 2 is the same as .
4 4 x 2 1
2
1
2 x (x 2) b a (a b),
a useful change in
2(x 2) 2(x 2) some problems.
1
1
2
2x3 2x2y 2xy2 x3 y3
(C)
x y xy3
3
x2
2xy y2
2 1 1
2x(x2 xy y2) (x y)2
xy(x y)(x y) (x y)(x2 xy y2)
y 1 1 1
2
y(x y)
MATCHED PROBLEM Perform the indicated operations and reduce to lowest terms.
3 (A)
12x2y3
2
y2 6y 9
3 2
(B) (4 x)
x2 16
2xy 6xy 3y 9y 5
3 3 3 2 2 3
m n mn mn mn
(C) 2 2 3 2 2 3
2m mn n 2m n mn
5. A-36 Appendix A A BASIC ALGEBRA REVIEW
Addition and Subtraction
Again, because we are restricting variable replacements to real numbers, addition
and subtraction of rational expressions follow the rules for adding and subtract-
ing real number fractions (Theorem 3 in Section A-1).
ADDITION AND SUBTRACTION
For a, b, and c real numbers with b 0:
a c a c x 2 x 2
1.
b b b x 3 x 3 x 3
a c a c x x 4 x (x 4)
2.
b b b 2xy 2 2xy 2 2xy 2
Thus, we add rational expressions with the same denominators by adding or
subtracting their numerators and placing the result over the common denomina-
tor. If the denominators are not the same, we raise the fractions to higher terms,
using the fundamental property of fractions to obtain common denominators, and
then proceed as described.
Even though any common denominator will do, our work will be simplified
if the least common denominator (LCD) is used. Often, the LCD is obvious, but
if it is not, the steps in the box describe how to find it.
THE LEAST COMMON DENOMINATOR (LCD)
The LCD of two or more rational expressions is found as follows:
1. Factor each denominator completely.
2. Identify each different prime factor from all the denominators.
3. Form a product using each different factor to the highest power that
occurs in any one denominator. This product is the LCD.
EXAMPLE Adding and Subtracting Rational Expressions
4 Combine into a single fraction and reduce to lowest terms.
3 5 11 4 5x
(A) (B) 1
10 6 45 9x 6y2
x 3 x 2 5
(C) 2
x 6x 9 x2 9 3 x
Solutions (A) To find the LCD, factor each denominator completely:
10 2 5
6 2 3 LCD 2 32 5 90
45 32 5
6. A-4 Rational Expressions: Basic Operations A-37
Now use the fundamental property of fractions to make each denominator
90:
3 5 11 9 3 15 5 2 11
10 6 45 9 10 15 6 2 45
27 75 22
90 90 90
27 75 22 80 8
90 90 9
9x 32x
(B) LCD 2 32xy2 18xy2
6y2 2 3y2
4 5x 2y2 4 3x 5x 18xy2
1
9x 6y2 2y2 9x 3x 6y2 18xy2
8y2 15x2 18xy2
18xy2
x 3 x 2 5 x 3 x 2 5
(C)
x2 6x 9 x2 9 3 x (x 3)2 (x 3)(x 3) x 3
5 5 5 We have again used
Note: the fact that
3 x (x 3) x 3
a b (b a).
The LCD (x 3)2(x 3). Thus,
(x 3)2 (x 3)(x 2) 5(x 3)(x 3)
(x 3)2(x 3) (x 3)2(x 3) (x 3)2(x 3)
(x2 6x 9) (x2 x 6) 5(x2 9) Be careful of sign
(x 3)2(x 3) errors here.
x2 6x 9 x2 x 6 5x2 45
(x 3)2(x 3)
5x2 7x 30
(x 3)2(x 3)
MATCHED PROBLEM Combine into a single fraction and reduce to lowest terms.
4 5 1 6 1 2x 1 3
(A) (B)
28 10 35 4x2 3x3 12x
y 3 y 2 2
(C)
y2 4 y 2
4y 4 2 y
7. A-38 Appendix A A BASIC ALGEBRA REVIEW
16
Explore/Discuss
What is the value of 4 ?
2
2 What is the result of entering 16
What is the difference between 16
4
(4
2 on a calculator?
2) and (16 4) 2?
How could you use fraction bars to distinguish between these two cases
16
when writing 4 ?
2
Compound Fractions
A fractional expression with fractions in its numerator, denominator, or both is
called a compound fraction. It is often necessary to represent a compound frac-
tion as a simple fraction—that is (in all cases we will consider), as the quotient
of two polynomials. The process does not involve any new concepts. It is a mat-
ter of applying old concepts and processes in the right sequence. We will illus-
trate two approaches to the problem, each with its own merits, depending on the
particular problem under consideration.
EXAMPLE Simplifying Compound Fractions
5 Express as a simple fraction reduced to lowest terms.
2
1
x
4
1
x2
Solution Method 1. Multiply the numerator and denominator by the LCD of all frac-
tions in the numerator and denominator—in this case, x2. (We are multiplying by
1 x2/x2).
2 2 1
x2 1 x2 x2
x x 2x x2 x(2 x)
4 4 4 x2 (2 x)(2 x)
x2 2 1 x2 2 x2
x x 1
x
2 x
8. A-4 Rational Expressions: Basic Operations A-39
Method 2. Write the numerator and denominator as single fractions. Then treat as
a quotient.
2 2 x 1 x
1
x x 2 x 4 x2 2 x x2
4 4 x2 x x2 x (2 x)(2 x)
1
x2 2
x 1 1
x
2 x
MATCHED PROBLEM Express as a simple fraction reduced to lowest terms. Use the two methods
5 described in Example 5.
1
1
x
1
x
x
EXAMPLE Simplifying Compound Fractions
6 Express as a simple fraction reduced to lowest terms.
y x
x2 y2
y x
x y
Solution Using the first method described in Example 5, we have
y x y x 1
x2y2 x2y2 x2y2
x2 y2 x2 y2 y3 x3 (y x)(y2 xy x2)
y x y x xy3 x3y xy(y x)(y x)
x2y2 x2y2 x2y2
x y x y 1
y2 xy x2
xy(y x)
MATCHED PROBLEM Express as a simple fraction reduced to lowest terms. Use the first method
6 described in Example 5.
a b
b a
a b
2
b a
9. A-40 Appendix A A BASIC ALGEBRA REVIEW
Answers to Matched Problems
3x 2 x2 2x 4 3(x2
1)2(x 1)(x 1) 5
1. (A) (B) 2. 3. (A) 2x (B) (C) mn
x 1 3x 4 x4 x 4
1 3x2 5x 4 2y2
9y 6 1 a b
4. (A) (B) (C) 5. 6.
4 12x3 ( y 2)2( y 2) x 1 a b
EXERCISE A-4 B
Problems 21–26 are calculus-related. Reduce each fraction to
lowest terms.
A
6x3(x2 2)2 2x(x2 2)3
21.
In Problems 1–20, perform the indicated operations and x 4
reduce answers to lowest terms. Represent any compound 4 2
fractions as simple fractions reduced to lowest terms. 4x (x 3) 3x2(x2 3)2
22. 6
x
d5 d2 a d5 d2 a
1. 2. 2x(1 3x)3 9x2(1 3x)2
3a 6a2 4d 3 3a 6a2 4d 3 23.
(1 3x)6
2y 1 y x2 x 1
3. 4. 2x(2x 3)4 8x2(2x 3)3
18 28 42 12 18 30 24.
(2x 3)8
3x 8 2x 1 5 4m 3 3 2m 1
5. 6. 2x(x 4)3 3(3 x2)(x 4)2
4x2 x3 8x 18m3 4m 6m2 25.
(x 4)6
2x2 7x 3 x2 9
7. (x 3) 8. (x2 x 12) 3x2(x 1)3 3(x3 4)(x 1)2
4x2 1 x2 3x 26.
(x 1)6
m n m2 mn
9.
m2 n2 m 2
2mn n2 In Problems 27–40, perform the indicated operations and
x 2
6x 9 x 2
2x 15 reduce answers to lowest terms. Represent any compound
10. fractions as simple fractions reduced to lowest terms.
x2 x 6 x2 2x
y 1 2
1 1 27.
11. 2 2 2 2 y2 y 2 y2 5y 14 y2 8y 7
a b a 2ab b
2
x x 1 1
3 2 28.
12. x2 2x 1 3x 3 6
x2 1 x2 2x 1
9 m2 m 2
m 1 x 1 29.
13. m 3 14. 1 m2
5m 6 m 3
m 2 x 1
2 x x2 4x 4
5 2 3 2 30.
15. 16. 2x x2 x2 4
x 3 3 x a 1 1 a
x 7 y 9
2 1 2y 31.
17. ax bx by ay
y 3 y 3 y2 9
c 2 c 2 c
2x 1 1 32.
18. 2 2 5c 5 3c 3 1 c
x y x y x y
2 2
x 16 x 13x 36
y2 3 33.
1 1 2x2 10x 8 x3 1
x2 x
19. 20.
y 9 x3 y3 y x2 xy y2
1 x 34. 3
x x y x y y2
10. A-5 Integer Exponents A-41
x2 xy x2 y2 x2 2xy y2 (x h)3 x3
35. 48. (x 1)3 x3 3x2 3x 1
xy y2 x 2
2xy y 2
xy 2
xy 2 h
x2 xy x2 y2 x2 2xy y2 x2 2x x2 2x x 2
36. 49. 2
x 2 1
xy y2 x 2
2xy y2
xy 2
xy 2 x x 2 x2 x 2
x 1 4 2 x 3 2x 2 x 3 1
37. 50.
x2 16 x 4 x 4 x 1 x2 1 x2 1 x 1
2 2 2
3 1 x 4 2x x 2x x 2x x
38. 51.
x 2 x 1 x 2 x2 4 x 2 x2 4 x 2
2 15 x y x 2 x x 2 2
1 2 52. x 2
x x2 y x x 3x 2 x2 3x 2 x 2
39. 40.
4 5 x y
1
x x2 y x
C
Problems 41–44 are calculus-related. Perform the indicated
operations and reduce answers to lowest terms. Represent any In Problems 53–56, perform the indicated operations and
compound fractions as simple fractions reduced to lowest reduce answers to lowest terms. Represent any compound
terms. fractions as simple fractions reduced to lowest terms.
1 1 1 1 y2 s2
y s
x h x (x h)2 x2 y x s t
41. 42. 53. 54.
h h x2 t2
1 2 2
t
y x s t
(x h)2 x2 2x 2h 3 2x 3
x h 2 x 2 x h x 1 1
43. 44. 55. 2 56. 1
h h 2 1
1 1
a 2 1
1
In Problems 45–52, imagine that the indicated “solutions” x
were given to you by a student whom you were tutoring in this
class.
(A) Is the solution correct? If the solution is incorrect, explain In Problems 57 and 58, a, b, c, and d represent real numbers.
what is wrong and how it can be corrected. 57. (A) Prove that d/c is the multiplicative inverse of c/d
(B) Show a correct solution for each incorrect solution. (c, d 0).
x2 5x 4 x2 5x (B) Use part A to prove that
45. x 5
x 4 x a c a d
b, c, d 0
x 2
2x 3 x2
2x b d b c
46. x 2
x 3 x 58. Prove that
2 2
(x h) x a c a c
47. (x 1)2 x2 2x 1 b 0
h b b b
Section A-5 Integer Exponents
Integer Exponents
Scientific Notation
The French philosopher/mathematician René Descartes (1596–1650) is generally
credited with the introduction of the very useful exponent notation “xn.” This