This document analyzes derivative sign patterns (DSPs) in three dimensions. The author begins by reviewing Clark's work on one-dimensional DSPs and Schilling's expansion to two dimensions. In two dimensions, Schilling found 4 possible DSPs. The author then analyzes all possible combinations of applying two-dimensional DSPs in three dimensions, finding only 8 valid arrangements. The DSP in three dimensions, like one and two dimensions, is determined by the original function and its first derivative. Extensions to other functions or domains are suggested for future research.
Rational exponents allow radicals to be written as fractional exponents. To write a radical with an index of n as a rational exponent, the index becomes the denominator and the exponent of the radicand becomes the numerator. The rules of exponents still apply to rational exponents. Expressions with rational exponents are simplified by having no negative exponents, no fractional exponents in the denominator, not being a complex fraction, and having the least possible index for any remaining radicals. Examples show how to evaluate, simplify, and perform operations on expressions with rational exponents.
This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.
MIT Math Syllabus 10-3 Lesson 3: Rational expressionsLawrence De Vera
This document provides information about rational expressions including:
- A rational expression is a fraction where the numerator and denominator are polynomials.
- The domain of a rational expression excludes any values that would cause division by zero.
- Rational expressions can be simplified by factoring the numerator and denominator and cancelling out common factors.
- The four arithmetic operations can be performed on rational expressions using similar properties as rational numbers.
- Complex fractions containing fractions in the numerator or denominator can be simplified by finding the least common denominator or multiplying the numerator and denominator by the reciprocal.
- Rational expressions can be used to solve applications involving averages, rates, or other fractional relationships.
1. A function is a relation between a set of inputs and a set of outputs such that each input is mapped to exactly one output.
2. Functions can be defined formally as a subset of the Cartesian product between the domain and codomain sets such that each element in the domain appears as the first element of exactly one ordered pair in the subset.
3. Examples of functions include one that maps shapes to their colors, one that maps natural numbers to integers by subtracting 4, and one that maps polygons to their number of vertices.
The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of x-values and the range is the set of y-values. A function is a relation where each domain value is paired with exactly one range value. The document provides examples of determining if a relation represents a function using the vertical line test and evaluating functions using function notation.
This document provides an overview of exponential and logarithmic functions. It covers composite and inverse functions, exponential functions, logarithmic functions, properties of logarithms, common logarithms, exponential and logarithmic equations, and natural exponential and logarithmic functions. Example problems are provided to illustrate each concept.
This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
The document discusses rational expressions and operations involving them. It covers evaluating, simplifying, multiplying, dividing, adding and subtracting rational expressions. It also discusses finding least common denominators to add or subtract rational expressions with unlike denominators. The sections include evaluating rational expressions, simplifying rational expressions, multiplying and dividing rational expressions, and adding and subtracting rational expressions with the same or different denominators.
Rational exponents allow radicals to be written as fractional exponents. To write a radical with an index of n as a rational exponent, the index becomes the denominator and the exponent of the radicand becomes the numerator. The rules of exponents still apply to rational exponents. Expressions with rational exponents are simplified by having no negative exponents, no fractional exponents in the denominator, not being a complex fraction, and having the least possible index for any remaining radicals. Examples show how to evaluate, simplify, and perform operations on expressions with rational exponents.
This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.
MIT Math Syllabus 10-3 Lesson 3: Rational expressionsLawrence De Vera
This document provides information about rational expressions including:
- A rational expression is a fraction where the numerator and denominator are polynomials.
- The domain of a rational expression excludes any values that would cause division by zero.
- Rational expressions can be simplified by factoring the numerator and denominator and cancelling out common factors.
- The four arithmetic operations can be performed on rational expressions using similar properties as rational numbers.
- Complex fractions containing fractions in the numerator or denominator can be simplified by finding the least common denominator or multiplying the numerator and denominator by the reciprocal.
- Rational expressions can be used to solve applications involving averages, rates, or other fractional relationships.
1. A function is a relation between a set of inputs and a set of outputs such that each input is mapped to exactly one output.
2. Functions can be defined formally as a subset of the Cartesian product between the domain and codomain sets such that each element in the domain appears as the first element of exactly one ordered pair in the subset.
3. Examples of functions include one that maps shapes to their colors, one that maps natural numbers to integers by subtracting 4, and one that maps polygons to their number of vertices.
The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of x-values and the range is the set of y-values. A function is a relation where each domain value is paired with exactly one range value. The document provides examples of determining if a relation represents a function using the vertical line test and evaluating functions using function notation.
This document provides an overview of exponential and logarithmic functions. It covers composite and inverse functions, exponential functions, logarithmic functions, properties of logarithms, common logarithms, exponential and logarithmic equations, and natural exponential and logarithmic functions. Example problems are provided to illustrate each concept.
This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
The document discusses rational expressions and operations involving them. It covers evaluating, simplifying, multiplying, dividing, adding and subtracting rational expressions. It also discusses finding least common denominators to add or subtract rational expressions with unlike denominators. The sections include evaluating rational expressions, simplifying rational expressions, multiplying and dividing rational expressions, and adding and subtracting rational expressions with the same or different denominators.
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 outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
This document provides an overview of rational expressions and equations. It discusses reviewing rational expressions, domains of rational expressions, writing rational expressions in lowest terms, multiplying and dividing rational expressions, complex fractions, adding and subtracting rational expressions, and simplifying complex fractions. Examples are provided to illustrate key concepts and steps for working with rational expressions.
The document discusses solving rational inequalities. It defines interval and set notation that can be used to represent the solutions to inequalities. It then presents the procedure for solving rational inequalities, which involves rewriting the inequality as a single fraction on one side of the inequality symbol and 0 on the other side, and determining the intervals where the fraction is positive or negative. Examples are provided to demonstrate solving rational inequalities and applying the solutions to word problems.
The document discusses various types of functions including:
- Constant functions which assign the same real number to every element of the domain.
- Linear functions which have a degree of 1 and are defined by the equation f(x)=mx+b.
- Quadratic functions which are polynomial functions of degree 2.
- Cubic/power functions which are polynomial functions of degree 3.
It also briefly describes identity, absolute value, rational, and algebraic functions. The document concludes with instructions for a group activity on identifying different function types from graphs.
1) The document discusses graphing and properties of exponential and logarithmic functions, including: graphing exponential functions by substituting values of the variable into the equation, graphing logarithmic functions using the change of base formula, and properties like the product, quotient, and power properties of logarithms.
2) Examples are provided of solving exponential and logarithmic equations using properties like changing bases to the same value, multiplying or dividing arguments using the product and quotient properties, and applying exponents using the power property.
3) Steps shown include using properties to isolate the variable, set arguments or exponents equal to each other, and solve the resulting equation.
This document discusses functions and their properties. It defines a function as a relation where each input is paired with exactly one output. Functions can be represented numerically in tables, visually with graphs, algebraically with explicit formulas, or verbally. The domain is the set of inputs, the codomain is the set of all possible outputs, and the range is the set of actual outputs. Functions can be one-to-one (injective) if each input maps to a unique output, or onto (surjective) if each possible output is the image of some input.
A function is a relation between a set of inputs (domain) and set of outputs (codomain) where each input is mapped to exactly one output. There are different types of functions such as one-to-one, onto, bijective, many-to-one, and inverse functions. Functions can be represented graphically or using function notation such as f(x). Common functions include polynomial, trigonometric, exponential, logarithmic, and composite functions which are the composition of two simpler functions.
The document describes the false position method for finding roots of equations. It involves using linear interpolation between two initial guesses that bracket the root to find a new, improved estimate of the root. The method iteratively calculates new estimates using a false position formula until converging to within a specified tolerance of the true root. While faster than bisection, false position may converge less precisely in some cases if the graph is convex down between the initial guesses.
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
The document discusses the bisection method, a numerical method for finding the roots of a polynomial function. The bisection method works by repeatedly bisecting an interval where the function changes sign and narrowing in on a root. It involves finding an interval [a,b] where f(a) and f(b) have opposite signs, taking the midpoint x1, and discarding half of the interval based on the sign of f(x1). This process is repeated, halving the interval width each time, until a root is isolated within the desired accuracy. An example applying the bisection method to find the roots of f(x) = x^3 + 3x - 5 on the interval [1,2
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.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
Simplifying expressions, solving, domain and rangejoannahstevens
This document appears to be notes from a math class covering simplifying expressions, solving equations, and domain and range. It includes examples of irrational and imaginary numbers, naming polynomials by degree and terms, multiplying binomials, homework assignments on the topics, and notes on using the vertical line test to determine if a relation is a function by checking if each x-value has exactly one y-value. Sketching graphs of functions given domains and ranges is also mentioned.
55 addition and subtraction of rational expressions alg1testreview
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.
Logarithmic functions relate the exponent of a number to the logarithm of its base. They are inverse functions of exponentials, where the domain and range are flipped. Key properties include:
1) Logarithmic graphs have a domain of positive real numbers and range of all real numbers, while being always increasing with a vertical asymptote at y=0.
2) Logarithms can have various bases, with common bases being base 10 (common log), base e (natural log), and base 2 or 10.
3) Logarithmic equations can be solved by converting them to exponential form and then solving the exponential equation.
Math functions, relations, domain & rangeRenee Scott
This document discusses math functions, relations, and their domains and ranges. It provides examples of relations and explains that the domain is the set of first numbers in ordered pairs, while the range is the set of second numbers. A function is defined as a relation where each x-value has only one corresponding y-value. It compares examples of relations that are and are not functions and describes how to evaluate functions by inserting values. Tests for identifying functions like the vertical line test are also outlined.
Functions are special types of relations where each input is paired with exactly one output. Functions can be expressed as sets of ordered pairs, tables, mappings, or graphs. A function expressed as a graph will pass the vertical line test, meaning a vertical line drawn on the graph will intersect it at only one point.
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.
The document provides examples and definitions of various types of functions to illustrate to beginning students that functions can take many forms beyond polynomial equations. It defines linear, quadratic, and polynomial functions precisely and provides examples of each. It also introduces other types of functions like absolute value, sequences, Tau, Sigma, the Fibonacci sequence, and arithmetic sequences. The overall purpose is to demonstrate the diversity of functions and convince students that the domain and rule of a function need not be numbers or equations.
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 outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
This document provides an overview of rational expressions and equations. It discusses reviewing rational expressions, domains of rational expressions, writing rational expressions in lowest terms, multiplying and dividing rational expressions, complex fractions, adding and subtracting rational expressions, and simplifying complex fractions. Examples are provided to illustrate key concepts and steps for working with rational expressions.
The document discusses solving rational inequalities. It defines interval and set notation that can be used to represent the solutions to inequalities. It then presents the procedure for solving rational inequalities, which involves rewriting the inequality as a single fraction on one side of the inequality symbol and 0 on the other side, and determining the intervals where the fraction is positive or negative. Examples are provided to demonstrate solving rational inequalities and applying the solutions to word problems.
The document discusses various types of functions including:
- Constant functions which assign the same real number to every element of the domain.
- Linear functions which have a degree of 1 and are defined by the equation f(x)=mx+b.
- Quadratic functions which are polynomial functions of degree 2.
- Cubic/power functions which are polynomial functions of degree 3.
It also briefly describes identity, absolute value, rational, and algebraic functions. The document concludes with instructions for a group activity on identifying different function types from graphs.
1) The document discusses graphing and properties of exponential and logarithmic functions, including: graphing exponential functions by substituting values of the variable into the equation, graphing logarithmic functions using the change of base formula, and properties like the product, quotient, and power properties of logarithms.
2) Examples are provided of solving exponential and logarithmic equations using properties like changing bases to the same value, multiplying or dividing arguments using the product and quotient properties, and applying exponents using the power property.
3) Steps shown include using properties to isolate the variable, set arguments or exponents equal to each other, and solve the resulting equation.
This document discusses functions and their properties. It defines a function as a relation where each input is paired with exactly one output. Functions can be represented numerically in tables, visually with graphs, algebraically with explicit formulas, or verbally. The domain is the set of inputs, the codomain is the set of all possible outputs, and the range is the set of actual outputs. Functions can be one-to-one (injective) if each input maps to a unique output, or onto (surjective) if each possible output is the image of some input.
A function is a relation between a set of inputs (domain) and set of outputs (codomain) where each input is mapped to exactly one output. There are different types of functions such as one-to-one, onto, bijective, many-to-one, and inverse functions. Functions can be represented graphically or using function notation such as f(x). Common functions include polynomial, trigonometric, exponential, logarithmic, and composite functions which are the composition of two simpler functions.
The document describes the false position method for finding roots of equations. It involves using linear interpolation between two initial guesses that bracket the root to find a new, improved estimate of the root. The method iteratively calculates new estimates using a false position formula until converging to within a specified tolerance of the true root. While faster than bisection, false position may converge less precisely in some cases if the graph is convex down between the initial guesses.
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
The document discusses the bisection method, a numerical method for finding the roots of a polynomial function. The bisection method works by repeatedly bisecting an interval where the function changes sign and narrowing in on a root. It involves finding an interval [a,b] where f(a) and f(b) have opposite signs, taking the midpoint x1, and discarding half of the interval based on the sign of f(x1). This process is repeated, halving the interval width each time, until a root is isolated within the desired accuracy. An example applying the bisection method to find the roots of f(x) = x^3 + 3x - 5 on the interval [1,2
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.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
Simplifying expressions, solving, domain and rangejoannahstevens
This document appears to be notes from a math class covering simplifying expressions, solving equations, and domain and range. It includes examples of irrational and imaginary numbers, naming polynomials by degree and terms, multiplying binomials, homework assignments on the topics, and notes on using the vertical line test to determine if a relation is a function by checking if each x-value has exactly one y-value. Sketching graphs of functions given domains and ranges is also mentioned.
55 addition and subtraction of rational expressions alg1testreview
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.
Logarithmic functions relate the exponent of a number to the logarithm of its base. They are inverse functions of exponentials, where the domain and range are flipped. Key properties include:
1) Logarithmic graphs have a domain of positive real numbers and range of all real numbers, while being always increasing with a vertical asymptote at y=0.
2) Logarithms can have various bases, with common bases being base 10 (common log), base e (natural log), and base 2 or 10.
3) Logarithmic equations can be solved by converting them to exponential form and then solving the exponential equation.
Math functions, relations, domain & rangeRenee Scott
This document discusses math functions, relations, and their domains and ranges. It provides examples of relations and explains that the domain is the set of first numbers in ordered pairs, while the range is the set of second numbers. A function is defined as a relation where each x-value has only one corresponding y-value. It compares examples of relations that are and are not functions and describes how to evaluate functions by inserting values. Tests for identifying functions like the vertical line test are also outlined.
Functions are special types of relations where each input is paired with exactly one output. Functions can be expressed as sets of ordered pairs, tables, mappings, or graphs. A function expressed as a graph will pass the vertical line test, meaning a vertical line drawn on the graph will intersect it at only one point.
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.
The document provides examples and definitions of various types of functions to illustrate to beginning students that functions can take many forms beyond polynomial equations. It defines linear, quadratic, and polynomial functions precisely and provides examples of each. It also introduces other types of functions like absolute value, sequences, Tau, Sigma, the Fibonacci sequence, and arithmetic sequences. The overall purpose is to demonstrate the diversity of functions and convince students that the domain and rule of a function need not be numbers or equations.
The document provides information about the BC Aware Campaign, an annual cybersecurity and privacy awareness initiative in British Columbia. It summarizes the goals and history of the campaign, which began in 2014 in response to growing cybersecurity threats and a lack of understanding around information security concepts. The campaign aims to educate both professionals and citizens through various events organized by ISACA Vancouver and other participating organizations. The bulk of the document consists of an agenda and session descriptions for the 2016 BC Aware Day event, which features over 20 presentations, panels and discussions on topics related to cybersecurity, privacy, risk management and compliance.
The document describes a large ship called "The Liberty" and provides details about its features and amenities. It notes there is one captain and describes the various areas of the ship including multiple decks, dining rooms, gyms, pools, and other facilities. Sections are repeated with different levels of detail.
Mangesh Pusdekar is seeking a position utilizing his engineering and management skills. He has experience as a Team Engineer at Finolex Industries, working on process control, reliability, and plant expansion. He has an MBA in Finance and Operations from RTMNU Nagpur and a B.E. in Chemical Engineering from Shivaji University Kolhapur. His strengths include being optimistic and believing in teamwork. He completed an internship at Rashtriya Chemicals & Fertilizers in their sulphuric acid plant.
Sandra Tooks is an experienced office administration professional seeking a new position. She has over 13 years of experience in bookkeeping, accounts payable/receivable, operations management, and administrative support. Currently she works as a Community Liaison for the New York City Department of Transportation, where her responsibilities include coordinating volunteer programs, creating reports, and resolving complaints. Previously she held administrative roles at New Testament Baptist Church and Wipe-Tex International Corp, where she managed accounts, budgets, and bookkeeping activities. She has strong communication, organizational, and computer skills.
Este documento describe las funciones y clasificaciones de los sistemas operativos. Explica que un sistema operativo controla la computadora y administra programas y recursos. Sus funciones principales incluyen la interfaz de usuario, administración de recursos, archivos y tareas, y proveer servicios de soporte. Los sistemas operativos se pueden clasificar como mono o multitarea, mono o multiusuario, de tiempo real o tiempo compartido.
Political parties and media organizations use data mining of voter databases to target voters more effectively during election campaigns. They combine disparate databases to gain clearer insights into voter preferences and characteristics to tailor get-out-the-vote efforts. The Election Commission also mines past election data to optimize polling booth locations and identify areas needing improved security or having low voter turnout. Data mining helps predict election outcomes through opinion and exit polls by identifying the most accurate predictive models.
This module discusses polynomial functions of degree greater than two. The key points are:
1. The graph of a third-degree polynomial has both a minimum and maximum point, while higher degree polynomials have one less turning point than their degree.
2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
The document discusses functions and relations. It defines functions as special relations where each element of the domain is mapped to only one element in the range. It provides examples of determining whether a relation represented by ordered pairs or a graph is a function. It also discusses determining the domain and range of functions from ordered pairs or graphs. Finally, it introduces mapping and functional notation used to represent functions.
This document discusses functions and relations. It defines a relation as a set of ordered pairs and provides examples. It then defines a function as a special type of relation where each element of the domain corresponds to exactly one element of the range, meaning no two ordered pairs can have the same first element. The document discusses different types of relations including one-to-one, one-to-many, many-to-one, and many-to-many. It also discusses how functions can be presented using arrow diagrams, tables, graphs, and ordered pairs. Finally, it discusses function notation and evaluating functions by substituting values into the function.
The document defines and provides properties of various mathematical functions including:
- Relations and sets including Cartesian products and relations.
- Functions including domain, co-domain, range, and the number of possible functions between sets.
- Types of functions such as polynomial, algebraic, transcendental, rational, exponential, logarithmic, and absolute value functions.
- Graphs of important functions are shown such as 1/x, sinx, logx, |x|, [x], and their key properties are described.
This document defines key terms related to functions such as domain, range, and piecewise functions. It provides examples of representing functions using tables, ordered pairs, graphs, and equations. It also discusses how to determine if a relation represents a function and describes piecewise functions as using more than one formula with separate domains.
The document defines key concepts related to functions and relations, including:
- Sets, set notation, and operations like intersection and union
- Different types of number sets like natural, integer, rational, and real numbers
- Ordered pairs and how they are used to define relations and functions
- The domain and range of relations and functions
- What defines a function versus a relation
- Examples of functions, their graphs, and evaluating functions for given inputs
This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
This document defines functions and discusses key concepts related to functions including:
- A function relates each element of its domain to a unique element of its range.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, equations, and graphs.
- The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
The document discusses the Cartesian coordinate system and functions. It explains that Cartesian coordinates use two number lines at right angles (x and y axes) to represent points in two dimensions. Functions are then defined as relations where each x-value is paired with only one y-value. The document provides examples of linear and parabolic functions graphed in the Cartesian plane and examines their domain, range, and whether they satisfy the definition of a function.
Grade 11-Strand(Concept of functions).pptxAlwinCAsuncion
This document discusses functions and their properties including:
1. Defining functions and related terms like domain and range
2. Determining if a relation represents a function
3. Defining piecewise functions and representing real-life situations using functions and piecewise functions.
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
This document provides an overview of character theory for finite groups and its analogy to representation theory for the infinite compact group S1. It discusses several key concepts:
1) Character theory characterizes representations of a finite group G by their characters, which are class functions that map G to complex numbers. Characters determine representations up to isomorphism.
2) The irreducible representations of S1 are 1-dimensional and in bijection with integers n, where each representation maps z to zn.
3) By analogy to Fourier analysis, the characters of S1 form an orthonormal basis for L2(S1) and decompose representations into irreducibles in the same way as for finite groups.
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
- The document is a lesson on identifying and graphing linear functions. It provides examples of determining if a graph or set of ordered pairs represents a linear function based on whether the relationship is constant.
- It also discusses writing linear equations in standard form and using linear equations to graph the line by choosing values for the variable and plotting the corresponding points.
- Real-world examples are given to show restricting the domain and range based on the context and graphing linear functions as discrete points rather than a continuous line.
1. Addition of functions: Add the outputs of two functions with the same inputs.
2. Subtraction of functions: Subtract the outputs of two functions with the same inputs.
3. Multiplication of functions: Multiply the outputs of two functions with the same inputs.
4. Division of functions: Divide the output of one function by the output of another function with the same inputs.
5. Composition of functions: Replace the inputs of one function with the outputs of another function.
The document defines functions and discusses:
- Functions are sets of ordered pairs with each first element paired to a unique second element.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, as equations, and as graphs.
- The domain of a function is the set of first elements and the range is the set of second elements.
1. Derivative Sign Patterns for Infinitely
Differentiable Functions in Three-Dimensions
Madeline Edwards
March 9, 2015
Abstract
A derivative sign pattern (DSP) is a sequence of positive and negative
signs that represent the signs of a function and its derivatives over its do-
main. Some infinitely differentiable functions have sign patterns, but not
all. Functions that fall into this category are trigonometric functions, ex-
ponential functions, logarithmic functions, and possibly others. Of these,
only exponential functions have sign patterns; the others take different
signs on their domains. As seen in Calculus I, certain functions can be
differentiated without eventually becoming 0. In the one-dimensional case
studied by Clark, ex
is an example of a function that has an infinite num-
ber of derivatives. In the domain of all real numbers, R, Clark determined
the DSPs and found example functions to match the pattern. In R, Clark
found only four valid sign patterns, all positives, positive and negative
signs alternating, and their negations. In the case of R, a function that
is infinitely differentiable that has a sign pattern can be determined from
the original function and the first derivative. Schilling expanded from the
one-dimensional case to the two-dimensional case for the entire plane. In
the domain of R × R with ordered pairs, Schilling found eight possible
DSPs. Building on Schilling’s Derivative Sign Pattern Theorem, the ex-
pansion to the three-dimensional case is analyzed. The specific case of
interest in three-dimensions is ordered triples of real numbers, R × R × R.
From Schilling’s research of matrix possibilities in two-dimensions, anal-
ysis of what is possible in three-dimensions can be constructed. In the
three-dimensional case, there is interesting geometry among the deriva-
tive sign patterns with only a finite number of possible combinations found
in R×R×R. While applications of DSPs in three-dimensions are limited,
the gained understanding of how functions and derivatives work within a
given domain is the greater purpose to this research.
Derivative Sign Patterns in domain R
In the one-dimensional case from Clark’s research and two-dimensional case from
Schilling’s research, there are only a finite number of derivative sign patterns
(DSP) in the domain of all real numbers, R. When determining all possible DSP
1
2. in three-dimensions, Clark found two possible DSPs and their negations [?] and
Schilling stated that there are four possible DSPs in two-dimensions, and their
negations [?]. By looking at combinations of DSP types, DSP A, DSP B, DSP
C, DSP D, and their negations, with single combinations calculations, there are
56 possible distinct arrangements of two-dimensional DSP in three-dimensions.
+ + + ...
+ + + ...
+ + + ...
...
...
...
(a)
+ − + ...
+ − + ...
+ − + ...
...
...
...
(b)
+ + + ...
− − − ...
+ + + ...
...
...
...
(c)
+ − + ...
− + − ...
+ − + ...
...
...
...
(d)
Figure 1: The valid two-dimensional DSP found by Schilling in R; types DSP A, DSP B, DSP C,
and DSP D.
Valid arrangements of DSPs are where positive and negative signs align with
every slice in three-dimensions with respect to the xy, xz, and yz planes. From
observing all possible combinations in three-dimensions of two-dimensional ar-
rangements, only 8 possible arrangements are valid in addition to their nega-
tions.
DSP Example Function Negation DSP Example Function
A, A, A ex+y+z
Neg. A, Neg. A, Neg. A −ex+y+z
C, C, A e−x+y+z
Neg. C, Neg. C, Neg. A −e−x+y+z
A, B, B ex+y−z
Neg. A, Neg. B, Neg. B −ex+y−z
C, D, B e−x+y−z
Neg. C, Neg. D, Neg. B −e−x+y−z
B, A, C ex−y+z
Neg. B, Neg. A, Neg. C −ex−y+z
D, C, C e−x−y+z
Neg. D, Neg. C, Neg. C −e−x−y+z
B, B, D ex−y−z
Neg. B, Neg. B, Neg. D −ex−y−z
D, D, D e−x−y−z
Neg. D, Neg. D, Neg. D −e−x−y−z
Table 1
The combinations from Schilling’s two-dimensional
cases of derivative sign patterns.
A single equation in one position is either positive or negative. At any given
position in three-dimension, if one perspective of a plane in a single position is
one sign, while another perspective of a different plane, but in the same position,
are of opposite signs, this shows a contradiction. Since a function cannot be both
positive and negative within one position, this contradiction shows an invalid
combination of a three-dimensional DSP.
To represent valid and invalid DSP’s, Figure 2 and Figure 3 display positive
and negative sign values with respect to x, y, z.
2
3. Figure 2: Valid DSP: B, A, C Figure 3: Invalid DSP: Negation B, A, C
The small grid works by creating cubes within the figure represent a single
position with a derivative taken with respect to x, y, and z. The different posi-
tions within the cube along the x, y, z axes demonstrate the various derivatives
taken with respect to each variable. If in a given sub-cube in the figure does
not have identical signs on all faces of the sub-cube, then the DSP is not valid
because there are conflicting signs within a single position.
Derivative Sign Pattern Determination
The derivative sign pattern is determined by knowing the sign pattern of the
original function and its first derivative. As seen in Clark’s research, if (an)
has a sign pattern in the domain R, then an = an+2 for all n. Thus the sign
pattern is completely determined after the original function and first derivative.
In two-dimensions, Schilling’s matrices the exhibit the DSPs are determined by
knowing the original function f(x, y), and the first derivatives, fx, fy. The sign
of fx,y is forced based on the other signs of the first and original functions signs.
+ + ...
+ fx,y ...
...
...
→
+ + ...
+ − ...
...
...
Figure 4
The invalid two-dimensional DSP found by Schilling in R; type E where this is not
possible, thus the fx,y is forced based on the other sign entries.
Specifically in the matrices, only the first 2 × 2 inputs are needed to know
how the DSPs will behave in R.
3
4. Three-Dimensional Derivative Sign Pattern Determination
In three-dimensions, the original function and the first derivative with respect
to x, y, z determine the sign pattern in R. If f(x, y, z) has a sign pattern in R,
then the sign pattern is completely determined by f, fx, fy, and fz.
f fx
fy fx,y
fz fx,z
fy,z fx,y,z
Figure 5
Placing ± signs at any node automatically force surrounding nodes to be a specific sign so
that it does not violate the two-dimensional DSP possibilities listed prior.
This only holds true because in the case of R, there are only a finite number
of patterns that are possible and the patterns remains constant throughout the
entire domain, thus fn = fn+2 ∀n ∈ N, to which n is the power that f is derived.
Thus, like in the one-dimensional case and two-dimensional case, the original
function and first derivative determine the DSP. All even and odd derivatives
have the same sign, respectively. Similarly to the two-dimensional case, in three
dimensions, the signs of fx,y, fx,z, fy,z, and fx,y,z are forced.
4
5. ± ±
± fx,y
± fx,z
fy,z fx,y,z
Figure 6
The known signs of f, fx, fy, fz force the signs of fx,y, fy,z, fx,z, and fx,y,z.
Assume that f, fx, fy, fz signs are known. By strictly looking at the x, y
plane, this forces fx,y to chose a specific sign based on the signs of f, fx, fy, fz
as this two-dimensional perspective must follow one of the valid DSP cases, A,
B, C, D or their negations. Now that fx,y is determined, fx,z is also forced. By
looking at the x, z plane, since f, fx, and fz are known, then fx,z is forced and
must follow a two-dimensional DSP cases. This then leads fx,y,z to be forced
because fx, fx,y, and fx,z are known. We are only left with the last determined
node, fy,z which is forced since fy, fx,y and fx,y,z are known. To determine each
node’s sign with respect to the specific derivative, the surround three nodes of a
single surface must be known. Thus determination follows this pattern strictly,
Known Signs Determines Planes of Cube
f, fx, fy, fz fx,y, fx,z x, y and x, z
f, fx, fy, fz, fx,y, fx,z fx,y,z x, y, z
f, fx, fy, fz, fx,y, fx,z, fx,y,z fy,z y, z
Table 2
Once three node signs are know, a single plane is determined.
Like the one and two-dimensional case, the DSPs in three dimensions is com-
pletely determined by the original function and the first derivative. The first
2 × 2 × 2 cube determines the entire DSP pattern in R × R × R. From the table
above, there are only 16 possibilities of valid DSPs in three-dimensions. Notice
that every example function matching a unique DSP is of the form ±e±x±y±z
.
With each node on the cube, there are two possibilities; either positive or nega-
tive signs. Since there are eight nodes on the 2 × 2 × 2 cube, there are a total of
16 choices of signs on the cube, representing the only possible 16 DSP found in
three dimensions. Looking at the example equations, ±e±x±y±z
, there are 24
5
6. possible combinations, or 16 total possibilities for the function, each uniquely
identifying with only one specific DSP.
Extensions of Research
Possible extensions of this research would be to look at other infinitely differ-
entiable functions that exhibit theses DSPs in derivative sign patterns. While
exponential functions have been noted previously in the table, it is possible that
there are many other types of functions with a DSP in three-dimensions. An-
other area of research would be to look at other domains and possible DSPs.
Domains include the unit cube, R+
× R+
× R+
, (0, 1) × R+
× R+
, as well as
others.
References
[1] Clark, Jeffrey. Derivative Sign Patterns, The College Mathemat-
ics Journal 42.5 (2011): 379-82. JSTOR. Web.20 Sept. 2014.
http://www.jstor.org/stable/10.4169/college.math.j.42.5.379?ref=no-x-
route453a5d25a141997a8acf14a65956bcf4
[2] Schilling, Kenneth. Derivative Sign Patterns in Two Dimensions. The Col-
lege Mathematics Journal 44.2 (2013): 102-08. JSTOR. Web. 20 Sept.
2014. http://www.jstor.org/stable/10.4169/college.math.j.44.2.102?ref=no-
x-route:937a49d3d27b9e5b683e99af4239ce50.
6