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اگر آپ تعلیمی نیوز، رجسٹریشن، داخلہ، ڈیٹ شیٹ، رزلٹ، اسائنمنٹ،جابز اور باقی تمام اپ ڈیٹس اپنے موبائل پر فری حاصل کرنا چاہتے ہیں ۔تو نیچے دیے گئے واٹس ایپ نمبرکو اپنے موبائل میں سیو کرکے اپنا نام لکھ کر واٹس ایپ کر دیں۔ سٹیٹس روزانہ لازمی چیک کریں۔
نوٹ : اس کے علاوہ تمام یونیورسٹیز کے آن لائن داخلے بھجوانے اور جابز کے لیے آن لائن اپلائی کروانے کے لیے رابطہ کریں۔
This document provides an overview of topics covered in a differential calculus course, including:
1. Limits and differential calculus concepts such as derivatives
2. Special functions and numbers used in calculus
3. A brief history of calculus and its founders Newton and Leibniz
4. Explanations and examples of key calculus concepts such as variables, constants, functions, and limits
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
The document discusses various mathematical concepts related to functions and graphs including:
1) Transformations of graphs such as translations, reflections, and rotations. It also discusses parent functions and their derivatives.
2) Examples of graphing functions after applying transformations to translate, scale, or reflect the original graphs. Equations are provided for the transformed graphs.
3) Theorems related to how statistics of data change after translations or scale changes. For example, the mean, median and mode change proportionally but variance, standard deviation, and range change in specific ways.
4) Concepts involving inverse functions, including using the horizontal line test to determine if an inverse is a function and notations for inverse functions
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
Here are the key steps to solve this problem algebraically:
Let x = number of units of product X
Let y = number of units of product Y
Equation for process A: 3x + y ≤ 1750
Equation for process B: 2x + 4y ≤ 4000
Solve the two equations simultaneously using elimination:
3x + y = 1750
2x + 4y = 4000
Eliminate y by subtracting the equations:
x = 1250
Substitute x = 1250 into one of the original equations to find y:
3(1250) + y = 1750
y = 500
Therefore, the maximum number of units of X is 1250 and
This document provides definitions and concepts related to mechanics, forces, and statics. It introduces coordinate systems, units of measurement, and numerical accuracy. Newton's laws of motion are defined. Vectors are described including operations like addition, subtraction, and dot and cross products. Forces are classified as concentrated or distributed. Statics deals with forces acting on bodies at rest.
Here are the key steps to solve this problem algebraically:
Let x = number of units of product X
Let y = number of units of product Y
Write equations relating the process hours to the number of units:
3x + 2x = Hours used in A
1y + 4y = Hours used in B
Solve the simultaneous equations to find the maximum number of each product that can be made.
AIOU Code 803 Mathematics for Economists Semester Spring 2022 Assignment 2.pptxZawarali786
Skilling Foundation
Download Free
Past Papers
Guess Papers
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Skilling.pk
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اگر آپ تعلیمی نیوز، رجسٹریشن، داخلہ، ڈیٹ شیٹ، رزلٹ، اسائنمنٹ،جابز اور باقی تمام اپ ڈیٹس اپنے موبائل پر فری حاصل کرنا چاہتے ہیں ۔تو نیچے دیے گئے واٹس ایپ نمبرکو اپنے موبائل میں سیو کرکے اپنا نام لکھ کر واٹس ایپ کر دیں۔ سٹیٹس روزانہ لازمی چیک کریں۔
نوٹ : اس کے علاوہ تمام یونیورسٹیز کے آن لائن داخلے بھجوانے اور جابز کے لیے آن لائن اپلائی کروانے کے لیے رابطہ کریں۔
This document provides an overview of topics covered in a differential calculus course, including:
1. Limits and differential calculus concepts such as derivatives
2. Special functions and numbers used in calculus
3. A brief history of calculus and its founders Newton and Leibniz
4. Explanations and examples of key calculus concepts such as variables, constants, functions, and limits
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
The document discusses various mathematical concepts related to functions and graphs including:
1) Transformations of graphs such as translations, reflections, and rotations. It also discusses parent functions and their derivatives.
2) Examples of graphing functions after applying transformations to translate, scale, or reflect the original graphs. Equations are provided for the transformed graphs.
3) Theorems related to how statistics of data change after translations or scale changes. For example, the mean, median and mode change proportionally but variance, standard deviation, and range change in specific ways.
4) Concepts involving inverse functions, including using the horizontal line test to determine if an inverse is a function and notations for inverse functions
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
Here are the key steps to solve this problem algebraically:
Let x = number of units of product X
Let y = number of units of product Y
Equation for process A: 3x + y ≤ 1750
Equation for process B: 2x + 4y ≤ 4000
Solve the two equations simultaneously using elimination:
3x + y = 1750
2x + 4y = 4000
Eliminate y by subtracting the equations:
x = 1250
Substitute x = 1250 into one of the original equations to find y:
3(1250) + y = 1750
y = 500
Therefore, the maximum number of units of X is 1250 and
This document provides definitions and concepts related to mechanics, forces, and statics. It introduces coordinate systems, units of measurement, and numerical accuracy. Newton's laws of motion are defined. Vectors are described including operations like addition, subtraction, and dot and cross products. Forces are classified as concentrated or distributed. Statics deals with forces acting on bodies at rest.
Here are the key steps to solve this problem algebraically:
Let x = number of units of product X
Let y = number of units of product Y
Write equations relating the process hours to the number of units:
3x + 2x = Hours used in A
1y + 4y = Hours used in B
Solve the simultaneous equations to find the maximum number of each product that can be made.
AIOU Code 803 Mathematics for Economists Semester Spring 2022 Assignment 2.pptxZawarali786
Skilling Foundation
Download Free
Past Papers
Guess Papers
Solved Assignments
Solved Thesis
Solved Lesson Plans
PDF Books
Skilling.pk
Other Websites
Diya.pk
Stamflay.com
Please Subscribe Our YouTube Channel
Skilling Foundation:https://bit.ly/3kEJI0q
WordPress Tutorials:https://bit.ly/3rqcgfE
Stamflay:https://bit.ly/2AoClW8
Please Contact at:
0314-4646739
0332-4646739
0336-4646739
اگر آپ تعلیمی نیوز، رجسٹریشن، داخلہ، ڈیٹ شیٹ، رزلٹ، اسائنمنٹ،جابز اور باقی تمام اپ ڈیٹس اپنے موبائل پر فری حاصل کرنا چاہتے ہیں ۔تو نیچے دیے گئے واٹس ایپ نمبرکو اپنے موبائل میں سیو کرکے اپنا نام لکھ کر واٹس ایپ کر دیں۔ سٹیٹس روزانہ لازمی چیک کریں۔
نوٹ : اس کے علاوہ تمام یونیورسٹیز کے آن لائن داخلے بھجوانے اور جابز کے لیے آن لائن اپلائی کروانے کے لیے رابطہ کریں۔
R can be used to analyze data and perform statistical analysis. Functions like help(), ? and help.start() provide information about other functions. Objects created in R sessions are stored by name and can be removed with rm(). Vectors like x=c(1,2,3,4,5) can be created and their length checked with length(x). Subsets of vectors can be selected using logical or integer indexes inside square brackets. Matrices are multi-dimensional generalizations of vectors that can be manipulated using operators like * and %. Data can be read into R from external files using functions like read.table() and read.delim(). Common statistical distributions like normal, uniform and exponential are available as functions in R for
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The rate of change of a function is found by taking the difference quotient, which is similar to the slope formula. The domain of a function is the set of all possible x-values, while the range is the resulting y-values. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.
Linear algebra concepts like vectors, matrices, and linear transformations are important for recommendation systems. Vectors represent items or users, matrices represent item-user preference data. Linear algebra allows analyzing this data to identify patterns and recommend new items. Key techniques include eigendecomposition to reduce dimensionality and identify important relationships in the data, and singular value decomposition to factor matrices for recommendations. These linear algebra concepts are essential mathematical tools for building personalized recommendation models.
This document provides information about Baraka Loibanguti, who is the author of an advanced mathematics book. It includes his contact information and some notes about copyright and permissions. The document then begins discussing functions, including definitions of domain, range, and different types of functions like linear, quadratic, cubic, and polynomial functions. It provides examples of how to graph different types of functions by creating tables of values or using intercepts.
1) The document discusses graphing and properties of exponential and logarithmic functions, including: graphing exponential functions by substituting values of the variable into the equation, graphing logarithmic functions using the change of base formula, and properties like the product, quotient, and power properties of logarithms.
2) Examples are provided of solving exponential and logarithmic equations using properties like changing bases to the same value, multiplying or dividing arguments using the product and quotient properties, and applying exponents using the power property.
3) Steps shown include using properties to isolate the variable, set arguments or exponents equal to each other, and solve the resulting equation.
R can be used to analyze data and perform statistical analysis. Key functions include help() and ? to get information on functions, and objects() to view stored objects. Vectors can be created with c() and manipulated using arithmetic operators. Matrices are two-dimensional arrays that can be operated on using *, /, and t(). Larger datasets are typically read from external files using read.table() or read.delim(). Common distributions can be explored using functions like dnorm(), pnorm(), and rnorm(). Statistical analysis includes commands like cov() and cor() to measure covariance and correlation between variables.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
1. The document discusses the concept of derivatives and how to calculate them. It defines key terms like increment, average rate of change, and instantaneous rate of change.
2. Methods are provided for calculating the derivative of various types of functions, including polynomials, rational functions, and functions with roots. Examples are worked through step-by-step.
3. The relationship between derivatives and tangent lines is explained, with derivatives defined as the instantaneous rate of change or slope of the curve at a point.
1. Physics deals with matter and energy through defining and characterizing interactions between the two.
2. Mechanics studies motion and forces, including quantities like speed, velocity, acceleration, force, momentum and Newton's Laws of Motion.
3. Solving physics problems involves identifying known and unknown quantities, selecting the appropriate equation, and using the correct units.
The document describes five main families of functions - linear, power, root, reciprocal, and absolute value functions. It provides the name, equation, domain and range for each type of function. It also discusses concepts like piecewise functions, average rate of change, transformations, combinations of functions, and variations.
The document discusses differentiation rules for various functions. It begins by discussing the derivatives of polynomials and exponential functions. The power rule is introduced, which states the derivative of x^n is nx^{n-1}. It then covers the derivatives of exponential functions f(x)=ax, proving the formula f'(x)=af(x). The product rule and quotient rule are also introduced. Finally, it discusses the derivatives of trigonometric functions, proving that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
- The document is a lesson on identifying and graphing linear functions. It provides examples of determining if a graph or set of ordered pairs represents a linear function based on whether the relationship is constant.
- It also discusses writing linear equations in standard form and using linear equations to graph the line by choosing values for the variable and plotting the corresponding points.
- Real-world examples are given to show restricting the domain and range based on the context and graphing linear functions as discrete points rather than a continuous line.
This document discusses three types of vectors: numeric vectors, geometric/physical vectors, and functions. Numeric vectors are lists of numbers. Geometric/physical vectors have magnitude and direction, like directed line segments representing displacements. Functions can also be viewed as vectors. All three types of vectors can be added, subtracted, and multiplied by numbers. Numeric vectors correspond to geometric vectors through their components in a coordinate system. Forces are represented as geometric vectors with magnitude and direction.
This document provides an introduction to matrix algebra concepts needed for a systems biology course, including matrices, determinants, inverses, eigenvalues and eigenvectors. It discusses how matrices first arose from solving systems of linear equations and how the modern approach is to transform linear systems into matrix equations. Key concepts introduced include matrix operations like addition and multiplication, properties of the matrix multiplication like it being non-commutative, and how determinants are important for solving linear systems. The document also notes how complex numbers allow solving equations that have no real number solutions.
Mc0079 computer based optimization methods--phpapp02Rabby Bhatt
This document discusses mathematical models and provides examples of different types of mathematical models. It begins by defining a mathematical model as a description of a system using mathematical concepts and language. It then classifies mathematical models in several ways, such as linear vs nonlinear, deterministic vs probabilistic, static vs dynamic, discrete vs continuous, and deductive vs inductive vs floating. The document provides examples and explanations of each type of model. It also discusses using finite queuing tables to analyze queuing systems with a finite population size. In summary, the document outlines different ways to classify mathematical models and provides examples of applying various types of models.
This document summarizes Andrew Ng's lecture notes on supervised learning and linear regression. It begins with examples of supervised learning problems like predicting housing prices from living area size. It introduces key concepts like training examples, features, hypotheses, and cost functions. It then describes using linear regression to predict prices from area and bedrooms. Gradient descent and stochastic gradient descent are introduced as algorithms to minimize the cost function. Finally, it discusses an alternative approach using the normal equations to explicitly minimize the cost function without iteration.
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
Linear regression models the relationship between two variables, where one variable is considered the dependent variable and the other is the independent variable. The linear regression line minimizes the sum of the squared distances between the observed dependent variable values and the predicted dependent variable values. This line can be used to predict the dependent variable value based on new independent variable values. Multiple linear regression extends this to model the relationship between a dependent variable and two or more independent variables. Other types of regression models include nonlinear, generalized linear, and exponential regression.
Impact of teamwork on social skills development and peer relationships among ...Zawarali786
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R can be used to analyze data and perform statistical analysis. Functions like help(), ? and help.start() provide information about other functions. Objects created in R sessions are stored by name and can be removed with rm(). Vectors like x=c(1,2,3,4,5) can be created and their length checked with length(x). Subsets of vectors can be selected using logical or integer indexes inside square brackets. Matrices are multi-dimensional generalizations of vectors that can be manipulated using operators like * and %. Data can be read into R from external files using functions like read.table() and read.delim(). Common statistical distributions like normal, uniform and exponential are available as functions in R for
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The rate of change of a function is found by taking the difference quotient, which is similar to the slope formula. The domain of a function is the set of all possible x-values, while the range is the resulting y-values. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.
Linear algebra concepts like vectors, matrices, and linear transformations are important for recommendation systems. Vectors represent items or users, matrices represent item-user preference data. Linear algebra allows analyzing this data to identify patterns and recommend new items. Key techniques include eigendecomposition to reduce dimensionality and identify important relationships in the data, and singular value decomposition to factor matrices for recommendations. These linear algebra concepts are essential mathematical tools for building personalized recommendation models.
This document provides information about Baraka Loibanguti, who is the author of an advanced mathematics book. It includes his contact information and some notes about copyright and permissions. The document then begins discussing functions, including definitions of domain, range, and different types of functions like linear, quadratic, cubic, and polynomial functions. It provides examples of how to graph different types of functions by creating tables of values or using intercepts.
1) The document discusses graphing and properties of exponential and logarithmic functions, including: graphing exponential functions by substituting values of the variable into the equation, graphing logarithmic functions using the change of base formula, and properties like the product, quotient, and power properties of logarithms.
2) Examples are provided of solving exponential and logarithmic equations using properties like changing bases to the same value, multiplying or dividing arguments using the product and quotient properties, and applying exponents using the power property.
3) Steps shown include using properties to isolate the variable, set arguments or exponents equal to each other, and solve the resulting equation.
R can be used to analyze data and perform statistical analysis. Key functions include help() and ? to get information on functions, and objects() to view stored objects. Vectors can be created with c() and manipulated using arithmetic operators. Matrices are two-dimensional arrays that can be operated on using *, /, and t(). Larger datasets are typically read from external files using read.table() or read.delim(). Common distributions can be explored using functions like dnorm(), pnorm(), and rnorm(). Statistical analysis includes commands like cov() and cor() to measure covariance and correlation between variables.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
1. The document discusses the concept of derivatives and how to calculate them. It defines key terms like increment, average rate of change, and instantaneous rate of change.
2. Methods are provided for calculating the derivative of various types of functions, including polynomials, rational functions, and functions with roots. Examples are worked through step-by-step.
3. The relationship between derivatives and tangent lines is explained, with derivatives defined as the instantaneous rate of change or slope of the curve at a point.
1. Physics deals with matter and energy through defining and characterizing interactions between the two.
2. Mechanics studies motion and forces, including quantities like speed, velocity, acceleration, force, momentum and Newton's Laws of Motion.
3. Solving physics problems involves identifying known and unknown quantities, selecting the appropriate equation, and using the correct units.
The document describes five main families of functions - linear, power, root, reciprocal, and absolute value functions. It provides the name, equation, domain and range for each type of function. It also discusses concepts like piecewise functions, average rate of change, transformations, combinations of functions, and variations.
The document discusses differentiation rules for various functions. It begins by discussing the derivatives of polynomials and exponential functions. The power rule is introduced, which states the derivative of x^n is nx^{n-1}. It then covers the derivatives of exponential functions f(x)=ax, proving the formula f'(x)=af(x). The product rule and quotient rule are also introduced. Finally, it discusses the derivatives of trigonometric functions, proving that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
- The document is a lesson on identifying and graphing linear functions. It provides examples of determining if a graph or set of ordered pairs represents a linear function based on whether the relationship is constant.
- It also discusses writing linear equations in standard form and using linear equations to graph the line by choosing values for the variable and plotting the corresponding points.
- Real-world examples are given to show restricting the domain and range based on the context and graphing linear functions as discrete points rather than a continuous line.
This document discusses three types of vectors: numeric vectors, geometric/physical vectors, and functions. Numeric vectors are lists of numbers. Geometric/physical vectors have magnitude and direction, like directed line segments representing displacements. Functions can also be viewed as vectors. All three types of vectors can be added, subtracted, and multiplied by numbers. Numeric vectors correspond to geometric vectors through their components in a coordinate system. Forces are represented as geometric vectors with magnitude and direction.
This document provides an introduction to matrix algebra concepts needed for a systems biology course, including matrices, determinants, inverses, eigenvalues and eigenvectors. It discusses how matrices first arose from solving systems of linear equations and how the modern approach is to transform linear systems into matrix equations. Key concepts introduced include matrix operations like addition and multiplication, properties of the matrix multiplication like it being non-commutative, and how determinants are important for solving linear systems. The document also notes how complex numbers allow solving equations that have no real number solutions.
Mc0079 computer based optimization methods--phpapp02Rabby Bhatt
This document discusses mathematical models and provides examples of different types of mathematical models. It begins by defining a mathematical model as a description of a system using mathematical concepts and language. It then classifies mathematical models in several ways, such as linear vs nonlinear, deterministic vs probabilistic, static vs dynamic, discrete vs continuous, and deductive vs inductive vs floating. The document provides examples and explanations of each type of model. It also discusses using finite queuing tables to analyze queuing systems with a finite population size. In summary, the document outlines different ways to classify mathematical models and provides examples of applying various types of models.
This document summarizes Andrew Ng's lecture notes on supervised learning and linear regression. It begins with examples of supervised learning problems like predicting housing prices from living area size. It introduces key concepts like training examples, features, hypotheses, and cost functions. It then describes using linear regression to predict prices from area and bedrooms. Gradient descent and stochastic gradient descent are introduced as algorithms to minimize the cost function. Finally, it discusses an alternative approach using the normal equations to explicitly minimize the cost function without iteration.
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
Linear regression models the relationship between two variables, where one variable is considered the dependent variable and the other is the independent variable. The linear regression line minimizes the sum of the squared distances between the observed dependent variable values and the predicted dependent variable values. This line can be used to predict the dependent variable value based on new independent variable values. Multiple linear regression extends this to model the relationship between a dependent variable and two or more independent variables. Other types of regression models include nonlinear, generalized linear, and exponential regression.
Similar to AIOU Code 803 Mathematics for Economists Semester Spring 2022 Assignment 1.pptx (20)
Impact of teamwork on social skills development and peer relationships among ...Zawarali786
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Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
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.
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.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
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
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.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...
AIOU Code 803 Mathematics for Economists Semester Spring 2022 Assignment 1.pptx
1. 0314-4646739 0336-4646739 0332-4646739
Course: Mathematics for Economists (803)
Semester: Spring, 2022
Assignment No. 1
Q.1 Differentiate among variables, constants and parameters. Also define endogenous as
well as exogenous variables.
A constant is something like a "number". It doesn't change as variables change. For example 3 is
a constant as is π.
A parameter is a constant that defines a class of equations.
(xa)2+(yb)2=1
is the general equation for an ellipse. aa and bb are constants in this equation, but if we want to
talk about the entire class of ellipses then they are also parameters -- because even though they
are constant for any particular ellipse, they can take any positive real values.
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2. A variable is an element of the domain or codomain of a relation. Remember that functions are
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function x↦ax+3x↦ax+3, then xx is a variable and aa is a parameter -- and thus a constant. 33 is
also a constant but it is not a parameter.
A "known" variable is typically a value that the conditions of the problem dictate the
variable must take. For example if we are discussing an object an free fall, then acceleration is a
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variable. But physics puts a constraint on the value that that variable may take -- acceleration in
free fall is a=g≈9.8. Thus, though aa may be defined as the input of a function, it must take a
"known" value. Thus it is a known variable.
The Pythagorean theorem states that a2+b2=c2 for sides a,b and hypotenuse cc of a right triangle.
These are parameters -- thus they are also constants.
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They are all the same sort of thing on different levels of abstraction/generalization. Setting a
value creates a more specialized (less general) version of the mathematical object (function,
optimization problem, etc.), and replacing a formerly exactly defined value by a symbol creates a
generalized problem (covering a whole family of the specific problems).
If you set aa to some value in ax+3ax+3, you get a more specific version, for example 5x+3.
If you further set xx to some value, you get a specific number out, like 5⋅6+3.
In the other direction, if you turn ax+3 into ax+tax+t, you can represent a whole family of
(parameterized) functions including ax+8 and ax+1.
tt is the highest level parameter, aa is one lower and xx is the lowest. Since we usually only use a
few such levels at a time, we like to use names for them instead of just saying "higher level"
parameter. Variables are usually those that get adjusted on the lowest level, parameters are a level
above and constants are those that we don't change or adjust in our current task, but they could be
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turned into even higher-level parameters (called hyperparameters) if we wanted to further
generalize our problem or object.
Any function with multiple parameters can be turned into a higher-level function that just takes
one parameter and gives you a new function which now takes one less parameter than the
original. This is called currying. So your f(a,x)=ax+3f(a,x)=ax+3 can be turned into a function
which gives a new function for each aa:
F=(a↦(x↦(ax+3)))
So F(7) would be a function itself, 7x+3.
If you are familiar with programming it is also similar to variable scoping, i.e. that values are
defined in nested contexts. Functional programming uses these concepts even more heavily.
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Which parameter you put on which level depends on the current problem at hand and the same
problem can often be analyzed in multiple ways, i.e. by swapping parameters across levels (like
in our example, interpreting aa as the lowest and xx as the higher-level parameter).
A variable is, of course, a quantity that is allowed to vary over its range of definition. For
example, f(x)=3x+5f(x)=3x+5 is a function, where x ranges over the real numbers.
Now, I think the difference between constants and parameters is a bit more subtle. First,
constants:
A constant is just something that doesn't vary. 3 is a constant value, π is a constant value. But
then, in the function f(x)=ax+b. a and b are arbitrary constants. So, for whatever reason, say we
want to study functions of the form f(x)=ax+b, where aa and bb are some fixed values, but we
don't really care if those values are 3, 42, or π, so we say aa and bb are constants. I think in that
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sense there's a distinction between specific constants, like π, and arbitrary ones, like a,b in the
previous formula.
Now, with parameters, in my experience, there's always some notion of partial application going
on. I think statistical distributions are a really good example of this. For example, take the normal
distribution, if we wanted to, we could think of it as a function of three variables, x,μ and σ, but
that's not what a normal distribution is! A normal distribution is the particular single-variable
function of x you get when you choose a particular σ and μ, as opposed to being arbitrary,
like aa and bb we talked about above.
Parameter=para+meter=against+measure to measure something against some other thing"
(against an object treated as a unit). So basicly a "parameter" is something which could be
measured with a ruler.
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For the sake of modelling some real life system we make up a mathematical object. Now let's
imagine that we are modelling our solar system. After many hours of mental labor we came up
with a model:
x=f(t)=at−bx=f(t)=at−b
where
x - a variable, position of the 3rd planet in the solar system under consideration,
t - a variable, time,
a - a parameter of the system, speed of the planet rotation,
b - a parameter of the system, planet's initial position relative to some point,
thus by plugging tt into f(t) we should get the future position of the planet in the solar system.
Now f(t)=at−b is a model of the system but it is a general model, not a model of our particular
solar system, but a framework for modelling any solar system there is! To
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render f(t)=at−bf(t)=at−b into OUR home system we should measure our
system's parameters: a,b. By measureing them we transform model of general solar system into
particular model of our solar system.
Now let's imagine that after taking measurments by using telescopes we get values for our
parameters: aa=500 km/h and bb=100 000 km. So f(t)=at−bf(t)=at−b transforms
into f(t)=500t−100000f(t)=500t−100000. Now we can calculate the variable x.
In the example above you should see that:
1.We do not measure variables! We calculate them or plug them in the model (function). We
calculate variables from measured parameters.
2.We do not calculate parameters, instead we measure them as the etymology of the word
suggest.
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3. By measuring parameters we select particular model f(t)=500t−100000 from the class of
models f(t)=at−b.
So in the end: If you measure something - it's a parameter. If you calculate something - it's a
variable.
In the field of mathematics, a variable defines as an element connected to a number, known as an
estimation of the variable that is self-estimated, not completely determined, or ambiguous. The
expression “variable” originates from the way that, when the argument (additionally called the
“variable of the Function”) changes, then the estimate changes accordingly.
2. Consider the market model.
Qd = 20-3P
Qs = 10P-2
a) Calculate the equilibrium price and quantity.
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equilibrium price and quantity after tax?
Qd = 20-3P
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Qd=Qs
20-3P = 10P-2
20+2 = 10P +13P
22 = 13P
P = 22 / 13 = 1.692
Qd = 20-3P
Qd = 20-3(22 / 13)
Qd = 20- 66/13
Qd = 194 / 13 = 14.92
b) Suppose government impose Rs 16 per unit tax on consumers. What will be the
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3P = 20 – Qd
P = 20/3 – Qd / 3
Qs = 10P-2
10P = Qs+2
P = Qs / 10 + 1 / 5 + 16
3. a) Define matrix, vector and scalar.
Scalars
These are direction independent quantities that can be fully described by a single number, and
are unaffected by rotations or changes in co-ordinate system. Examples of physical properties
that are scalars: Energy, Temperature, Mass.
For this TLP scalars will be written in italics.
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Vectors
These are objects that possess a magnitude and a direction, and are referenced to a particular
set of axes known as a basis. A basis is a set of unit vectors (vectors with a magnitude of 1)
from which any other vector can be constructed by multiplication and addition.
The vector is referenced to the basis by its components. If possible, the maths is simplified by
using an orthonormal base with orthogonal (mutually perpendicular) unit vectors. Examples
of physical properties that are described by vectors: Mechanical force, Heat flow, Electric
field.
Vectors will be written in bold and components of a vector, say x, will be written as xi
Matrices
A matrix is a mathematical object that contains a rectangular array of numbers that can be
added and multiplied (according to matrix multiplication rules). They are very useful in many
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applications, for example in reducing a set of linear equations into a single equation, storing
the coefficients of linear transformations (e.g. rotations), and as we shall see, in describing
tensors.
The components of matrix A are written aij where i refers to the row element and j refers to
the column element.
Scalar products
For two vectors: a = (a1, a2, a3) and b = (b1, b2, b3) The scalar product (also known as the dot
product)
and so, for
is defined as: a.b = a1b1 + a2b2 + a3b3
example, the vectors (1, 4, −3) and (2, 5, 1) have a scalar product
of 1×2 + 4×5 − 3×1 = 19.
The scalar product is related to θ, the angle between the two vectors, and can equivalently be
written as: a.b = |a||b|cosθ.
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For vectors of unit length, we can see that the scalar product is equal to the cosine of the
angle between them.
Matrix multiplication
If we have two matrices,
A =
a11 a12 a13
a21 a22 a23
a31 a32 a33
and B =
b11 b12 b13
b21 b22 b23
b31 b32 b33
Then the product C = AB is found by ∑3
k=1 aikbkj where i, j and k are indices that represent the
position of the element in the matrix.
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ROW COLUMN
× = RC = "Race Car" or "Really Cool!" or make up your own acronym to
remember it. This is also useful to remember the conventional order of suffices, where the first
suffix indicates the row and the second indicates the column.
You can use the following activity to practice more matrix multiplication.
b) Using Cramer’s rule find the value of ‘x’, ‘y’ and ‘t’
4x + 2y+t= 2
5x + y+2t= 1
3x + 4y+6t= 4
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20. Q.4 Write a detailed note on Jacobean determinants, also give economic interpretation of
total differentiation.
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In vector calculus, the Jacobian matrix of a vector-valued function of several variables is
the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the
function takes the same number of variables as input as the number of vector components of its
output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if
applicable) the determinant are often referred to simply as the Jacobian in literature.
The Jacobian of a vector-valued function in several variables generalizes the gradient of a scalar-
valued function in several variables, which in turn generalizes the derivative of a scalar-valued
function of a single variable. In other words, the Jacobian matrix of a scalar-valued function in
several variables is (the transpose of) its gradient and the gradient of a scalar-valued function of a
single variable is its derivative.
At each point where a function is differentiable, its Jacobian matrix can also be thought of as
describing the amount of "stretching", "rotating" or "transforming" that the function imposes
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locally near that point. For example, if (x′, y′) = f(x, y) is used to smoothly transform an image,
the Jacobian matrix Jf(x, y), describes how the image in the neighborhood of (x, y) is
transformed.
If a function is differentiable at a point, its differential is given in coordinates by the Jacobian
matrix. However a function does not need to be differentiable for its Jacobian matrix to be
defined, since only its first-order partial derivatives are required to exist.
If f is differentiable at a point p in Rn, then its differential is represented by Jf(p). In this case,
the linear transformation represented by Jf(p) is the best linear approximation of f near the
point p, in the sense that
where o(‖x − p‖) is a quantity that approaches zero much
the distance between x and p does as x approaches p. This approximation
faster than
specializes to the
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approximation of a scalar function of a single variable by its Taylor polynomial of degree one,
namely
In this sense, the Jacobian may be regarded as a kind of "first-order derivative" of a vector-valued
function of several variables. In particular, this means that the gradient of a scalar-valued function
of several variables may too be regarded as its "first-order derivative".
Composable differentiable functions f : Rn → Rm and g : Rm → Rk satisfy the chain rule,
namely for x in Rn.
If m = n, then f is a function from Rn to itself and the Jacobian matrix is a square matrix. We can
then form its determinant, known as the Jacobian determinant. The Jacobian determinant is
sometimes simply referred to as "the Jacobian".
The Jacobian determinant at a given point gives important information about the behavior
of f near that point. For instance, the continuously differentiable function f is invertible near a
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point p ∈ Rn if the Jacobian determinant at p is non-zero. This is the inverse function theorem.
Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it
is negative, f reverses orientation. The absolute value of the Jacobian determinant at p gives us
the factor by which the function f expands or shrinks volumes near p; this is why it occurs in the
general substitution rule.
The Jacobian determinant is used when making a change of variables when evaluating a multiple
integral of a function over a region within its domain. To accommodate for the change of
coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the
integral. This is because the n-dimensional dV element is in general a parallelepiped in the new
coordinate system, and the n-volume of a parallelepiped is the determinant of its edge vectors.
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25. The Jacobian can also be used to determine the stability of equilibria for systems of differential
equations by approximating behavior near an equilibrium point. Its applications include
determining the stability of the disease-free equilibrium in disease modelling.
Q.5 Find the derivatives of the following:
a) Y= (24x2+4) (23x+11)
b) Y= (12x2 + 14) (4x-1 -3)
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