Grade 12 Functions and Inverses
This is a build-up from the foundation laid in earlier grades. The concept of the function introduced in grade 10 forms an important foundation for this topic.
This document discusses bilingualism and some key issues related to being bilingual. It defines bilingualism as using two languages that differ in sounds, vocabulary and syntax. It distinguishes between additive bilingualism, where a second language is acquired without loss to the first language, and subtractive bilingualism, where the new language replaces the first. The document also discusses advantages of bilingualism like increased cognitive flexibility and sensitivity to language pragmatics, as well as potential minor disadvantages like slightly slower language processing. Finally, it covers the concept of code switching, where bilinguals switch between languages when speaking to other bilinguals.
2016, UNIVERSITY OF SELANGOR SPEECH ERROR : SLIPS OF TONGUE IN PSYCHOLINGUISTICNURUL AQILAH MUSARI
This document discusses speech errors known as slips of the tongue. It provides examples and classifications of different types of speech errors:
1) Substitution - when one unit in a sentence is replaced by another, like "the queer old dean" instead of "the dear old queen".
2) Several researchers have studied slips of the tongue and believe they occur when the brain and tongue fail to coordinate during language production due to unconscious processes.
3) Speech errors can provide insights into the cognitive processes involved in language production, such as where a speaker pauses to think. The document provides 10 classifications of speech errors and examples of each.
Approaches and Methods for Language Teachingvblori
This document summarizes 14 language teaching methods: Grammar Translation Method, Direct Method, Natural Approach, Audio-Lingual Method, Total Physical Response, Silent Way, Desuggestopedia, Community Language Learning, Communicative Language Teaching, Participatory Approaches, Content Based, Task Based, Learning Strategy, Cooperative Learning, and Multiple Intelligences. For each method, it provides a brief overview of the key principles and techniques used.
This document discusses the debate between whether innate grammar or emergent grammar through conversation drives language acquisition. It presents the arguments for the innate grammar view (APG) versus the emergent grammar view (EG) that sees grammar as a byproduct of discourse. It also discusses connectionist models of language learning that see the brain as accumulating statistics from a communicative environment to develop language, without innate grammatical structures.
This document discusses neurofunctional theories of language, including brain lateralization and the roles of Broca's and Wernicke's areas in language processing. It describes different types of aphasia that can result from damage to these language areas. Theories around brain equipotentiality vs invariance in language acquisition are presented. Evidence from sign language, split-brain patients, and bilingualism research is discussed.
The document discusses several theories of first language acquisition:
1) Behavioral approaches focus on observable responses and reinforcement/punishment, while the nativist approach sees language acquisition as innate with a language acquisition device.
2) The functional approach views language as a tool for interacting with the world and communicating socially.
3) Issues in acquisition include the interplay between nature and nurture, universals versus variability across languages, and the influence of language on thought and cognition.
Cognitive Approaches to Second Language AcquisitionOla Sayed Ahmed
This document provides an overview of cognitive approaches to second language acquisition (SLA). It discusses two main groups of cognitive theorists: processing approaches and emergentist/constructionist approaches. Processing approaches investigate how learners process linguistic information and develop this ability over time, focusing on computational dimensions of language learning. Emergentist approaches see language development as driven by associative learning from communicative needs and patterns in language input. Specific cognitive models discussed include McLaughlin's information processing model, Anderson's ACT model, and Pienemann's processability theory. The document also covers Slobin's perceptual saliency approach and its operating principles for first and second language acquisition.
The document discusses language production and summarizes key points in 3 sentences:
Language production involves conceptualizing thoughts, formulating linguistic plans by selecting words and structures, and implementing plans through articulation. Evidence from eye movements, slips of the tongue, and self-repairs suggests language production involves parallel planning at multiple linguistic levels from meaning to sounds. Models of speech production propose different views on whether planning proceeds incrementally from smaller units or begins with larger syntactic structures.
This document discusses bilingualism and some key issues related to being bilingual. It defines bilingualism as using two languages that differ in sounds, vocabulary and syntax. It distinguishes between additive bilingualism, where a second language is acquired without loss to the first language, and subtractive bilingualism, where the new language replaces the first. The document also discusses advantages of bilingualism like increased cognitive flexibility and sensitivity to language pragmatics, as well as potential minor disadvantages like slightly slower language processing. Finally, it covers the concept of code switching, where bilinguals switch between languages when speaking to other bilinguals.
2016, UNIVERSITY OF SELANGOR SPEECH ERROR : SLIPS OF TONGUE IN PSYCHOLINGUISTICNURUL AQILAH MUSARI
This document discusses speech errors known as slips of the tongue. It provides examples and classifications of different types of speech errors:
1) Substitution - when one unit in a sentence is replaced by another, like "the queer old dean" instead of "the dear old queen".
2) Several researchers have studied slips of the tongue and believe they occur when the brain and tongue fail to coordinate during language production due to unconscious processes.
3) Speech errors can provide insights into the cognitive processes involved in language production, such as where a speaker pauses to think. The document provides 10 classifications of speech errors and examples of each.
Approaches and Methods for Language Teachingvblori
This document summarizes 14 language teaching methods: Grammar Translation Method, Direct Method, Natural Approach, Audio-Lingual Method, Total Physical Response, Silent Way, Desuggestopedia, Community Language Learning, Communicative Language Teaching, Participatory Approaches, Content Based, Task Based, Learning Strategy, Cooperative Learning, and Multiple Intelligences. For each method, it provides a brief overview of the key principles and techniques used.
This document discusses the debate between whether innate grammar or emergent grammar through conversation drives language acquisition. It presents the arguments for the innate grammar view (APG) versus the emergent grammar view (EG) that sees grammar as a byproduct of discourse. It also discusses connectionist models of language learning that see the brain as accumulating statistics from a communicative environment to develop language, without innate grammatical structures.
This document discusses neurofunctional theories of language, including brain lateralization and the roles of Broca's and Wernicke's areas in language processing. It describes different types of aphasia that can result from damage to these language areas. Theories around brain equipotentiality vs invariance in language acquisition are presented. Evidence from sign language, split-brain patients, and bilingualism research is discussed.
The document discusses several theories of first language acquisition:
1) Behavioral approaches focus on observable responses and reinforcement/punishment, while the nativist approach sees language acquisition as innate with a language acquisition device.
2) The functional approach views language as a tool for interacting with the world and communicating socially.
3) Issues in acquisition include the interplay between nature and nurture, universals versus variability across languages, and the influence of language on thought and cognition.
Cognitive Approaches to Second Language AcquisitionOla Sayed Ahmed
This document provides an overview of cognitive approaches to second language acquisition (SLA). It discusses two main groups of cognitive theorists: processing approaches and emergentist/constructionist approaches. Processing approaches investigate how learners process linguistic information and develop this ability over time, focusing on computational dimensions of language learning. Emergentist approaches see language development as driven by associative learning from communicative needs and patterns in language input. Specific cognitive models discussed include McLaughlin's information processing model, Anderson's ACT model, and Pienemann's processability theory. The document also covers Slobin's perceptual saliency approach and its operating principles for first and second language acquisition.
The document discusses language production and summarizes key points in 3 sentences:
Language production involves conceptualizing thoughts, formulating linguistic plans by selecting words and structures, and implementing plans through articulation. Evidence from eye movements, slips of the tongue, and self-repairs suggests language production involves parallel planning at multiple linguistic levels from meaning to sounds. Models of speech production propose different views on whether planning proceeds incrementally from smaller units or begins with larger syntactic structures.
The document discusses function composition, which is combining two or more functions into a new function by plugging the output of one function into the input of another. Specifically, it provides an example of composing the functions f(x) = 3x + 2 and g(x) = x - 1, finding the value of f(g(3)) by first applying g(3) to get 2, then applying f(2) to get the final answer of 8. It also explains that the second function listed is always plugged into the x variable of the first function to perform the composition.
- word2vec is a neural network model that learns word embeddings from large amounts of text by predicting words from their context. It has two implementations: Continuous Bag of Words (CBOW) and Skip-Gram.
- CBOW predicts a target word based on surrounding context words, while Skip-Gram predicts surrounding context words given the target word.
- Both models are trained using backpropagation and stochastic gradient descent to maximize the log-likelihood of predicting correct word-context pairs in a corpus.
https://arxiv.org/abs/2006.07865
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and e-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" e-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and e-commutativity are studied.
The document discusses differentiation rules for products and quotients of functions. It begins by introducing the product rule, which states that the derivative of a product of two functions f and g is equal to f times the derivative of g plus g times the derivative of f. Next, it derives the quotient rule through a similar process, concluding that the derivative of a quotient of two functions u and v is equal to the denominator v times the derivative of the numerator u minus the numerator u times the derivative of the denominator v, all over the square of the denominator v squared. Several examples are provided to demonstrate applying these rules to find derivatives.
This document discusses transformations of functions, including rigid and non-rigid transformations. Rigid transformations include vertical and horizontal shifts and reflections, which change the position of the graph but not its shape. Non-rigid transformations include vertical and horizontal stretches and shrinks, which alter the shape of the graph. Examples of each type of transformation are presented and explained using quadratic functions.
This document discusses transformations of functions, including rigid and non-rigid transformations. Rigid transformations include vertical and horizontal shifts as well as reflections, which change the position of the graph but not its shape. Non-rigid transformations include vertical and horizontal stretches and shrinks, which alter the shape of the graph. Examples of each type of transformation are presented and discussed.
The document discusses continuity in calculus, including the definition of continuity, types of continuity and discontinuity, and continuity of composite functions and intervals. It defines a continuous function as one where small changes in the input variable result in only small changes in the output variable. There are two main types of discontinuity: jump discontinuities, where the left and right limits exist but are not equal, and infinite discontinuities where the limits do not exist. A function is continuous over an interval if its graph can be drawn without lifting the pencil from that interval.
1) Index notation is a useful tool for performing vector algebra using subscript notation. Vectors can be expressed as sums of their coordinate components multiplied by basis vectors.
2) Key concepts include: the Kronecker delta symbol δij, Einstein notation where repeated indices imply summation, and the Levi-Civita symbol ijk which determines the sign of cross products.
3) Using these tools, vector operations like the dot product and cross product can be written concisely in index notation. For example, the dot product of two vectors ~a and ~b is aibi and their cross product is ijkaibjêk.
This document discusses inverse functions. It begins by explaining that an inverse function reverses the mapping of a regular function by mapping from the output set back to the input set. Only one-to-one functions, where each output is paired with exactly one input, will have inverse functions. The document provides examples of one-to-one and many-to-one functions through graphs and equations. It then outlines the steps to find the inverse of a function by replacing the function with y, swapping x and y, and solving for y. An example inverse function is worked through. The document emphasizes that the inverse undoes the original function when composed together.
This document discusses functions and their graphs. It defines relations, functions, and how to write and graph relations and functions. It discusses domain and range, and how to identify whether a relation is a function using the vertical line test. It provides examples of how to graph linear and non-linear equations, evaluate functions, and find the domain and range of functions.
Reduced Order Observer (DGO) based State Variable Design of Two Loop Lateral ...IJERD Editor
International Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
Reduced Order Observer (DGO) based State Variable Design of Two Loop Lateral ...IJERD Editor
International Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
The document defines key concepts relating to functions and relations:
- A relation is a set of ordered pairs where the domain is the set of all x-values and the range is the set of all y-values.
- A function is a special type of relation where each x-value is assigned to exactly one y-value.
- Function notation uses f(x) to represent the output of a function f when the input is x.
- The domain of a function is the set of all valid input values that do not result in undefined outputs like division by zero or square roots of negative numbers.
This document discusses transformations of functions. It begins by introducing common parent functions like linear, quadratic, absolute value, cubic and square root functions using xy-tables and graphs. It then demonstrates how adding or subtracting values like 1, 5, or 2 to a quadratic function f(x)=x^2 results in a rigid transformation that shifts the graph up, down, left or right but maintains the same basic shape. The document defines rigid transformations as those that only change the position of the graph and lists the three types as vertical shifts, horizontal shifts, and reflections, providing formulas for shifting graphs vertically and horizontally.
The document discusses derivatives and their applications. It begins by introducing derivatives and defining them as the rate of change of a function near an input value. It then discusses rules for finding derivatives such as the constant multiple rule, sum and difference rules, product rule, and quotient rule. Examples are given to illustrate applying these rules. The document also covers composite functions, inverse functions, second derivatives, and applications of derivatives in physics for problems involving velocity and acceleration.
The document provides an outline for a lesson on functions and relations. It includes:
- A review of functions as machines, tables of values, graphs, and the vertical line test.
- How functions can represent real-life situations, including piecewise functions.
- An example of using a piecewise function to model the temperature of water as heat is added.
- The lesson aims to represent real-life situations using functions and solve problems involving functions.
A function is a relation where each input is paired with exactly one output. Functions are commonly represented using function notation with an independent variable x and dependent variable y, written as f(x). Functions can be represented verbally, numerically in a table, visually in a graph, or algebraically with an explicit formula. The domain is the set of inputs, while the range is the set of outputs. Functions can be one-to-one, onto, many-to-one, or into. Operations like addition, subtraction, multiplication, and division can be performed on functions if they have overlapping domains. The composition of two functions is written as f(g(x)). Common functions include linear, square, cubic, and absolute
The document discusses functional programming in Swift. It defines functional programming as avoiding mutable data and state. This means that in functional programming, variables are immutable and do not change value once assigned, and functions have no side effects or dependence on external state. The advantages of this approach include cleaner, more modular code with no hidden state, referential transparency allowing parallelization and memoization, and easier debugging. Functional concepts like immutable objects, higher order functions, lazy evaluation, and recursion are demonstrated in Swift examples.
Bab 4 dan bab 5 algebra and trigonometryCiciPajarakan
This document discusses exponential functions and their properties. Exponential functions have the form f(x) = ax, where a is the base. They model phenomena like population growth, compound interest, and radioactive decay. Exponential functions either increase or decrease rapidly depending on whether the base a is greater than or less than 1. Their graphs have a characteristic shape, with the x-axis as a horizontal asymptote. Logarithmic functions are the inverses of exponential functions and can be used to solve equations involving exponential models.
Gr 8/9 Whole Numbers: RATIOS and RATES.pptxVukile Xhego
1. The document discusses ratios and rates, with the objectives being able to compare quantities, share ratios where a whole is given, and compare different quantities as rates.
2. It provides examples of simplifying ratios, expressing distances as ratios, finding missing quantities in ratios, dividing numbers in given ratios, and expressing decimals as ratios of whole numbers.
3. Terminology defined includes ratio, highest common factor, and examples provided include ratios of apples, distances, running distances in a relay race, and expressing decimal ratios as whole number ratios.
The document discusses exam preparation for Grade 12 National Senior Certificate (NSC) students. It covers assessment in Grade 12, including the weighting of topics and cognitive levels per exam paper. The document also elaborates on exam content/topics for various subjects, such as providing acceptable reasons for proofs in Euclidean geometry. General guidelines are provided for marking the Grade 12 NSC exams.
The document discusses function composition, which is combining two or more functions into a new function by plugging the output of one function into the input of another. Specifically, it provides an example of composing the functions f(x) = 3x + 2 and g(x) = x - 1, finding the value of f(g(3)) by first applying g(3) to get 2, then applying f(2) to get the final answer of 8. It also explains that the second function listed is always plugged into the x variable of the first function to perform the composition.
- word2vec is a neural network model that learns word embeddings from large amounts of text by predicting words from their context. It has two implementations: Continuous Bag of Words (CBOW) and Skip-Gram.
- CBOW predicts a target word based on surrounding context words, while Skip-Gram predicts surrounding context words given the target word.
- Both models are trained using backpropagation and stochastic gradient descent to maximize the log-likelihood of predicting correct word-context pairs in a corpus.
https://arxiv.org/abs/2006.07865
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and e-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" e-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and e-commutativity are studied.
The document discusses differentiation rules for products and quotients of functions. It begins by introducing the product rule, which states that the derivative of a product of two functions f and g is equal to f times the derivative of g plus g times the derivative of f. Next, it derives the quotient rule through a similar process, concluding that the derivative of a quotient of two functions u and v is equal to the denominator v times the derivative of the numerator u minus the numerator u times the derivative of the denominator v, all over the square of the denominator v squared. Several examples are provided to demonstrate applying these rules to find derivatives.
This document discusses transformations of functions, including rigid and non-rigid transformations. Rigid transformations include vertical and horizontal shifts and reflections, which change the position of the graph but not its shape. Non-rigid transformations include vertical and horizontal stretches and shrinks, which alter the shape of the graph. Examples of each type of transformation are presented and explained using quadratic functions.
This document discusses transformations of functions, including rigid and non-rigid transformations. Rigid transformations include vertical and horizontal shifts as well as reflections, which change the position of the graph but not its shape. Non-rigid transformations include vertical and horizontal stretches and shrinks, which alter the shape of the graph. Examples of each type of transformation are presented and discussed.
The document discusses continuity in calculus, including the definition of continuity, types of continuity and discontinuity, and continuity of composite functions and intervals. It defines a continuous function as one where small changes in the input variable result in only small changes in the output variable. There are two main types of discontinuity: jump discontinuities, where the left and right limits exist but are not equal, and infinite discontinuities where the limits do not exist. A function is continuous over an interval if its graph can be drawn without lifting the pencil from that interval.
1) Index notation is a useful tool for performing vector algebra using subscript notation. Vectors can be expressed as sums of their coordinate components multiplied by basis vectors.
2) Key concepts include: the Kronecker delta symbol δij, Einstein notation where repeated indices imply summation, and the Levi-Civita symbol ijk which determines the sign of cross products.
3) Using these tools, vector operations like the dot product and cross product can be written concisely in index notation. For example, the dot product of two vectors ~a and ~b is aibi and their cross product is ijkaibjêk.
This document discusses inverse functions. It begins by explaining that an inverse function reverses the mapping of a regular function by mapping from the output set back to the input set. Only one-to-one functions, where each output is paired with exactly one input, will have inverse functions. The document provides examples of one-to-one and many-to-one functions through graphs and equations. It then outlines the steps to find the inverse of a function by replacing the function with y, swapping x and y, and solving for y. An example inverse function is worked through. The document emphasizes that the inverse undoes the original function when composed together.
This document discusses functions and their graphs. It defines relations, functions, and how to write and graph relations and functions. It discusses domain and range, and how to identify whether a relation is a function using the vertical line test. It provides examples of how to graph linear and non-linear equations, evaluate functions, and find the domain and range of functions.
Reduced Order Observer (DGO) based State Variable Design of Two Loop Lateral ...IJERD Editor
International Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
Reduced Order Observer (DGO) based State Variable Design of Two Loop Lateral ...IJERD Editor
International Journal of Engineering Research and Development is an international premier peer reviewed open access engineering and technology journal promoting the discovery, innovation, advancement and dissemination of basic and transitional knowledge in engineering, technology and related disciplines.
The document defines key concepts relating to functions and relations:
- A relation is a set of ordered pairs where the domain is the set of all x-values and the range is the set of all y-values.
- A function is a special type of relation where each x-value is assigned to exactly one y-value.
- Function notation uses f(x) to represent the output of a function f when the input is x.
- The domain of a function is the set of all valid input values that do not result in undefined outputs like division by zero or square roots of negative numbers.
This document discusses transformations of functions. It begins by introducing common parent functions like linear, quadratic, absolute value, cubic and square root functions using xy-tables and graphs. It then demonstrates how adding or subtracting values like 1, 5, or 2 to a quadratic function f(x)=x^2 results in a rigid transformation that shifts the graph up, down, left or right but maintains the same basic shape. The document defines rigid transformations as those that only change the position of the graph and lists the three types as vertical shifts, horizontal shifts, and reflections, providing formulas for shifting graphs vertically and horizontally.
The document discusses derivatives and their applications. It begins by introducing derivatives and defining them as the rate of change of a function near an input value. It then discusses rules for finding derivatives such as the constant multiple rule, sum and difference rules, product rule, and quotient rule. Examples are given to illustrate applying these rules. The document also covers composite functions, inverse functions, second derivatives, and applications of derivatives in physics for problems involving velocity and acceleration.
The document provides an outline for a lesson on functions and relations. It includes:
- A review of functions as machines, tables of values, graphs, and the vertical line test.
- How functions can represent real-life situations, including piecewise functions.
- An example of using a piecewise function to model the temperature of water as heat is added.
- The lesson aims to represent real-life situations using functions and solve problems involving functions.
A function is a relation where each input is paired with exactly one output. Functions are commonly represented using function notation with an independent variable x and dependent variable y, written as f(x). Functions can be represented verbally, numerically in a table, visually in a graph, or algebraically with an explicit formula. The domain is the set of inputs, while the range is the set of outputs. Functions can be one-to-one, onto, many-to-one, or into. Operations like addition, subtraction, multiplication, and division can be performed on functions if they have overlapping domains. The composition of two functions is written as f(g(x)). Common functions include linear, square, cubic, and absolute
The document discusses functional programming in Swift. It defines functional programming as avoiding mutable data and state. This means that in functional programming, variables are immutable and do not change value once assigned, and functions have no side effects or dependence on external state. The advantages of this approach include cleaner, more modular code with no hidden state, referential transparency allowing parallelization and memoization, and easier debugging. Functional concepts like immutable objects, higher order functions, lazy evaluation, and recursion are demonstrated in Swift examples.
Bab 4 dan bab 5 algebra and trigonometryCiciPajarakan
This document discusses exponential functions and their properties. Exponential functions have the form f(x) = ax, where a is the base. They model phenomena like population growth, compound interest, and radioactive decay. Exponential functions either increase or decrease rapidly depending on whether the base a is greater than or less than 1. Their graphs have a characteristic shape, with the x-axis as a horizontal asymptote. Logarithmic functions are the inverses of exponential functions and can be used to solve equations involving exponential models.
Gr 8/9 Whole Numbers: RATIOS and RATES.pptxVukile Xhego
1. The document discusses ratios and rates, with the objectives being able to compare quantities, share ratios where a whole is given, and compare different quantities as rates.
2. It provides examples of simplifying ratios, expressing distances as ratios, finding missing quantities in ratios, dividing numbers in given ratios, and expressing decimals as ratios of whole numbers.
3. Terminology defined includes ratio, highest common factor, and examples provided include ratios of apples, distances, running distances in a relay race, and expressing decimal ratios as whole number ratios.
The document discusses exam preparation for Grade 12 National Senior Certificate (NSC) students. It covers assessment in Grade 12, including the weighting of topics and cognitive levels per exam paper. The document also elaborates on exam content/topics for various subjects, such as providing acceptable reasons for proofs in Euclidean geometry. General guidelines are provided for marking the Grade 12 NSC exams.
The document provides an overview of key concepts in probability, including definitions of terms like sample space, event, and probability of an event. It also covers rules for calculating probabilities, such as the addition rule, complementary rule, and product rule for independent and dependent events. Examples are given to demonstrate calculating probabilities using these rules for events like coin tosses, card draws, and dice rolls.
This document introduces the fundamental counting principle for determining the number of possible outcomes when performing multiple independent tasks. It states that if there are m ways to perform one task and n ways to perform another, unrelated task, the total number of ways to perform both tasks is m × n. It provides examples of applying this principle to problems involving arrangements of objects and letters. The document also covers factorial notation and examples involving probability.
Revision of distance, midpoint and gradient between two points.
Inclination of a line and between two lines.
Equation of a straight line.
Application in Triangles and Quadrilaterals
This document discusses linear and quadratic number patterns. It begins by reviewing linear patterns with terms of the form Tn = bn + c. It then introduces quadratic patterns, which have terms of the form Tn = an^2 + bn + c. An example quadratic pattern of Tn = n^2 + 1 is generated to show the constant second difference of 2a. It explains that while linear patterns have a constant first difference, quadratic patterns have a constant second difference. The document then derives the expressions for determining the nth term of any given quadratic pattern from the coefficients a, b, and c.
The document discusses arithmetic and geometric sequences and series. It defines arithmetic and geometric sequences, and provides the formulas for calculating the nth term of each type of sequence. Specifically, it states that the formula for the nth term of an arithmetic sequence is Tn = a + (n-1)d, where a is the first term and d is the common difference. The formula for the nth term of a geometric sequence is Tn = arn-1, where a is the first term and r is the common ratio. The document also provides examples of using these formulas to find terms, common differences or ratios, and to write the formulas for given sequences.
The document discusses various topics related to interest rates and compound interest calculations. It begins by reviewing Grade 11 work including simple and compound interest, nominal and effective interest rates, and timelines. It then outlines topics to be covered in Grade 12, such as calculating investment periods using logarithms, future value annuities, present value annuities, and choosing better investment options. Various examples of compound interest, simple interest, and annuity calculations are provided.
The document discusses simple and compound interest. It defines both types of interest and provides formulas to calculate future value under each. An example is shown where a learner named Steve invests R300 at 10% interest annually under simple and compound interest over 3 years. Compound interest provides a higher return due to interest earning interest each period. The document encourages choosing investments that use compound versus simple interest.
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.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
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.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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.
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.
6. REVISION OF THE CONCEPT OF A FUNCTION
A function is a rule by means of which each element of the domain is
associated with only one element of the range.
For a function, two or more elements of the domain may be associated
with the same element of the range as well.
However, a relation is not a function if one element of the domain is
associated with more than one element of the range.
VUKILE XHEGO (GDE)
7. REVISION OF THE CONCEPT OF A FUNCTION
Consider the following relations.
VUKILE XHEGO (GDE)
10. The Vertical and Horizontal Line Tests
We can use a ruler to perform the “vertical line test” on a graph to see
whether it is a function or not.
Hold a clear plastic ruler parallel to the y-axis, i.e. vertical.
Move it from left to right over the axes.
If the ruler only ever cuts the curve at one place as the ruler moves
from left to right across the graph, then the graph is a function.
If the ruler at any stage cuts the graph at more than one place, then the
graph is not a function. This is because the same x-value will be
associated with more than one y-value.
VUKILE XHEGO (GDE)
11. The Vertical and Horizontal Line Tests
Once a graph passes the “vertical line test”, the “horizontal line test” is
used to determine the type of function.
We have two types of functions:
1. One-to-one function
2. Many-to-one function
VUKILE XHEGO (GDE)
12. The Vertical and Horizontal Line Tests
Procedure of “horizontal line test”:
Place the ruler horizontally so that it is parallel to the x-axis
Move the ruler up and down
If the ruler only cuts the curve in one place as the ruler moves up and
down the graph, then the graph is a one-to-one function.
If the ruler at any stage cuts the graph in more than one place, then the
graph is a many-to-one function.
VUKILE XHEGO (GDE)
40. THE INVERSE OF THE EXPONENTIAL FUNCTION
Introduction
VUKILE XHEGO (GDE)
41. THE INVERSE OF THE EXPONENTIAL FUNCTION
Finding the equation of the Inverse
Step 1: let f(x)=y
𝑦 = 2𝑥
Step 2: interchange (swap) x and y
𝑥 = 2𝑦
We have a problem…
None of the methods learned so far will help make y the subject
VUKILE XHEGO (GDE)
42. THE INVERSE OF THE
EXPONENTIAL
FUNCTION
A Scottish mathematician named John
Napier (1550-1617) devised a clever way
of making y the subject of the formula.
He introduced a notation referred to as a
logarithm.
We will now discuss the concept of a
logarithm and then later on develop the
theory of logarithms in more detail.
VUKILE XHEGO (GDE)
43. Introduction to Logarithms
If a number is written in exponential form, then the exponent is called the
logarithm of the number.
For example, the number 8 can be written in exponential form as 8 = 23
.
Clearly, the exponent in this example is 3 and the base is 2.
We can then say that the logarithm of 8 to base 2 is 3.
This can be written as log2 8 = 3. The base 2 is written as a sub-script
between the “log” and the number 8.
VUKILE XHEGO (GDE)