The document discusses inequalities and graphs. It provides examples of solving inequalities like x^2 ≤ 1 and discusses the graphs of functions like y = x. It shows that y = x is an increasing function for x ≥ 0. It also uses integrals to calculate the area under the graph of y = x and proves results using mathematical induction.
This document provides steps for graphing a function based on analyzing its derivatives. It uses the function f(x) = 1/x + 2 as a detailed example. The steps are:
1) Find where the function is positive, negative, zero, and undefined
2) Analyze the first derivative f' to determine maxima/minima and intervals of increase/decrease
3) Analyze the second derivative f'' to determine concavity and points of inflection
4) Combine the analyses into a chart and graph the function
This document lists multiplication facts for the number 2. It shows that 2 multiplied by single digit numbers 1 through 12 equals the products 2 through 24 in order.
This document discusses search problems and exploring state spaces. It provides examples of search problems like the 8-puzzle and 15-puzzle. It explains that search involves exploring a state space to find a path from an initial state to a goal state. The state space grows enormously large for puzzles like the 15-puzzle. Effective search requires constructing solutions by exploring only a small portion of the total state space, which is typically represented as a search tree. The document outlines the key components of formulating a problem as a search problem and searching the state space to find a solution.
A Terra, Estudos e representações_População e Povoamentomarise brandao
Este documento descreve um projeto para alunos do 7o e 8o ano usarem mapas digitais para localizar lugares, calcular distâncias e analisar a distribuição espacial de indicadores demográficos. Os alunos irão usar software de SIG online para completar as atividades planejadas e serão avaliados pelo professor com base na observação em sala de aula e no trabalho final apresentado.
This poem addresses a woman of Africa, questioning why she is shy about speaking her wisdom and history. It notes that history is not just about kings, queens and foreign conquests, and that as a woman she is surrounded by hair and beauty products from companies that do not make products for her. It encourages her to speak her wisdom in cold lands heated by machines, where blackness is not accepted and beauty is defined as thin. It asks why she is shy about her blessed body that gives nurture and comfort, while sexuality is represented by dressed-up images made by exploited children. It questions why she does not use her wisdom when her men are stereotyped through ages of ignorance that only she can dispel.
This document provides steps for graphing a function based on analyzing its derivatives. It uses the function f(x) = 1/x + 2 as a detailed example. The steps are:
1) Find where the function is positive, negative, zero, and undefined
2) Analyze the first derivative f' to determine maxima/minima and intervals of increase/decrease
3) Analyze the second derivative f'' to determine concavity and points of inflection
4) Combine the analyses into a chart and graph the function
This document lists multiplication facts for the number 2. It shows that 2 multiplied by single digit numbers 1 through 12 equals the products 2 through 24 in order.
This document discusses search problems and exploring state spaces. It provides examples of search problems like the 8-puzzle and 15-puzzle. It explains that search involves exploring a state space to find a path from an initial state to a goal state. The state space grows enormously large for puzzles like the 15-puzzle. Effective search requires constructing solutions by exploring only a small portion of the total state space, which is typically represented as a search tree. The document outlines the key components of formulating a problem as a search problem and searching the state space to find a solution.
A Terra, Estudos e representações_População e Povoamentomarise brandao
Este documento descreve um projeto para alunos do 7o e 8o ano usarem mapas digitais para localizar lugares, calcular distâncias e analisar a distribuição espacial de indicadores demográficos. Os alunos irão usar software de SIG online para completar as atividades planejadas e serão avaliados pelo professor com base na observação em sala de aula e no trabalho final apresentado.
This poem addresses a woman of Africa, questioning why she is shy about speaking her wisdom and history. It notes that history is not just about kings, queens and foreign conquests, and that as a woman she is surrounded by hair and beauty products from companies that do not make products for her. It encourages her to speak her wisdom in cold lands heated by machines, where blackness is not accepted and beauty is defined as thin. It asks why she is shy about her blessed body that gives nurture and comfort, while sexuality is represented by dressed-up images made by exploited children. It questions why she does not use her wisdom when her men are stereotyped through ages of ignorance that only she can dispel.
Este documento presenta biografías breves de varias personas notables. Nir Ofir creó el BlogDay en 2005 para que los blogueros puedan conocerse unos a otros. Yoani Sánchez es una periodista cubana conocida por su blog Generación Y donde critica la realidad de Cuba a pesar del acoso del gobierno. Michelle Malkin es una comentarista política conservadora estadounidense cuyas columnas aparecen en varios periódicos.
The document provides an overview of Ryan Leach's project to compile and visualize historic drillhole data from the Dominique Property in Yarmouth County, Nova Scotia. The project evolved from constructing a 3D resource model to a data collection and compilation exercise due to insufficient available data. Key steps included researching reports, georeferencing maps, digitizing drillhole collars and local grids, compiling assay data, creating cross-sections and 3D models in MapInfo, and addressing issues around missing or outdated drillhole data formats. The end result was a compilation of available data but no meaningful 3D model or interpretation could be completed without complete downhole data.
Miles de personas conversando simultaneamente en más de 130 ubicaciones para resolver un desafio compartido. Buscar las soluciones -medidas concretas- que permitan salir antes de la crisis. Promovida por Innobasque, la metologia Worls Cafe -de probada eficacia- aplicada en BBVA. Asi vivimos la experiencia.
The document describes the benefits of joining the Association of Information Technology Professionals (AITP) student organization at Kent State University, which allows students to network with IT professionals, learn leadership skills through organizing activities, and have fun while giving back to the community through food drives and other events. Members can also attend regional and national conferences on technology and certification testing.
This document is an Arizona nonresident personal income tax return form for the 2006 tax year. It requests basic identifying information about the taxpayer such as name, address, social security number. It also requests filing status, exemptions, income and deduction amounts to calculate Arizona adjusted gross income, taxable income, tax amount, payments and credits, and any refund or tax due. Parts A through E of an attached schedule request additional information about dependents, income sources, additions and subtractions to calculate the Arizona percentage of total income.
El documento proporciona información sobre varios niños y adolescentes desaparecidos en Argentina. Se incluyen sus nombres, edades, fechas y lugares de desaparición, así como datos de contacto para aquellos que puedan tener información sobre su paradero.
This document provides information about Greek art motifs and discusses details versus whole pieces of art. It includes examples of art details showing common Greek myths and legends. Students are instructed to select and copy an art detail showing a monster from a piece of Greek art into their sketchbook. The details will then be enlarged and combined to create a frieze decoration for a Halloween party.
Quiksilver Universitário de Surf 2012 - 2ª etapaibrasurf
1. O concurso Garota Roxy da Universidade de São Paulo realizou ativações na praia de Santos para divulgar o evento.
2. Estudantes da faculdade participaram de aulas de ioga e distribuíram material de divulgação do concurso na praia.
3. Marcas como Quiksilver, Roxy e Peugeot patrocinaram as ativações na praia para promover seus produtos.
1) O documento discute a iniciativa "Novas Oportunidades" em Portugal, que tem como objetivo fornecer educação e treinamento para adultos.
2) Os dados mostram um aumento constante no número de adultos inscritos e certificados ao longo dos anos.
3) Vários desafios são discutidos, como melhorar os encaminhamentos para formação adicional e assegurar padrões elevados de qualidade ao mesmo tempo que se atingem as metas quantitativas.
Este documento proporciona instrucciones detalladas sobre cómo navegar por el blog y acceder a su contenido. Explica que el blog contiene varias secciones, y cómo hacer clic en estas secciones o enlaces para ver artículos y fotos. También describe cómo usar atajos de teclado y el ratón para aumentar o disminuir el tamaño del texto y las imágenes, y cómo usar las flechas para navegar entre páginas dentro del blog.
Este documento describe un proyecto para disminuir las necesidades educativas y transitorias en estudiantes del ciclo I en el colegio Venecia. El objetivo general es reducir estas necesidades y los objetivos específicos incluyen realizar investigaciones sobre las diferentes problemáticas, diseñar nuevas metodologías de comportamiento para los niños, y analizar y organizar detalladamente la información sobre las necesidades educativas y transitorias.
XXIII Torneo Internacional De Voley Playavpsuances
Este boletín de inscripción proporciona información sobre el XXIII Torneo Internacional de Voley Playa "Villa de Suances" que se celebrará del 25 al 26 de julio de 2009. Los equipos deben completar el boletín con sus datos y pagar la cuota de inscripción antes del 23 de julio. La inscripción solo se podrá realizar por fax y se confirmará una vez recibido el pago. El número de equipos en la categoría B estará limitado a 140.
Franz Heinrich Louis Corinth Giohanna B. Rodriguez Garciaeskuadrooon
Lovis Corinth nació en Alemania en 1858 y estudió arte en varias academias en Alemania y Francia. Se convirtió en uno de los artistas más importantes del movimiento Sezession en Berlín donde enseñó en una academia de arte para mujeres. Su estilo evolucionó del impresionismo al expresionismo. Realizó numerosos autorretratos y obras de temática mitológica y religiosa hasta su muerte en 1925.
Programación Neurolingüística permite el control o dominio de las inteligencias múltiples, es decir, desarrollo y aplicación del cerebro triuno, pensar, actuar y sentir.
El documento resume el manga, incluyendo su definición como historieta japonesa, su amplio alcance y géneros, sus características como leerse de derecha a izquierda, su origen en la combinación del arte gráfico y la historieta japonesa y occidental, y su importancia en Japón y expansión internacional.
An accidental confluence of old interests and new techniques led a few scientists in the 1950s to realize that human activity might be changing the world’s climate. While the idea of human-caused global warming was first proposed in 1896 by Svante Arrhenius, it was largely ignored for over half a century. By the early 1960s, many scientists had become seriously concerned that warming was not just a natural cycle but could be accelerating and caused by human emissions. This shift in scientific understanding of global warming as a potential threat may be one of the pivotal developments of the century, though it resulted largely from work on unrelated questions.
The document discusses concavity and turning points. It states that concavity is measured by the second derivative and describes how the sign of the second derivative indicates whether a curve is concave up or down. It also explains that turning points occur at stationary points, where the first derivative is equal to zero. The minimum and maximum turning points can be identified by whether the second derivative is less than or greater than zero at that point. An example finds the turning points of a cubic function by taking its first and second derivatives and setting them equal to zero.
This document provides an overview of Gaussian elimination for solving systems of linear equations. It begins with examples of single and multi-variable linear equations. It then introduces the concept of representing a system of equations using an augmented matrix and describes row operations that can be performed on the matrix without changing the solution. The document walks through using Gaussian elimination to solve a sample system of 3 equations in 3 variables. It concludes with more details on the elimination process, including performing a "backward pass" to solve for the variables.
Este documento presenta biografías breves de varias personas notables. Nir Ofir creó el BlogDay en 2005 para que los blogueros puedan conocerse unos a otros. Yoani Sánchez es una periodista cubana conocida por su blog Generación Y donde critica la realidad de Cuba a pesar del acoso del gobierno. Michelle Malkin es una comentarista política conservadora estadounidense cuyas columnas aparecen en varios periódicos.
The document provides an overview of Ryan Leach's project to compile and visualize historic drillhole data from the Dominique Property in Yarmouth County, Nova Scotia. The project evolved from constructing a 3D resource model to a data collection and compilation exercise due to insufficient available data. Key steps included researching reports, georeferencing maps, digitizing drillhole collars and local grids, compiling assay data, creating cross-sections and 3D models in MapInfo, and addressing issues around missing or outdated drillhole data formats. The end result was a compilation of available data but no meaningful 3D model or interpretation could be completed without complete downhole data.
Miles de personas conversando simultaneamente en más de 130 ubicaciones para resolver un desafio compartido. Buscar las soluciones -medidas concretas- que permitan salir antes de la crisis. Promovida por Innobasque, la metologia Worls Cafe -de probada eficacia- aplicada en BBVA. Asi vivimos la experiencia.
The document describes the benefits of joining the Association of Information Technology Professionals (AITP) student organization at Kent State University, which allows students to network with IT professionals, learn leadership skills through organizing activities, and have fun while giving back to the community through food drives and other events. Members can also attend regional and national conferences on technology and certification testing.
This document is an Arizona nonresident personal income tax return form for the 2006 tax year. It requests basic identifying information about the taxpayer such as name, address, social security number. It also requests filing status, exemptions, income and deduction amounts to calculate Arizona adjusted gross income, taxable income, tax amount, payments and credits, and any refund or tax due. Parts A through E of an attached schedule request additional information about dependents, income sources, additions and subtractions to calculate the Arizona percentage of total income.
El documento proporciona información sobre varios niños y adolescentes desaparecidos en Argentina. Se incluyen sus nombres, edades, fechas y lugares de desaparición, así como datos de contacto para aquellos que puedan tener información sobre su paradero.
This document provides information about Greek art motifs and discusses details versus whole pieces of art. It includes examples of art details showing common Greek myths and legends. Students are instructed to select and copy an art detail showing a monster from a piece of Greek art into their sketchbook. The details will then be enlarged and combined to create a frieze decoration for a Halloween party.
Quiksilver Universitário de Surf 2012 - 2ª etapaibrasurf
1. O concurso Garota Roxy da Universidade de São Paulo realizou ativações na praia de Santos para divulgar o evento.
2. Estudantes da faculdade participaram de aulas de ioga e distribuíram material de divulgação do concurso na praia.
3. Marcas como Quiksilver, Roxy e Peugeot patrocinaram as ativações na praia para promover seus produtos.
1) O documento discute a iniciativa "Novas Oportunidades" em Portugal, que tem como objetivo fornecer educação e treinamento para adultos.
2) Os dados mostram um aumento constante no número de adultos inscritos e certificados ao longo dos anos.
3) Vários desafios são discutidos, como melhorar os encaminhamentos para formação adicional e assegurar padrões elevados de qualidade ao mesmo tempo que se atingem as metas quantitativas.
Este documento proporciona instrucciones detalladas sobre cómo navegar por el blog y acceder a su contenido. Explica que el blog contiene varias secciones, y cómo hacer clic en estas secciones o enlaces para ver artículos y fotos. También describe cómo usar atajos de teclado y el ratón para aumentar o disminuir el tamaño del texto y las imágenes, y cómo usar las flechas para navegar entre páginas dentro del blog.
Este documento describe un proyecto para disminuir las necesidades educativas y transitorias en estudiantes del ciclo I en el colegio Venecia. El objetivo general es reducir estas necesidades y los objetivos específicos incluyen realizar investigaciones sobre las diferentes problemáticas, diseñar nuevas metodologías de comportamiento para los niños, y analizar y organizar detalladamente la información sobre las necesidades educativas y transitorias.
XXIII Torneo Internacional De Voley Playavpsuances
Este boletín de inscripción proporciona información sobre el XXIII Torneo Internacional de Voley Playa "Villa de Suances" que se celebrará del 25 al 26 de julio de 2009. Los equipos deben completar el boletín con sus datos y pagar la cuota de inscripción antes del 23 de julio. La inscripción solo se podrá realizar por fax y se confirmará una vez recibido el pago. El número de equipos en la categoría B estará limitado a 140.
Franz Heinrich Louis Corinth Giohanna B. Rodriguez Garciaeskuadrooon
Lovis Corinth nació en Alemania en 1858 y estudió arte en varias academias en Alemania y Francia. Se convirtió en uno de los artistas más importantes del movimiento Sezession en Berlín donde enseñó en una academia de arte para mujeres. Su estilo evolucionó del impresionismo al expresionismo. Realizó numerosos autorretratos y obras de temática mitológica y religiosa hasta su muerte en 1925.
Programación Neurolingüística permite el control o dominio de las inteligencias múltiples, es decir, desarrollo y aplicación del cerebro triuno, pensar, actuar y sentir.
El documento resume el manga, incluyendo su definición como historieta japonesa, su amplio alcance y géneros, sus características como leerse de derecha a izquierda, su origen en la combinación del arte gráfico y la historieta japonesa y occidental, y su importancia en Japón y expansión internacional.
An accidental confluence of old interests and new techniques led a few scientists in the 1950s to realize that human activity might be changing the world’s climate. While the idea of human-caused global warming was first proposed in 1896 by Svante Arrhenius, it was largely ignored for over half a century. By the early 1960s, many scientists had become seriously concerned that warming was not just a natural cycle but could be accelerating and caused by human emissions. This shift in scientific understanding of global warming as a potential threat may be one of the pivotal developments of the century, though it resulted largely from work on unrelated questions.
The document discusses concavity and turning points. It states that concavity is measured by the second derivative and describes how the sign of the second derivative indicates whether a curve is concave up or down. It also explains that turning points occur at stationary points, where the first derivative is equal to zero. The minimum and maximum turning points can be identified by whether the second derivative is less than or greater than zero at that point. An example finds the turning points of a cubic function by taking its first and second derivatives and setting them equal to zero.
This document provides an overview of Gaussian elimination for solving systems of linear equations. It begins with examples of single and multi-variable linear equations. It then introduces the concept of representing a system of equations using an augmented matrix and describes row operations that can be performed on the matrix without changing the solution. The document walks through using Gaussian elimination to solve a sample system of 3 equations in 3 variables. It concludes with more details on the elimination process, including performing a "backward pass" to solve for the variables.
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses properties of the graph of y=x, showing that it is increasing for x≥0. It uses this to prove inequalities relating sums and integrals. Finally, it introduces a proof by mathematical induction to show an inequality relating sums and fractions is true for all integers n≥1.
X2 T08 01 inequalities and graphs (2010)Nigel Simmons
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
1) 0! equals 1 based on the algebraic definition of the factorial function and properties of the gamma function.
2) The variance formula s^2 is derived from the definition of variance and using algebraic manipulations to simplify the expression in terms of sums of the data values.
3) It is shown that the sample mean x can be expressed as the population mean x' plus the sum of the standardized deviations from the mean, divided by n.
4) The equations for the slope b and y-intercept a of the linear regression line are derived by taking expectations of both sides of the linear equation and rearranging terms involving sums of the data values.
The document discusses velocity and acceleration in terms of position x. It provides equations showing that acceleration is equal to the derivative of velocity with respect to time, and the derivative of velocity with respect to x. It also gives examples of using these relationships to find velocity and position as functions of x and time for particles where acceleration is given.
This document discusses function operations and composition of functions. It defines operations that can be performed on functions like addition, subtraction, multiplication, and division. It also discusses finding the difference quotient of a function, which is the slope of the secant line. The document concludes by defining function composition as applying one function to the output of another, and gives examples of evaluating composite functions and determining their domains.
This document contains:
1) An announcement about an assigned problem set due November 28th and office hours.
2) A summary of the regression theorem for finding local maxima, minima, and saddle points of functions with two variables.
3) An example of classifying critical points of a function.
4) A discussion of finding the line of best fit to a set of data points by minimizing the sum of squared errors between the data points and fitted line.
The document discusses matrices and linear algebra concepts such as:
- A matrix is a set of elements organized into rows and columns. Basic matrix operations include addition, subtraction, and multiplication.
- Vectors can be represented as matrices and operations like the dot product and cross product are used to describe relationships between vectors.
- Important matrix properties include inverses, determinants, and homogeneous matrices which allow translations and perspective transforms.
- An orthonormal basis is a set of orthogonal vectors that form a coordinate system where the magnitude of each basis vector is 1.
This presentation is a witty journey to boredom. In an excellent way.
Holding attention is important in storytelling.
Storytelling is important in a presentation.
presentation is important in any project.
No matter how boring the topic is.
Identify the transformations to the graph of a quadratic function
Change a quadratic function from general form to vertex form
Identify the axis and vertex of a parabola
Lesson 21: Curve Sketching II (Section 10 version)Matthew Leingang
The document provides an outline and examples for graphing functions. It includes a checklist for graphing a function, which involves finding where the function is positive, negative, zero or undefined. It then discusses finding the first and second derivatives to determine monotonicity and concavity. Examples are provided to demonstrate this process, including graphing the function f(x) = x + √|x| and f(x) = xe-x^2. Key aspects like asymptotes, points of non-differentiability, and putting the analysis together into a graph are also covered.
This document provides an outline and review for a midterm exam in Math 20. It covers topics like vectors, matrices, vector and matrix algebra, geometry of lines and planes, and determinants. There will be a midterm exam on October 18th from 7-8:30pm in Hall A. Old exams and solutions are available online, and there are review sessions being held on Wednesday.
The document describes the process of integration by partial fractions. It explains that when the degree of the numerator is greater than or equal to the denominator, division is performed. Otherwise, the denominator is factored. For each linear factor, the numerator is written as a sum of terms divided by that factor. For multiple linear factors, the numerator is written as a sum of terms divided by powers of that factor. Examples are provided to demonstrate these steps.
Lesson 21: Curve Sketching II (Section 4 version)Matthew Leingang
The document provides guidance on graphing functions by outlining a checklist process involving 4 steps: 1) finding signs of the function, 2) taking the derivative to determine monotonicity and local extrema, 3) taking the second derivative to determine concavity, and 4) combining the information into a graph. An example function is then graphed in detail to demonstrate the full process.
This document provides information on various mathematical topics including:
1. Graphs of polynomial functions in factorized form such as quadratics, cubics, and quartics.
2. Transformations of functions including translations, reflections, dilations, and their effects on graphs.
3. Exponential, logarithmic, and trigonometric functions and their graphs.
4. Relations, functions, and tests to determine if a relation is a function and if a function is one-to-one or many-to-one.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
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The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
12. Inequalities & Graphs
x2
e.g. i Solve 1 x2
x2 1
x2
x2 x 2
x2 x 2 0
x 2 x 1 0
x 2 or x 1
13. Inequalities & Graphs
x2
e.g. i Solve 1 x2
x2 1
x2
x2 x 2
x2 x 2 0
x 2 x 1 0
x 2 or x 1
x2
1
x2
x 2 or 1 x 2
16. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
17. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x
dy 1
dx 2 x
18. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x
dy 1
dx 2 x
dy
0 for x 0
dx
19. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x at x 0, y 0
dy 1
dx 2 x
dy
0 for x 0
dx
20. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x at x 0, y 0
dy 1 when x 0, y 0
dx 2 x
dy
0 for x 0
dx
21. (ii) (1990)
Consider the graph y x
a) Show that the graph is increasing for all x 0
dy
Curve is increasing when 0
dx
y x at x 0, y 0
dy 1 when x 0, y 0
dx 2 x
dy
0 for x 0 curve is increasing for x 0
dx
22. b) Hence show that;
n
2
1 2 n xdx n n
0
3
23. b) Hence show that;
n
2
1 2 n xdx n n
0
3
24. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
25. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
n
1 2 n xdx
0
26. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
n
1 2 n xdx
0
n
2 x x
3 0
2
n n
3
27. b) Hence show that;
n
2
1 2 n xdx n n
0
3
As x is increasing;
Area outer rectangles Area under curve
n
1 2 n xdx
0
n
2 x x
3 0
2
n n
3
n
2
1 2 n xdx n n
0
3
28. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
29. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
30. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
L.H .S 1
1
31. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
32. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
33. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
34. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
35. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
4k 3
i.e. 1 2 k k
6
36. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
4k 3
i.e. 1 2 k k
6
Prove the result is true for n k 1
37. c) Use mathematical induction to show that;
4n 3
1 2 n n for all integers n 1
6
Test: n = 1
41 3
L.H .S 1 R.H .S 1
6
1
7
6
L.H .S R.H .S
Hence the result is true for n = 1
Assume the result is true for n k where k is a positive integer
4k 3
i.e. 1 2 k k
6
Prove the result is true for n k 1
4k 7
i.e. Prove 1 2 k 1 k 1
6
40. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
41. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
42. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
43. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
44. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
k 14k 12 1 6 k 1
6
45. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
k 14k 12 1 6 k 1
6
k 14k 1 6 k 1
2
6
46. Proof: 1 2 k 1 1 2 k k 1
4k 3
k k 1
6
4k 3 k 6 k 1
2
6
16k 3 24k 2 9k 6 k 1
6
k 116k 2 8k 1 1 6 k 1
6
k 14k 12 1 6 k 1
6
k 14k 1 6 k 1
2
6
4k 1 k 1 6 k 1
6
4k 7 k 1
6
47. Hence the result is true for n = k +1 if it is also true for n =k
48. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
49. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
50. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
51. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
52. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
666700 1 2 10000 666700
53. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
666700 1 2 10000 666700
1 2 10000 666700 to the nearest hundred
54. Hence the result is true for n = k +1 if it is also true for n =k
Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2,
and since the result is true for n = 2 then it is also true for n =2+1 i.e.
n=3, and so on for all positive integral values of n
d) Use b) and c) to estimate;
1 2 10000 to the nearest hundred
2 4n 3
n n 1 2 n n
3 6
2 410000 3
10000 10000 1 2 10000 10000
3 6
666700 1 2 10000 666700
1 2 10000 666700 to the nearest hundred
Exercise 10F