1) The document discusses differential calculus and introduces concepts like gradients, tangents, and derivatives.
2) It explains that the gradient of a function at a point is defined as the gradient of the tangent line to the function at that point.
3) The document demonstrates finding gradients of linear functions using the two-point formula, and how to approximate the gradient of a nonlinear function like a parabola by considering secants that approach the tangent line.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
1) The document outlines a teaching plan for quadratic equations and functions over several weeks. It includes learning objectives, outcomes, suggested activities and points to note for teachers.
2) Key concepts covered are quadratic equations, functions, graphs, maximum/minimum values, and solving simultaneous equations. Suggested activities include using graphing calculators, computer software and real-world examples.
3) The document provides detailed guidance for teachers on topics, skills, strategies and values to focus on for each area of learning.
The document discusses linear transformations and their applications in mathematics for artificial intelligence. It begins by introducing linear transformations and how matrices can be used to define functions. It describes how a matrix A can define a linear transformation fA that maps vectors in Rn to vectors in Rm. It also defines key concepts for linear transformations like the kernel, range, row space, and column space. The document will continue exploring topics like the derivative of transformations, linear regression, principal component analysis, and singular value decomposition.
This document provides an outline and introduction to a course on mathematics for artificial intelligence, with a focus on vector spaces and linear algebra. It discusses:
1. A brief history of linear algebra, from ancient Babylonians solving systems of equations to modern definitions of matrices.
2. The definition of a vector space as a set that can be added and multiplied by elements of a field, with properties like closure under addition and scalar multiplication.
3. Examples of using matrices and vectors to model systems of linear equations and probabilities of transitions between web pages.
4. The importance of linear algebra concepts like bases, dimensions, and eigenvectors/eigenvalues for machine learning applications involving feature vectors and least squares error.
The document proposes a new method for approximating matrix finite impulse response (FIR) filters using lower order infinite impulse response (IIR) filters. The method is based on approximating descriptor systems and requires only standard linear algebraic routines. Both optimal and suboptimal cases are addressed in a unified treatment. The solution is derived using only the Markov parameters of the FIR filter and can be expressed in state-space or transfer function form. The effectiveness of the method is illustrated with a numerical example and additional applications are discussed.
This document discusses lines and planes in Rn (n-dimensional space). It defines a line as the set of points tv + p, where v is a direction vector, t is a scalar, and p is a point. Similarly, it defines a plane as the set of points tv + sw + p, where v and w are linearly independent direction vectors and t and s are scalars. It provides examples of finding vector and parametric equations for lines and planes. It also discusses concepts like parallel and perpendicular lines, as well as finding the shortest distance from a point to a line or plane.
The document discusses applying alternating direction implicit (ADI) methods to solve tensor structured equations. ADI methods were originally developed to solve linear systems related to Poisson problems on uniform grids. The document outlines how ADI can be generalized to solve systems with tensor structure, such as those arising from tensor train decompositions of multidimensional problems. By exploiting the tensor structure, the ADI method can solve large problems with significantly less computational cost and storage than solving the equivalent vectorized problem directly.
1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source.
2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation.
3) Specific solutions include dark/bright solitary waves described by a sech-squared profile, as well as periodic solutions.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
1) The document outlines a teaching plan for quadratic equations and functions over several weeks. It includes learning objectives, outcomes, suggested activities and points to note for teachers.
2) Key concepts covered are quadratic equations, functions, graphs, maximum/minimum values, and solving simultaneous equations. Suggested activities include using graphing calculators, computer software and real-world examples.
3) The document provides detailed guidance for teachers on topics, skills, strategies and values to focus on for each area of learning.
The document discusses linear transformations and their applications in mathematics for artificial intelligence. It begins by introducing linear transformations and how matrices can be used to define functions. It describes how a matrix A can define a linear transformation fA that maps vectors in Rn to vectors in Rm. It also defines key concepts for linear transformations like the kernel, range, row space, and column space. The document will continue exploring topics like the derivative of transformations, linear regression, principal component analysis, and singular value decomposition.
This document provides an outline and introduction to a course on mathematics for artificial intelligence, with a focus on vector spaces and linear algebra. It discusses:
1. A brief history of linear algebra, from ancient Babylonians solving systems of equations to modern definitions of matrices.
2. The definition of a vector space as a set that can be added and multiplied by elements of a field, with properties like closure under addition and scalar multiplication.
3. Examples of using matrices and vectors to model systems of linear equations and probabilities of transitions between web pages.
4. The importance of linear algebra concepts like bases, dimensions, and eigenvectors/eigenvalues for machine learning applications involving feature vectors and least squares error.
The document proposes a new method for approximating matrix finite impulse response (FIR) filters using lower order infinite impulse response (IIR) filters. The method is based on approximating descriptor systems and requires only standard linear algebraic routines. Both optimal and suboptimal cases are addressed in a unified treatment. The solution is derived using only the Markov parameters of the FIR filter and can be expressed in state-space or transfer function form. The effectiveness of the method is illustrated with a numerical example and additional applications are discussed.
This document discusses lines and planes in Rn (n-dimensional space). It defines a line as the set of points tv + p, where v is a direction vector, t is a scalar, and p is a point. Similarly, it defines a plane as the set of points tv + sw + p, where v and w are linearly independent direction vectors and t and s are scalars. It provides examples of finding vector and parametric equations for lines and planes. It also discusses concepts like parallel and perpendicular lines, as well as finding the shortest distance from a point to a line or plane.
The document discusses applying alternating direction implicit (ADI) methods to solve tensor structured equations. ADI methods were originally developed to solve linear systems related to Poisson problems on uniform grids. The document outlines how ADI can be generalized to solve systems with tensor structure, such as those arising from tensor train decompositions of multidimensional problems. By exploiting the tensor structure, the ADI method can solve large problems with significantly less computational cost and storage than solving the equivalent vectorized problem directly.
1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source.
2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation.
3) Specific solutions include dark/bright solitary waves described by a sech-squared profile, as well as periodic solutions.
1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source.
2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation.
3) Specific solutions include periodic solutions and bright/dark solitary wave solutions, with the intensity profiles of the bright solitary wave shown.
- Daubechies wavelets are a family of orthogonal wavelets that provide the highest number of vanishing moments for a given width, defined through recursive equations.
- They are approximately localized in both time and frequency domains. The wavelets and scaling functions are not defined by closed-form equations, but are instead generated numerically through an iterative process.
- Properties include orthogonality, localization, and a maximal number of vanishing moments for a given support width, with more coefficients providing more moments. They are widely used for problems involving signal discontinuities or self-similarity.
The document describes directed graphs and algorithms for analyzing them, including:
- Representing directed graphs using adjacency matrices and relations
- Determining reachability between nodes using the transitive closure of the adjacency relation or powers of the adjacency matrix
- Warshall's algorithm, which computes the transitive closure and reachability by taking the disjunction of the adjacency matrix with its powers up to the number of nodes.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Opti...Reza Rahimi
The document provides an overview of linear programming and its usage in approximation algorithms for NP-hard optimization problems. It discusses linear programming formulations, the complexity classes P and NP, approximation algorithms, and two case studies on the minimum weight vertex cover problem and the MAXSAT problem. Randomized rounding techniques are used to generate approximation algorithms for these problems from their linear programming relaxations.
1. The document discusses Rao-Blackwellisation, a technique to improve the efficiency of Metropolis-Hastings algorithms.
2. Rao-Blackwellisation can be applied to any Hastings Metropolis algorithm and results in asymptotically more efficient sampling than standard MCMC with a controlled computational cost.
3. The document outlines the Metropolis-Hastings algorithm and then discusses how Rao-Blackwellisation can reduce variance and improve asymptotic performance through partitioning variables.
Distributed Parallel Process Particle Swarm Optimization on Fixed Charge Netw...Corey Clark, Ph.D.
The document presents a dynamically distributed binary particle swarm optimization (BPSO) approach for solving fixed-charge network flow problems. The approach distributes the BPSO algorithm across a cluster of devices using a distributed accelerated analytics platform. Testing showed the distributed BPSO approach found better solutions faster than serial BPSO and optimization approaches for various problem sizes, demonstrating the benefits of dynamic distributed computing for difficult mixed integer programs.
This document discusses energy detection of unknown signals in fading environments. It proposes modeling the received signal power distribution under combined slow and fast fading. This allows deriving the distribution of the detector's decision variable in closed form. Specifically:
1) It models the received signal as the sum of the signal and noise, scaled by a complex channel amplitude representing fast and slow fading.
2) It derives an expression for the sufficient statistic at the detector's output and simplifies it under assumptions of high sample numbers and independent samples.
3) It expresses the distribution of the decision variable as an integral of the distribution for a fixed SNR, averaged over the SNR distribution due to fading.
4) It provides the specific
Do we need a logic of quantum computation?Matthew Leifer
1) The document discusses whether quantum computing needs a formal logic in the same way that classical computing is understood through classical logic. It examines previous proposals for "quantum logics" and focuses on Sequential Quantum Logic (SQL).
2) SQL models sequences of quantum measurements and operations through projection operators and sequential conjunction. The document proposes testing SQL propositions through a quantum algorithm that prepares an encoded "history state" and applies renormalization operations.
3) The proposed algorithm could test SQL propositions with exponentially small probability of success. Several open questions are raised about generalizing and improving SQL as a logic for quantum computing.
The document defines a circle as all points equidistant from a center point. It gives the standard and general forms of a circle equation, and examples of writing the equations of circles with specified radii and center points. It also defines exponential functions as having a constant base and a variable exponent, and gives examples of sketching the graphs of exponential functions with specified bases and exponents.
This document presents a new approach for reducing high order discrete time interval systems to lower order systems using least squares methods and time moment matching. The method transforms the original interval system into fixed transfer functions using Kharitonov's theorem. It then determines the time moments and Markov parameters of the shifted original system. These are used to generate equations that are solved using least squares to determine the coefficients of the reduced system denominator. The reduced numerator is then determined by moment matching. The method is illustrated with a numerical example of reducing a 3rd order discrete interval system.
The document discusses online handwritten character recognition in the Devanagari script using a hierarchical partitioned hidden Markov model approach. Key steps include preprocessing strokes, extracting directional features, using single linkage clustering to select prototypes, and building a two-layer model with bottom HMMs for clusters and an upper attribute graph layer. Mathematical foundations show that pruning points does not impact the dynamic time warping distance measure between strokes. The approach achieves a recognition rate of 91.24% on a test dataset.
Dragisa Zunic - Classical computing with explicit structural rules - the *X c...Dragisa Zunic
The document discusses the ∗X calculus, which provides an explicit computational interpretation of classical logic proofs represented in sequent calculus. The ∗X calculus makes weakening and contraction explicit through terms corresponding to proofs. Terms are built from names and represent proofs with explicit erasure and duplication operations corresponding to weakening and contraction.
This document presents a methodology for mapping multidimensional transforms onto reconfigurable architectures like FPGAs. The methodology uses tensor product decompositions and permutation matrices to express transforms recursively in terms of lower-order blocks. This allows large transforms to be computed by combining many parallel, smaller transform blocks. Specific examples are given for mapping one-dimensional linear convolution and discrete cosine transforms. The overall goal is to provide a unified framework and design process for implementing multidimensional transforms in a modular, parallel architecture.
This document provides instructions for analyzing statically indeterminate beams using the matrix force method. It discusses:
1. Formulating the problem by selecting redundant reactions and determining the deflections of the primary structure due to applied loads.
2. Assembling the flexibility matrix relating deflections to applied forces at various points.
3. Setting up compatibility equations equating total deflections to zero and solving the equations in matrix form to determine redundant reactions.
4. Using the determined redundant reactions and equilibrium equations to calculate all reactions and internal forces for a given statically indeterminate beam problem.
Two example problems are provided to demonstrate applying the procedure.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
Kernelization algorithms for graph and other structure modification problemsAnthony Perez
The document discusses kernelization algorithms for graph modification problems. It begins by introducing graph modification problems, which take as input a graph and property and output the minimum number of modifications to the graph to satisfy the property. It then discusses using parameterized complexity to more efficiently solve NP-hard graph modification problems. In particular, it covers the concept of kernels, which are polynomial-time algorithms that reduce an instance to an equivalent instance of size bounded by a function of the parameter. The document provides an overview of generic reduction rules and the concept of branches that can be applied to graph modification problems. It also introduces the specific problem of proper interval completion and known results about its parameterized complexity.
This document discusses image segmentation and deformable segmentation techniques. It introduces image segmentation as involving partitioning an image into meaningful regions to provide a higher-level representation. Deformable segmentation uses models that can deform to find object boundaries, including active contours and level set methods. The document focuses on deformable segmentation approaches, comparing explicit and implicit representations, and discussing challenges such as computational complexity, topology changes, and numerical solutions.
Direct variational calculation of second-order reduced density matrix : appli...Maho Nakata
Presented at GCOE interdisciplinary workshop on numerical methods for many-body correlations, https://sites.google.com/a/cns.s.u-tokyo.ac.jp/shimizu/gcoe
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
This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.
IB Maths. Turning points. First derivative testestelav
By the end of the lesson, students will be able to use derivatives to find maximum and minimum points of a function, and use second derivatives to determine the nature of stationary points and points of inflection. Specifically, they will learn that: (1) if the first derivative is zero at a point, it is a stationary point; (2) the second derivative test can determine if it is a local max, min or point of inflection; and (3) points of inflection occur when the curve changes concavity. Students will apply these concepts to find the stationary points of sample functions and classify their nature.
1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source.
2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation.
3) Specific solutions include periodic solutions and bright/dark solitary wave solutions, with the intensity profiles of the bright solitary wave shown.
- Daubechies wavelets are a family of orthogonal wavelets that provide the highest number of vanishing moments for a given width, defined through recursive equations.
- They are approximately localized in both time and frequency domains. The wavelets and scaling functions are not defined by closed-form equations, but are instead generated numerically through an iterative process.
- Properties include orthogonality, localization, and a maximal number of vanishing moments for a given support width, with more coefficients providing more moments. They are widely used for problems involving signal discontinuities or self-similarity.
The document describes directed graphs and algorithms for analyzing them, including:
- Representing directed graphs using adjacency matrices and relations
- Determining reachability between nodes using the transitive closure of the adjacency relation or powers of the adjacency matrix
- Warshall's algorithm, which computes the transitive closure and reachability by taking the disjunction of the adjacency matrix with its powers up to the number of nodes.
Linear Programming and its Usage in Approximation Algorithms for NP Hard Opti...Reza Rahimi
The document provides an overview of linear programming and its usage in approximation algorithms for NP-hard optimization problems. It discusses linear programming formulations, the complexity classes P and NP, approximation algorithms, and two case studies on the minimum weight vertex cover problem and the MAXSAT problem. Randomized rounding techniques are used to generate approximation algorithms for these problems from their linear programming relaxations.
1. The document discusses Rao-Blackwellisation, a technique to improve the efficiency of Metropolis-Hastings algorithms.
2. Rao-Blackwellisation can be applied to any Hastings Metropolis algorithm and results in asymptotically more efficient sampling than standard MCMC with a controlled computational cost.
3. The document outlines the Metropolis-Hastings algorithm and then discusses how Rao-Blackwellisation can reduce variance and improve asymptotic performance through partitioning variables.
Distributed Parallel Process Particle Swarm Optimization on Fixed Charge Netw...Corey Clark, Ph.D.
The document presents a dynamically distributed binary particle swarm optimization (BPSO) approach for solving fixed-charge network flow problems. The approach distributes the BPSO algorithm across a cluster of devices using a distributed accelerated analytics platform. Testing showed the distributed BPSO approach found better solutions faster than serial BPSO and optimization approaches for various problem sizes, demonstrating the benefits of dynamic distributed computing for difficult mixed integer programs.
This document discusses energy detection of unknown signals in fading environments. It proposes modeling the received signal power distribution under combined slow and fast fading. This allows deriving the distribution of the detector's decision variable in closed form. Specifically:
1) It models the received signal as the sum of the signal and noise, scaled by a complex channel amplitude representing fast and slow fading.
2) It derives an expression for the sufficient statistic at the detector's output and simplifies it under assumptions of high sample numbers and independent samples.
3) It expresses the distribution of the decision variable as an integral of the distribution for a fixed SNR, averaged over the SNR distribution due to fading.
4) It provides the specific
Do we need a logic of quantum computation?Matthew Leifer
1) The document discusses whether quantum computing needs a formal logic in the same way that classical computing is understood through classical logic. It examines previous proposals for "quantum logics" and focuses on Sequential Quantum Logic (SQL).
2) SQL models sequences of quantum measurements and operations through projection operators and sequential conjunction. The document proposes testing SQL propositions through a quantum algorithm that prepares an encoded "history state" and applies renormalization operations.
3) The proposed algorithm could test SQL propositions with exponentially small probability of success. Several open questions are raised about generalizing and improving SQL as a logic for quantum computing.
The document defines a circle as all points equidistant from a center point. It gives the standard and general forms of a circle equation, and examples of writing the equations of circles with specified radii and center points. It also defines exponential functions as having a constant base and a variable exponent, and gives examples of sketching the graphs of exponential functions with specified bases and exponents.
This document presents a new approach for reducing high order discrete time interval systems to lower order systems using least squares methods and time moment matching. The method transforms the original interval system into fixed transfer functions using Kharitonov's theorem. It then determines the time moments and Markov parameters of the shifted original system. These are used to generate equations that are solved using least squares to determine the coefficients of the reduced system denominator. The reduced numerator is then determined by moment matching. The method is illustrated with a numerical example of reducing a 3rd order discrete interval system.
The document discusses online handwritten character recognition in the Devanagari script using a hierarchical partitioned hidden Markov model approach. Key steps include preprocessing strokes, extracting directional features, using single linkage clustering to select prototypes, and building a two-layer model with bottom HMMs for clusters and an upper attribute graph layer. Mathematical foundations show that pruning points does not impact the dynamic time warping distance measure between strokes. The approach achieves a recognition rate of 91.24% on a test dataset.
Dragisa Zunic - Classical computing with explicit structural rules - the *X c...Dragisa Zunic
The document discusses the ∗X calculus, which provides an explicit computational interpretation of classical logic proofs represented in sequent calculus. The ∗X calculus makes weakening and contraction explicit through terms corresponding to proofs. Terms are built from names and represent proofs with explicit erasure and duplication operations corresponding to weakening and contraction.
This document presents a methodology for mapping multidimensional transforms onto reconfigurable architectures like FPGAs. The methodology uses tensor product decompositions and permutation matrices to express transforms recursively in terms of lower-order blocks. This allows large transforms to be computed by combining many parallel, smaller transform blocks. Specific examples are given for mapping one-dimensional linear convolution and discrete cosine transforms. The overall goal is to provide a unified framework and design process for implementing multidimensional transforms in a modular, parallel architecture.
This document provides instructions for analyzing statically indeterminate beams using the matrix force method. It discusses:
1. Formulating the problem by selecting redundant reactions and determining the deflections of the primary structure due to applied loads.
2. Assembling the flexibility matrix relating deflections to applied forces at various points.
3. Setting up compatibility equations equating total deflections to zero and solving the equations in matrix form to determine redundant reactions.
4. Using the determined redundant reactions and equilibrium equations to calculate all reactions and internal forces for a given statically indeterminate beam problem.
Two example problems are provided to demonstrate applying the procedure.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
Kernelization algorithms for graph and other structure modification problemsAnthony Perez
The document discusses kernelization algorithms for graph modification problems. It begins by introducing graph modification problems, which take as input a graph and property and output the minimum number of modifications to the graph to satisfy the property. It then discusses using parameterized complexity to more efficiently solve NP-hard graph modification problems. In particular, it covers the concept of kernels, which are polynomial-time algorithms that reduce an instance to an equivalent instance of size bounded by a function of the parameter. The document provides an overview of generic reduction rules and the concept of branches that can be applied to graph modification problems. It also introduces the specific problem of proper interval completion and known results about its parameterized complexity.
This document discusses image segmentation and deformable segmentation techniques. It introduces image segmentation as involving partitioning an image into meaningful regions to provide a higher-level representation. Deformable segmentation uses models that can deform to find object boundaries, including active contours and level set methods. The document focuses on deformable segmentation approaches, comparing explicit and implicit representations, and discussing challenges such as computational complexity, topology changes, and numerical solutions.
Direct variational calculation of second-order reduced density matrix : appli...Maho Nakata
Presented at GCOE interdisciplinary workshop on numerical methods for many-body correlations, https://sites.google.com/a/cns.s.u-tokyo.ac.jp/shimizu/gcoe
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
This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.
IB Maths. Turning points. First derivative testestelav
By the end of the lesson, students will be able to use derivatives to find maximum and minimum points of a function, and use second derivatives to determine the nature of stationary points and points of inflection. Specifically, they will learn that: (1) if the first derivative is zero at a point, it is a stationary point; (2) the second derivative test can determine if it is a local max, min or point of inflection; and (3) points of inflection occur when the curve changes concavity. Students will apply these concepts to find the stationary points of sample functions and classify their nature.
This document discusses stationary points (SPs) and how to identify and classify them. It explains that SPs occur when the derivative of a function is equal to 0. It provides examples of finding SPs by taking the derivative, setting it equal to 0, and solving for x. It also introduces the concept of using a nature table to determine whether a SP is a maximum or minimum turning point by examining the sign of the derivative just before and after the SP. The document demonstrates this process on several examples, finding the SPs and using the nature table to classify them.
This document contains exercises for differential calculus that involve finding zeroes of functions, evaluating operations on functions, determining continuity and discontinuity, calculating derivatives using rules like the three-step limit rule, product rule and quotient rule, and applying derivatives. The exercises cover topics like inverse functions, limits of functions, and applications of derivatives.
This document discusses how to sketch the graph of a function by:
1) Finding the stationary points (SPs) where the derivative is equal to 0
2) Determining where the graph intersects the x-axis and y-axis
3) Identifying the dominant term for large positive and negative values of x to understand how the graph behaves as x approaches infinity
For graphs of mathematical functions, see Graph of a function. For other uses, see Graph (disambiguation). A drawing of a graph. In mathematics graph theory is the study of graphs, which are mathematical structures used.In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any acyclic connected graph is a tree. A forest is a disjoint union of trees.
The document provides examples and explanations for sketching the graphs of various types of functions, including:
1) Linear functions, which produce straight lines. The slope and y-intercept determine the graph.
2) Quadratic functions, which produce parabolas. The direction of opening and intercepts are used to sketch the graph.
3) Cubic functions, which produce S-shaped curves. The direction of turning and intercepts are considered.
4) Reciprocal functions, which produce hyperbolas. The direction and intercepts are the key factors for the graph.
Step-by-step methods are outlined for accurately sketching graphs of each function type based on their defining characteristics.
Introduction to basic differential calculus for algebra students familiar with the concept of the slope of a line. See my blog at http://roundyeducationblog.blogspot.com/2015_11_01_archive.html
The document provides examples of common mistakes made when working with exponents, radicals, polynomials, rational expressions, and solving equations in pre-calculus. It demonstrates incorrect work and explains the errors, such as adding instead of multiplying terms with exponents or failing to square both sides of an equation when taking the square root of each side. Key concepts are reviewed, such as factoring polynomials before simplifying rational expressions or dividing terms. Helpful formulas are also listed, such as the quadratic formula.
This document discusses integral and differential calculus and their applications. It covers the fundamental theorems of calculus, how calculus helps understand space, time and motion, and how it provides tools to resolve paradoxes. It also notes that calculus is used in physics, chemistry, engineering, economics, medicine and other fields, and can find optimal solutions and approximate equations. It encourages learning more about calculus.
The document discusses parabolas and their key properties:
- A parabola is the set of all points equidistant from a fixed line called the directrix and a fixed point called the focus.
- The standard equation of a parabola depends on the orientation of its axis and vertex.
- Key properties include the axis of symmetry, direction of opening, and the length of the latus rectum.
The document discusses drawing graphs of different types of functions, including linear, quadratic, cubic, and reciprocal functions. It provides the general forms of each type of function, describes the steps to draw their graphs which include constructing a table of values, plotting points, and joining the points. As an example, it shows the graph of a reciprocal function f(x) = 1/x for -1 ≤ x ≤ 1.5, which forms a hyperbola shape.
This problem set covers differential calculus and requires students to find the derivatives of given functions and solve the equation 2=x+e-x. Students are instructed to write their solutions neatly and orderly.
Number series refers to numbers arranged in succession one after the other. Trigonometry is a branch of mathematics involving the study of relationships between lengths and angles of triangles, emerging from the Hellenistic world during the 3rd century BC.
This document discusses key concepts of integration including defining the integral as representing the whole or area under a curve, using integrals to symbolize the difference in values of a function, and how indefinite integrals symbolize a function whose derivative is another function. It also covers properties such as the bounds property for flipping integral values and adding a negative sign to simplify solving, and the integral sum property for breaking up integrated equations into separate parts.
The document discusses various applications of the definite integral, including finding the area under a curve, the area between two curves, and the volume of solids of revolution. It provides examples of calculating each of these, such as finding the area between the curves y=x and y=x5 from x=-1 to x=0. It also explains how to set up definite integrals to calculate volumes when rotating an area about the x- or y-axis. In conclusion, it states that integrals can represent areas or generalized areas and are fundamental objects in calculus, along with derivatives.
Dokumen tersebut membahas tentang parabola, termasuk definisi matematis dan geometris parabola, contoh persamaan parabola berdasarkan posisi pusat dan fokusnya, serta cara menentukan persamaan garis singgung dan normal pada suatu parabola.
This document provides an introduction to differentiation and calculus concepts including:
- Functional thinking and how changing one variable can impact another
- Determining the gradient of straight and curved lines by calculating the rate of change between two points
- How the gradient of a curve at a single point is equal to the gradient of the tangent line at that point
- Algebraic methods for determining the derivative of various functions like powers of x, trigonometric functions, products, quotients, and composite functions using rules like the chain rule.
Tall-and-skinny QR factorizations in MapReduce architecturesDavid Gleich
This document describes using MapReduce to perform a tall-and-skinny QR factorization. It discusses dividing the input matrix into blocks and using mappers to perform QR on each block. The reducers then take the R factors from each block and perform a second stage QR factorization. This allows computing the QR factorization in MapReduce without excessive communication. The document also discusses ways to recover the full Q factor and ensure numerical stability, such as using iterative refinement.
Direct tall-and-skinny QR factorizations in MapReduce architecturesDavid Gleich
This document discusses tall-and-skinny QR factorizations in MapReduce. It begins by reviewing the basics of QR factorization and how it can be used for regression. It then describes how tall-and-skinny matrices arise in applications like regression with many samples, iterative methods, and model reduction. The document proposes using MapReduce to compute the QR factorization of very large tall-and-skinny matrices from applications like dynamic mode decomposition on large-scale simulations. It provides a brief overview of MapReduce and gives examples of computing variance in MapReduce.
This document discusses key concepts in calculus including differentiation, integration, graphing functions, and calculating total vs net distance. It defines the derivative as the rate of change of a variable and outlines differentiation rules. Integration is defined as the area under a curve and the fundamental theorem of calculus is explained. Graphing techniques are outlined including finding zeros, critical points, concavity, and inflection points. Total distance is defined as the absolute distance traveled regardless of path while net distance is the displacement from the original position.
تطبيقات المشتقات والدوال المثلثية و في الحيباة اليويمة و طرق الاشتقاقzeeko4
1. The document discusses various applications of derivatives including determining if a function is increasing or decreasing, finding maxima and minima, determining concavity, and locating points of inflection.
2. Methods for finding critical points include taking the first derivative and setting it equal to zero. The second derivative test is used to determine concavity.
3. Examples demonstrate finding critical points, maxima/minima, and points of inflection for various functions by taking the first and second derivatives.
This document discusses how to identify and analyze polynomial functions. It defines polynomials as functions of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0 where the exponents are whole numbers and the coefficients are real numbers. It explains how to determine the degree, type, leading coefficient, and end behavior of polynomial functions. Examples are provided for evaluating polynomials, sketching graphs, and identifying properties from the graphs like the degree, leading coefficient, and number of bumps.
Advanced functions ppt (Chapter 1) part iTan Yuhang
The document discusses power functions and characteristics of polynomial functions.
It defines power functions as functions of the form f(x)= xa where a is a fixed number. It also discusses key features of polynomial functions such as reflection, local/absolute maximum/minimum points, and the relationship between finite differences and the polynomial equation.
The document discusses power functions and characteristics of polynomial functions.
It defines power functions as functions of the form f(x)= xa where a is a fixed number. It also discusses key features of polynomial functions such as reflection, local/absolute maximum/minimum points, and the relationship between finite differences and the polynomial equation.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
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.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Assessment and Planning in Educational technology.pptxKavitha Krishnan
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
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.
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.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
1. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
If I have seen further…..
…. it is because I have stood on the shoulders of giants.
ISaac Newton to Robert Hooke in 1675
ISAAC NEWTON 1643 -1727
2. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
3. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
1
4. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
5. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction The two-point formula Gradients of secants
The concept of gradient is so important for a thorough
understanding of differential calculus.
The graphs of some linear functions are steep with a positive slope >
The graphs of some linear functions are less steep >
… and others have negative slopes >
Gradient is a measure of this steepness or slope.
It is defined as the ratio of the rise to the run.
The gradient of the green function is 2.
Check the gradient of the red function is 1/3
And the blue line has a gradient of-1
2
6. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction The two-point formula Gradients of secants
The given line passes through the points P and Q where:
P = ( 2, 3 )
P
Q = ( 1, 1 ) X
In the interval PQ:
rise
rise = 3 - 1
run = 2 - 1
Q
X
run
Generally, the straight line passing through the two points; ,
Has a gradient given by:
3
7. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction The two-point formula Gradients of secants
Using the formula for the gradient of a lineHere is the graph of the function
through two points, we have: And here is a secant PQ
where and .
4
8. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
9. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Case study An algebraic approach Limiting process
The gradient of the graph of a linear function is easy to find; we can use the two-
point formula as shown in the previous section. But how can we find the gradient
at different points on a non-linear function, such as the one shown here?
Clearly the parabola gets steeper as the x-values increase…
… but how can we measure the actual gradient at any particular point on the curve?
Gradient of the function at the point ( 3, 9)
>
The gradient at a point P on the
DEFINITION
curve is defined as the gradient
of the tangent to the curve at Gradient of the function at the point ( 2, 4)
that point.
5
10. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Case study An algebraic approach Limiting process
Our goal here is means finding the gradient of the tangent
By definition thisto find exactly the gradient of the function to the curve the point
at at that point…
As a first approximation, As a second approximation, For a third approximation
consider the secant AP consider the secant BP we will need to zoom in
and consider secant CP….
Note now how close the
tangent is to the curve
C
P
6
11. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Case study An algebraic approach Limiting process
Our goal here is to find exactly the gradient of the function at the point
By definition this means finding the gradient of the tangent to the curve at that point…
As a first approximation, consider the secant
h
x+h
7
12. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Case study An algebraic approach Limiting process
Using first principles find the derived function for
h
x+h
8
13. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Case study An algebraic approach Limiting process
Using first principles find the derived function for Using first principles find the derived function for SUMMARY
8
14. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
15. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Equation of a tangent Equation of a normal ACTIVITY 1
A tangent to a curve is a straight line touching the curve at a single point. A normal is a straight line, perpendicular to the tangent.
FIGURE 1 FIGURE 2 FACT SHEET
This diagram shows the TANGENT to the This diagram shows the NORMAL to the • A TANGENT touches a curve at a single
curve curve point
at the point (1, -2) at the point (1, -2)
• It’s gradient, , is given by the gradient or derived
function at the value
• Its equation is given by
NORMAL
• The NORMAL at a point is perpendicular to the
tangent at that point. (It’s at 90 degrees)
TANGENT • It’s gradient, , is found using the gradient of the
tangent and the fact that
• It’s equation is given by
9
16. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Equation of a tangent Equation of a normal ACTIVITY 1
Find the equation of the tangent to the curve at the point (2, 0)
METHOD
1 Differentiate to obtain the
gradient function
2 Find the gradient of the
function at x=2
3 Substitute the gradient,
m = 1 and the coordinates
of the point into the point/
gradient form of a straight
line.
10
17. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Equation of a tangent Equation of a normal ACTIVITY 1
Find the equation of the normal to the curve at the point (2, 0)
METHOD
1 Differentiate to obtain the
gradient function
2 Find the gradient of the
function at x=2
3 Find the gradient of the
normal at x=2
4 Substitute the gradient,
m = -1 and the
coordinates of the point into
the point/ gradient form of a
straight line.
11
18. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Introduction Equation of a tangent Equation of a normal ACTIVITY
19. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
20. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
21. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
22. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
23. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals
2
Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
24. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
25. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
26. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
27. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
28. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
MAXIMUM MINIMUM INFLEXION
STATIONARY POINT A STATIONARY POINT B POINT C A
LOCATED AT: LOCATED AT: LOCATED AT: C
B
FIRST DERIVATIVE IS ZERO FIRST DERIVATIVE IS ZERO
SECOND DERIVATIVE IS NEGATIVE SECOND DERIVATIVE IS POSITIVE SECOND DERIVATIVE IS ZERO
29. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
FIGURE 1 FIGURE 2 FACT SHEET
This diagram shows the STATIONARY POINT on This diagram shows the STATIONARY POINTS on • Stationary points lie on the graphs of functions where
the graph of the quadratic function the graph of the cubic function the gradient is zero. (The tangents to the curve
are horizontal at these points; the function is
neither increasing nor decreasing.)
• The stationary point in figure 1 is called a maxima.
• Figure 2 shows a function having both a maxima and
minima
30. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Find the stationary point on the graph of the function
METHOD
1 Find the gradient
function
by differentiating
2 Find the x-value for
which the gradient is
zero
by solving the equation
3 Find the y-value of the
function at x = 1.5
by substitution into the
original function
4 Write down the
coordinates of the stationary
point
31. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
Find the stationary points on the function and determine their nature
METHOD
1 Differentiate the given
function to obtain the
gradient function
2 Find the x-values for
which the gradient
function is zero.
3 Substitute these x-values
into the original function to
determine the stationary
points.
4 Find the sign of the
second derivative to
determine the
nature of each stationary
point.
32. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
33. DIFFERENTIAL CALCULUS
Home Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points