The document discusses graphing trigonometric functions like sine and cosine curves. It explains that a sine curve is defined by the equation y = a sin(bx + c), where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. A worked example graphs y = 5 sin(9x - π/2) with a period of 2π/9 units and amplitude of 5 units, shifted π/18 units to the right. The cosine curve is similarly defined by y = a cos(bx + c) with the same period, amplitude, and shift properties. An example graphs y = -4 cos((x + π)/8 + 2
12 x1 t04 06 integrating functions of time (2012)Nigel Simmons
The document discusses integrating functions of time to determine changes in displacement, distance, velocity, and speed. It explains that the integral of position over time equals displacement, while subtracting integrals of position over different time intervals equals distance. Similarly, the integral of velocity over time equals speed, while the integral of acceleration over time equals velocity. Graphs of functions and their derivatives are also presented, showing the relationships between integration and differentiation.
12 x1 t04 06 integrating functions of time (2013)Nigel Simmons
The document discusses integrating functions of time to calculate changes in displacement, distance, velocity, and speed based on position, velocity, and acceleration graphs over time. It provides examples of how integrating areas under curves relates to these physical quantities. Derivative graphs and their relationships are also summarized, along with how different function types integrate or differentiate into other graph types. An example problem calculating the times when a particle is at rest and its maximum velocity is also worked through.
The document discusses inferring gene regulatory networks from time-course gene expression data. It presents the problem of inferring interactions between genes from high-dimensional and sparse time-course microarray data. It proposes using a Gaussian graphical model and introducing biologically grounded priors, such as sparsity and latent clustering of networks, to help address the scarcity of data. Several statistical models and algorithms are described for performing regularized inference on the networks with and without using a known or inferred latent clustering structure. The methods are evaluated on simulated time-course data and a real E. coli S.O.S DNA repair network.
Dsp U Lec07 Realization Of Discrete Time Systemstaha25
This document provides an overview of discrete-time systems and digital signal processing. It discusses discrete-time system components like unit delays and adders. It also covers discrete system networks including FIR and IIR networks. Various realizations of discrete systems are presented, including direct form I and II, cascaded, and parallel realizations. Digital filters are defined and the advantages and disadvantages as well as types (FIR and IIR) are discussed. Design steps and specifications for digital filters are also outlined.
Nonlinear Stochastic Programming by the Monte-Carlo methodSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 4.
More info at http://summerschool.ssa.org.ua
The document describes curve registration through nonparametric testing of Fourier coefficients. It presents a model where curves are represented by their Fourier coefficients with added noise. The hypotheses are that the curves either match up to a shift, or are minimally distant overall after any shift. A generalized likelihood ratio test statistic is developed that minimizes the squared differences between the Fourier coefficients of the two curves over all possible shift values.
Aristidis Likas, Associate Professor and Christoforos Nikou, Assistant Professor, University of Ioannina, Department of Computer Science , Mixture Models for Image Analysis
12 x1 t04 06 integrating functions of time (2012)Nigel Simmons
The document discusses integrating functions of time to determine changes in displacement, distance, velocity, and speed. It explains that the integral of position over time equals displacement, while subtracting integrals of position over different time intervals equals distance. Similarly, the integral of velocity over time equals speed, while the integral of acceleration over time equals velocity. Graphs of functions and their derivatives are also presented, showing the relationships between integration and differentiation.
12 x1 t04 06 integrating functions of time (2013)Nigel Simmons
The document discusses integrating functions of time to calculate changes in displacement, distance, velocity, and speed based on position, velocity, and acceleration graphs over time. It provides examples of how integrating areas under curves relates to these physical quantities. Derivative graphs and their relationships are also summarized, along with how different function types integrate or differentiate into other graph types. An example problem calculating the times when a particle is at rest and its maximum velocity is also worked through.
The document discusses inferring gene regulatory networks from time-course gene expression data. It presents the problem of inferring interactions between genes from high-dimensional and sparse time-course microarray data. It proposes using a Gaussian graphical model and introducing biologically grounded priors, such as sparsity and latent clustering of networks, to help address the scarcity of data. Several statistical models and algorithms are described for performing regularized inference on the networks with and without using a known or inferred latent clustering structure. The methods are evaluated on simulated time-course data and a real E. coli S.O.S DNA repair network.
Dsp U Lec07 Realization Of Discrete Time Systemstaha25
This document provides an overview of discrete-time systems and digital signal processing. It discusses discrete-time system components like unit delays and adders. It also covers discrete system networks including FIR and IIR networks. Various realizations of discrete systems are presented, including direct form I and II, cascaded, and parallel realizations. Digital filters are defined and the advantages and disadvantages as well as types (FIR and IIR) are discussed. Design steps and specifications for digital filters are also outlined.
Nonlinear Stochastic Programming by the Monte-Carlo methodSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 4.
More info at http://summerschool.ssa.org.ua
The document describes curve registration through nonparametric testing of Fourier coefficients. It presents a model where curves are represented by their Fourier coefficients with added noise. The hypotheses are that the curves either match up to a shift, or are minimally distant overall after any shift. A generalized likelihood ratio test statistic is developed that minimizes the squared differences between the Fourier coefficients of the two curves over all possible shift values.
Aristidis Likas, Associate Professor and Christoforos Nikou, Assistant Professor, University of Ioannina, Department of Computer Science , Mixture Models for Image Analysis
The document discusses rigidity, gap theorems, and maximum principles for Ricci solitons. It defines a Ricci soliton as rigid if it has the form of N ×Γ Rk, where N is an Einstein manifold and Γ acts freely on N and orthogonally on Rk. It presents theorems showing that compact Ricci solitons that are locally conformally flat or have harmonic Weyl tensor are Einstein. It also discusses results for complete noncompact shrinking Ricci solitons, showing they are rigid if the Weyl tensor is harmonic.
This document discusses the Wiener filter, which is used to estimate an original signal x(n) from a distorted signal y(n). It presents the basic formulation of the Wiener filter, which minimizes the mean square error between the estimated signal x(n) and the original signal by calculating the filter coefficients h(i). It also discusses modeling the distorted signal y(n) using different models, such as a linear model where y(n) is a linear transformation of x(n) with added noise, and convolution models. This allows calculating the filter coefficients without directly observing the original signal x(n).
This algorithm solves the assignment problem for rectangular matrices by finding an optimal one-to-one matching between rows and columns that minimizes the total cost. It begins by preprocessing the matrix to subtract the minimum value from each row and column. It then performs the Hungarian algorithm, labeling matched and unmatched rows and columns, to find the optimal assignment. The time complexity is O(n^3) for an n by n matrix.
1) Cepstral coefficients are commonly used in speech recognition as they are good at distinguishing phonemes and have approximately Gaussian distributions for each phoneme.
2) Line spectrum frequencies are often used in speech coding as they can be quantized to low precision without distorting the spectrum too much.
3) The cepstrum is calculated by taking the inverse Fourier transform of the log spectrum, and the cepstral coefficients describe the rates at which the spectrum changes with frequency.
X2 T08 01 inequalities and graphs (2010)Nigel Simmons
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
This document summarizes research on modeling the resting brain using multi-subject models. It discusses using spatial independent component analysis (ICA) to decompose brain activity into spatial maps that are consistent across subjects. It also discusses estimating functional connectivity networks by imposing sparsity on inverse covariance matrices estimated across subjects. Multi-subject dictionary learning approaches that estimate shared spatial patterns across subjects while modeling subject variability are presented. These approaches aim to overcome challenges from small sample sizes by leveraging information across subjects.
Learning and comparing multi-subject models of brain functional connecitivityGael Varoquaux
High-level brain function arises through functional interactions. These can be mapped via co-fluctuations in activity observed in functional imaging.
First, I first how spatial maps characteristic of on-going activity in a population of subjects can be learned using multi-subject decomposition models extending the popular Independent Component Analysis. These methods single out spatial atoms of brain activity: functional networks or brain regions. With a probabilistic model of inter-subject variability, they open the door to building data-driven atlases of on-going activity.
Subsequently, I discuss graphical modeling of the interactions between brain regions. To learn highly-resolved large scale individual
graphical models models, we use sparsity-inducing penalizations introducing a population prior that mitigates the data scarcity at the subject-level. The corresponding graphs capture better the community structure of brain activity than single-subject models or group averages.
Finally, I address the detection of connectivity differences between subjects. Explicit group variability models of the covariance structure can be used to build optimal edge-level test statistics. On stroke patients resting-state data, these models detect patient-specific functional connectivity perturbations.
ST (Spatial Temporal) Math®: Impact on student progressMarianne McFadden
An action research studying how two middle schools implement the ST Math® program and the level of effectiveness with regard to standardized test results, overall confidence,and academic achievement.
Trigonometric graphs have key vocabulary including amplitude (height), frequency (number of curves in a period), period (time for one curve), domain (x-values or angle in radians), and range (y-values or trig value).
Identify isosceles and equilateral triangles by side length and angle measure
Use the Isosceles Triangle Theorem to solve problems
Use the Equilateral Triangle Corollary to solve problems
* Classify triangles by sides and by angles
* Find the measures of missing angles of right and equiangular triangles
* Find the measures of missing remote interior and exterior angles
This document discusses inverse trigonometric functions. It defines inverse functions as the relation obtained by interchanging the x and y values of a function. It notes that the inverse of a one-to-one function will also be a function, and the domain of a function is the range of its inverse. Graphs of the inverse sine, cosine, and tangent functions are shown between -π and π. Examples are given of evaluating inverse trigonometric functions at exact values and approximate values. Practice problems for students are listed at the end.
1. The document discusses finding values of inverse trigonometric functions by restricting their domains to make them one-to-one functions with inverses. It gives examples of restricting the domains of tan x, sin x, and cos x and finding the inverse values of tan-1(-1), sin-1(-1), and cos-1(-0.5) without using a calculator.
This document discusses inverse trigonometric functions and their differentiation. It begins by defining the six inverse trig functions and explaining that they restrict the domains of the trig functions to make them inverses. It then provides examples of evaluating inverse trig functions. The document notes that while inverse trig functions are transcendental, their derivatives are algebraic. It provides examples of differentiating inverse trig functions. Finally, it discusses analyzing the graph of an inverse trig function by examining its critical points, asymptotes, and points of inflection.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
The document provides information about trigonometry at the foundation and pass levels. At the foundation level, students should be able to solve problems using right-angled triangles, Pythagoras' theorem, and calculating trig functions between 0-90 degrees. At pass level, students must also be able to use trigonometry to calculate triangle area, use sine and cosine rules to solve 2D problems, define trig functions for all values, and calculate areas of circles and arcs. The document then explains how to graph trig functions like sine, cosine, and tangent, noting their periodic properties.
The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.
The document discusses rigidity, gap theorems, and maximum principles for Ricci solitons. It defines a Ricci soliton as rigid if it has the form of N ×Γ Rk, where N is an Einstein manifold and Γ acts freely on N and orthogonally on Rk. It presents theorems showing that compact Ricci solitons that are locally conformally flat or have harmonic Weyl tensor are Einstein. It also discusses results for complete noncompact shrinking Ricci solitons, showing they are rigid if the Weyl tensor is harmonic.
This document discusses the Wiener filter, which is used to estimate an original signal x(n) from a distorted signal y(n). It presents the basic formulation of the Wiener filter, which minimizes the mean square error between the estimated signal x(n) and the original signal by calculating the filter coefficients h(i). It also discusses modeling the distorted signal y(n) using different models, such as a linear model where y(n) is a linear transformation of x(n) with added noise, and convolution models. This allows calculating the filter coefficients without directly observing the original signal x(n).
This algorithm solves the assignment problem for rectangular matrices by finding an optimal one-to-one matching between rows and columns that minimizes the total cost. It begins by preprocessing the matrix to subtract the minimum value from each row and column. It then performs the Hungarian algorithm, labeling matched and unmatched rows and columns, to find the optimal assignment. The time complexity is O(n^3) for an n by n matrix.
1) Cepstral coefficients are commonly used in speech recognition as they are good at distinguishing phonemes and have approximately Gaussian distributions for each phoneme.
2) Line spectrum frequencies are often used in speech coding as they can be quantized to low precision without distorting the spectrum too much.
3) The cepstrum is calculated by taking the inverse Fourier transform of the log spectrum, and the cepstral coefficients describe the rates at which the spectrum changes with frequency.
X2 T08 01 inequalities and graphs (2010)Nigel Simmons
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
This document summarizes research on modeling the resting brain using multi-subject models. It discusses using spatial independent component analysis (ICA) to decompose brain activity into spatial maps that are consistent across subjects. It also discusses estimating functional connectivity networks by imposing sparsity on inverse covariance matrices estimated across subjects. Multi-subject dictionary learning approaches that estimate shared spatial patterns across subjects while modeling subject variability are presented. These approaches aim to overcome challenges from small sample sizes by leveraging information across subjects.
Learning and comparing multi-subject models of brain functional connecitivityGael Varoquaux
High-level brain function arises through functional interactions. These can be mapped via co-fluctuations in activity observed in functional imaging.
First, I first how spatial maps characteristic of on-going activity in a population of subjects can be learned using multi-subject decomposition models extending the popular Independent Component Analysis. These methods single out spatial atoms of brain activity: functional networks or brain regions. With a probabilistic model of inter-subject variability, they open the door to building data-driven atlases of on-going activity.
Subsequently, I discuss graphical modeling of the interactions between brain regions. To learn highly-resolved large scale individual
graphical models models, we use sparsity-inducing penalizations introducing a population prior that mitigates the data scarcity at the subject-level. The corresponding graphs capture better the community structure of brain activity than single-subject models or group averages.
Finally, I address the detection of connectivity differences between subjects. Explicit group variability models of the covariance structure can be used to build optimal edge-level test statistics. On stroke patients resting-state data, these models detect patient-specific functional connectivity perturbations.
ST (Spatial Temporal) Math®: Impact on student progressMarianne McFadden
An action research studying how two middle schools implement the ST Math® program and the level of effectiveness with regard to standardized test results, overall confidence,and academic achievement.
Trigonometric graphs have key vocabulary including amplitude (height), frequency (number of curves in a period), period (time for one curve), domain (x-values or angle in radians), and range (y-values or trig value).
Identify isosceles and equilateral triangles by side length and angle measure
Use the Isosceles Triangle Theorem to solve problems
Use the Equilateral Triangle Corollary to solve problems
* Classify triangles by sides and by angles
* Find the measures of missing angles of right and equiangular triangles
* Find the measures of missing remote interior and exterior angles
This document discusses inverse trigonometric functions. It defines inverse functions as the relation obtained by interchanging the x and y values of a function. It notes that the inverse of a one-to-one function will also be a function, and the domain of a function is the range of its inverse. Graphs of the inverse sine, cosine, and tangent functions are shown between -π and π. Examples are given of evaluating inverse trigonometric functions at exact values and approximate values. Practice problems for students are listed at the end.
1. The document discusses finding values of inverse trigonometric functions by restricting their domains to make them one-to-one functions with inverses. It gives examples of restricting the domains of tan x, sin x, and cos x and finding the inverse values of tan-1(-1), sin-1(-1), and cos-1(-0.5) without using a calculator.
This document discusses inverse trigonometric functions and their differentiation. It begins by defining the six inverse trig functions and explaining that they restrict the domains of the trig functions to make them inverses. It then provides examples of evaluating inverse trig functions. The document notes that while inverse trig functions are transcendental, their derivatives are algebraic. It provides examples of differentiating inverse trig functions. Finally, it discusses analyzing the graph of an inverse trig function by examining its critical points, asymptotes, and points of inflection.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
The document provides information about trigonometry at the foundation and pass levels. At the foundation level, students should be able to solve problems using right-angled triangles, Pythagoras' theorem, and calculating trig functions between 0-90 degrees. At pass level, students must also be able to use trigonometry to calculate triangle area, use sine and cosine rules to solve 2D problems, define trig functions for all values, and calculate areas of circles and arcs. The document then explains how to graph trig functions like sine, cosine, and tangent, noting their periodic properties.
The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.
t5 graphs of trig functions and inverse trig functionsmath260
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The geometric mean is a type of average that indicates the central tendency of a set of numbers using their product, as opposed to the arithmetic mean which uses their sum, and it is calculated by taking the nth root of the product of the numbers. The geometric mean is more appropriate than the arithmetic mean for describing proportional growth and ratios, and it has various applications in fields like optics, signal processing, geometry, and finance.
The document discusses trigonometric graphs and identities. It covers graphing sine and cosine functions including amplitude, period, translations involving phase shift and vertical shift. It also provides an example of using trigonometric functions to model real-life tidal data and finds the depth at specific times as well as time periods when a boat can safely dock.
The document discusses how to graph trigonometric functions like sine and cosine. It explains that the sine function has a range from -1 to 1 and is decreasing in quadrants 1 and 2. The cosine function has the same range and properties as sine, except its zeros occur at odd integer multiples of π/2. Both functions have an amplitude of 1 and a period of 2π. The document also describes how changing the coefficients in trigonometric functions affects their amplitude, period, and horizontal or vertical shifts.
Tissues are groups of cells that work together to perform specific functions. The main tissue types are epithelial, connective, muscle and nervous tissue. Epithelial tissue covers organs and forms glands, and is made up of closely packed cells. Connective tissue connects and supports other tissues and includes bone, cartilage and blood. Muscle tissue includes cardiac, skeletal and smooth muscle that contract to produce movement.
This document discusses exponential functions and graphs. It provides examples of exponential growth and decay functions, shows how to write exponential functions based on patterns in tables, and how to graph exponential functions. Key aspects of exponential functions covered are their domains of all real numbers, ranges of positive values for growth and negative values for decay, and how the base affects whether the graph is a vertical stretch or shrink. Sample homework problems are also presented on graphing and analyzing exponential decay functions.
The document provides information about trigonometric graphs including:
- The graphs of y=sinx and y=cosx and their key properties like period, amplitude, maximum and minimum values.
- How the period of a trig function is determined based on its argument.
- A generalized form for sinusoidal functions showing how the amplitude, period, horizontal and vertical shifts affect the graph.
- Examples of specific sinusoidal functions are discussed step-by-step and their graphs depicted based on analyzing the amplitude, period, and shift values.
Key graphs and concepts about trigonometric functions and sinusoidal functions are summarized within 3 sentences.
The document discusses numerical integration methods for calculating ship geometrical properties. It introduces trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule for numerical integration. Simpson's 1st rule is recommended for calculating properties like waterplane area, sectional area, submerged volume, and centers of floatation and buoyancy which involve integrating curves related to the ship's shape. Detailed steps are provided for applying Simpson's 1st rule to calculate these properties numerically.
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type of curve, it gives the standard form of the equation and notes on identifying features like intercepts, vertices, and the overall shape of the graph. Sketching curves involves finding intercepts and using a table of values to plot points if needed.
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type of curve, it gives the standard form of the equation and notes on identifying features like intercepts, vertices, and asymptotes to help sketch the graph. It emphasizes using factorized forms, intercepts, and tables of values to determine a curve's shape.
The document discusses how to sketch graphs based on their equations. It provides the following key points:
- Numbers on axes must be evenly spaced. The y-intercept occurs when x=0 and the x-intercept occurs when y=0.
- Common curves include straight lines, parabolas, cubics, and higher order polynomials. Parabolas have x-intercepts found by solving the equation for where it equals 0.
- Higher order polynomials become flatter at the base and steeper on the sides as the power increases. Hyperbolas can be defined by equations like y=1/x or xy=1.
Once intercepts are found, curves can be sketched by
The document provides information on sketching graphs of basic curves. It lists 8 types of basic curves: (1) straight lines, (2) parabolas, (3) cubics, (4) higher powers, (5) hyperbolas, (6) circles, (7) exponentials, and (8) roots. For each type it provides the standard form of the equation and notes on identifying features like intercepts, vertices, and behavior as powers increase. It emphasizes using standard forms, intercepts, and factoring to determine a curve's shape.
The document discusses shifting curves by multiplying the function f(x) by a constant k. It states that if k > 1, the curve is steeper, and if 0 < k < 1, the curve is shallower. It also discusses shifting the curve by replacing x with kx, which stretches the curve horizontally. Examples are given of shifting the curves y = x, y = x2, and y = 1/x.
The document discusses how shifting curves vertically and horizontally by multiplying or dividing the independent and dependent variables by constants. Vertically shifting a curve by multiplying y by a constant k stretches the curve vertically, leaving the domain unchanged but altering the range. Horizontally shifting by multiplying x by k stretches the curve horizontally, altering the domain but leaving the range unchanged. Examples of shifting simple curves like lines and parabolas are shown to illustrate these transformations.
The document discusses how shifting curves vertically and horizontally by multiplying or dividing the independent and dependent variables by constants. Vertically shifting a curve by multiplying y by a constant k stretches the curve vertically, leaving the domain unchanged but altering the range. Horizontally shifting by multiplying x by k stretches the curve horizontally, altering the domain but leaving the range unchanged. Examples of shifting simple curves like lines and parabolas are shown to illustrate these transformations.
The document discusses how shifting curves vertically and horizontally affects their shape and properties. Vertically shifting a curve by a factor of k, where k>1 makes the curve steeper and k<1 makes it shallower. Horizontally shifting by a factor of k also affects the steepness in the same way and changes the domain. Examples of shifting simple curves like y=x are shown to illustrate these effects on the curve shape, intercepts and asymptotes.
The document describes properties of trigonometric functions including sine, cosine, and tangent. It discusses key features of their graphs such as period, amplitude, domain, range, and intercepts. Examples are provided to demonstrate how to sketch the graphs of trigonometric functions using these properties. Key points, periods, and asymptotes are calculated and graphs are drawn.
The document describes properties of trigonometric functions including sine, cosine, and tangent. It discusses key features of their graphs such as period, amplitude, domain, range, and intercepts. Examples are provided to demonstrate how to sketch the graphs of trigonometric functions using these properties. Key points, periods, and asymptotes are calculated and graphs are drawn.
This document discusses Infinite Impulse Response (IIR) filters. IIR filters are more computationally efficient than FIR filters as they require fewer coefficients due to using feedback. The document covers IIR filter concepts, properties, design procedures including specification, coefficient calculation, structure selection, and implementation. It also provides examples of coefficient calculation methods like pole-zero placement and the bilinear transform method for converting analog filters to digital IIR filters.
This document contains solutions to exercises from a pre-calculus textbook on radical functions.
1) It provides tables, graphs and explanations for various radical functions such as √x, √x+3, and their relation to other functions.
2) Students are asked to sketch graphs of radical functions based on given quadratic, cubic or other functions, and identify domains and ranges.
3) Radical equations are solved by graphing related functions and finding the x-intercept(s).
This document contains a summary of key concepts in algebra, geometry, and trigonometry:
1) Algebra topics include arithmetic operations, factoring, exponents, binomials, and the quadratic formula.
2) Geometry topics cover lines, triangles, circles, spheres, cones, cylinders, sectors, and trapezoids including formulas for area, perimeter, volume, and surface area.
3) Trigonometry definitions and formulas are provided for sine, cosine, tangent, cotangent, addition, subtraction, and half-angle identities.
This document contains a summary of key concepts in algebra, geometry, and trigonometry:
1) Algebra topics include arithmetic operations, factoring, exponents, binomials, and the quadratic formula.
2) Geometry topics cover lines, triangles, circles, spheres, cones, cylinders, sectors, and trapezoids including formulas for area, perimeter, volume, and surface area.
3) Trigonometry definitions and formulas are provided for sine, cosine, tangent, cotangent, addition, subtraction, and half-angle identities.
The document discusses exponential functions of the form f(x) = a^x where a is a constant. It provides examples of exponential functions with a = 2 and a = 1/2. It describes the domain, range, and common point of exponential functions. The document also discusses transformations of exponential functions by adding or subtracting constants, and provides examples of sketching and describing transformed exponential functions. Finally, it lists common exponential expressions.
This document provides a summary of key concepts in algebra, geometry, trigonometry and their definitions. It includes formulas and properties for lines, polynomials, exponents, trig functions, triangles, circles, spheres, cones, cylinders, distance and the quadratic formula. Key topics covered are factoring, binomials, slope-intercept form, trig ratios, trig identities, trig reciprocals and the Pythagorean identities.
This document provides solutions to exercises about analyzing and graphing rational functions. Some key points summarized:
- The exercises involve identifying vertical and horizontal asymptotes, holes, and domains by factoring rational functions. Oblique asymptotes are also determined.
- Graphing technology is used to verify characteristics like asymptotes and holes, and determine zeros of related functions.
- Students are asked to match rational functions to their graphs based on identified characteristics like asymptotes, holes, and behavior near non-permissible values.
Similar to 12X1 T03 02 graphing trig functions (20)
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
4. Graphing Trig
1) Sine Curve
Functions
y
1
2 2 3 4 x
-1
5. Graphing Trig
1) Sine Curve
Functions
y
1
2 2 3 4 x
-1
6. Graphing Trig
1) Sine Curve
Functions
y
1
2 2 3 4 x
-1
7. Graphing Trig
1) Sine Curve
Functions
y
1
2 2 3 4 x
-1
8. Graphing Trig
1) Sine Curve
Functions
y
1 y sin x
2 2 3 4 x
-1
9. Graphing Trig
1) Sine Curve
Functions
y
1 y sin x
2 2 3 4 x
-1
domain : all real x
10. Graphing Trig
1) Sine Curve
Functionsy
1 y sin x
2 2 3 4 x
-1
domain : all real x
range : - 1 y 1
11. Graphing Trig
1) Sine Curve
Functionsy
period
1 y sin x
2 2 3 4 x
-1
domain : all real x
range : - 1 y 1
12. Graphing Trig
1) Sine Curve
Functionsy
period
1 y sin x
2 2 3 4 x
-1
In general;
domain : all real x y a sin bx c
range : - 1 y 1 2
period units
b
13. Graphing Trig
1) Sine Curve
Functions
y
period
1 y sin x
amplitude
2 2 3 4 x
-1
In general;
domain : all real x y a sin bx c
range : - 1 y 1 2
period units
b
14. Graphing Trig
1) Sine Curve
Functions
y
period
1 y sin x
amplitude
2 2 3 4 x
-1
In general;
domain : all real x y a sin bx c
range : - 1 y 1 2
period units
b
amplitude a units
15. Graphing Trig
1) Sine Curve
Functions
y
period
1 y sin x
amplitude
2 2 3 4 x
-1
In general;
domain : all real x y a sin bx c
range : - 1 y 1 2 period
period units divisions
b 4
amplitude a units
16. Graphing Trig
1) Sine Curve
Functions
y
period
1 y sin x
amplitude
2 2 3 4 x
-1
In general;
domain : all real x y a sin bx c
range : - 1 y 1 2 period
period units divisions
b 4
c
amplitude a units shift to left
b
18.
e.g. y 5 sin 9 x
2
2 period units
9
19.
e.g. y 5 sin 9 x
2
2 period units
9
amplitude 5 units
20.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units
21.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units shift to right
18
22.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units shift to right
18
y
5
2
x
9 -5 9 9
23.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units shift to right
18
y
5
2
x
9 -5 9 9
24.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units shift to right
18
y
5
2
x
9 -5 9 9
25.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units shift to right
18
y
5
2
x
9 -5 9 9
26.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units shift to right
18
y
5
2
x
9 -5 9 9
27.
e.g. y 5 sin 9 x
2
2 period units divisions
9 18
amplitude 5 units shift to right
18
y 9x
y 5 sin
5 2
2
x
9 -5 9 9
31. 2) Cosine Curve y a cosbx c
2
period units
b
amplitude a units
32. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
amplitude a units
33. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
34. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2
e.g. y 4 cos
8
35. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period
8 1
8
16
36. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period
8 1
8
16
amplitude 4
37. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8
16
amplitude 4
38. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8 shift 8 to left, 2 up,
16 upside down
amplitude 4
39. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8 shift 8 to left, 2 up,
16 upside down
y
amplitude 4
6
2
8 8 16 x
-2
40. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8 shift 8 to left, 2 up,
16 upside down
y
amplitude 4
6
2
8 8 16 x
-2
41. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8 shift 8 to left, 2 up,
16 upside down
y
amplitude 4
6
2
8 8 16 x
-2
42. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8 shift 8 to left, 2 up,
16 upside down
y
amplitude 4
6
2
8 8 16 x
-2
43. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8 shift 8 to left, 2 up,
16 upside down
y
amplitude 4
6
2
8 8 16 x
-2
44. 2) Cosine Curve y a cosbx c
2 period
period units divisions
b 4
c
amplitude a units shift to left
b
x 2 2
e.g. y 4 cos period divisions 4
8 1
8 shift 8 to left, 2 up,
16 upside down
y
amplitude 4
y 4 cos 2
x
6
8
2
8 8 16 x
-2
48. 3) Tangent Curve y a tan bx c divisions
period
2
period units
b
49. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
50. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2
51. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2
period
1
52. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2 1
period divisions
2
1
53. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2 1
period divisions
2
1 shift 2 to right
54. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2 1
period divisions
2
1 shift 2 to right
y
1 2 x
55. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2 1
period divisions
2
1 shift 2 to right
y
1 2 x
56. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2 1
period divisions
2
1 shift 2 to right
y
1 2 x
57. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2 1
period divisions
2
1 shift 2 to right
y
y e tan x 2
1 2 x
58. 3) Tangent Curve y a tan bx c divisions
period
c
2
period units shift to left
b b
e.g. y e tan x 2 1
period divisions
2
1 shift 2 to right
y
y e tan x 2
Exercise 14C; 2b, 3b,
1 2 x 4b, 5bce, 8, 9, 10b, 13,
15, 16, 17, 20