This document provides information on various mathematical topics including:
1. Graphs of polynomial functions in factorized form such as quadratics, cubics, and quartics.
2. Transformations of functions including translations, reflections, dilations, and their effects on graphs.
3. Exponential, logarithmic, and trigonometric functions and their graphs.
4. Relations, functions, and tests to determine if a relation is a function and if a function is one-to-one or many-to-one.
Bai giang ham so kha vi va vi phan cua ham nhieu bienNhan Nguyen
This document introduces differentials in functions of several variables. It begins with a review of differentials in two variables using differentials dx and dy. It then extends the concept to functions of several variables, where the total differential dz is defined as the sum of its partial derivatives with respect to each variable times the differentials of those variables. Examples are provided to demonstrate calculating total differentials and comparing them to actual changes. The relationship between differentiability and continuity is also discussed.
This document provides instructions for a MATLAB assignment with two parts. Part I involves constructing Lagrange interpolants for a given function. Students are asked to create MATLAB function files for Lagrange interpolation and for defining a test function, as well as a script file to test the interpolation. Part II involves solving a system of linear ordinary differential equations and constructing the solution at discrete time points. Students are asked to create a function file to solve the ODE using eigenvalues and eigenvectors, and a script file to test it on a sample problem. Detailed hints are provided for both parts.
This document provides a calculus cheat sheet covering key topics in limits, derivatives, and integrals. It defines limits, including one-sided limits and limits at infinity. Properties of limits are listed. Derivatives are defined and basic rules like the power, constant multiple, sum, difference, and chain rules are covered. Common derivatives are provided. Higher order derivatives and the second derivative are defined. Evaluation techniques like L'Hospital's rule, polynomials at infinity, and piecewise functions are summarized.
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
BBMP1103 - Sept 2011 exam workshop - part 8Richard Ng
This document summarizes steps to solve constrained optimization problems using Lagrange multipliers. It provides an example of finding the minimum value of the function f(x,y)=5x^2-6y^2-xy subject to the constraint x+2y=24. The steps are: [1] Express the constraint as g(x,y)=0, [2] Form the Lagrange function F(x,y,λ)=f(x,y)-λg(x,y), [3] Take partial derivatives and set equal to 0, [4] Solve the system of equations for a minimum of (6,9). Additional practice problems and questions are also presented.
Bai giang ham so kha vi va vi phan cua ham nhieu bienNhan Nguyen
This document introduces differentials in functions of several variables. It begins with a review of differentials in two variables using differentials dx and dy. It then extends the concept to functions of several variables, where the total differential dz is defined as the sum of its partial derivatives with respect to each variable times the differentials of those variables. Examples are provided to demonstrate calculating total differentials and comparing them to actual changes. The relationship between differentiability and continuity is also discussed.
This document provides instructions for a MATLAB assignment with two parts. Part I involves constructing Lagrange interpolants for a given function. Students are asked to create MATLAB function files for Lagrange interpolation and for defining a test function, as well as a script file to test the interpolation. Part II involves solving a system of linear ordinary differential equations and constructing the solution at discrete time points. Students are asked to create a function file to solve the ODE using eigenvalues and eigenvectors, and a script file to test it on a sample problem. Detailed hints are provided for both parts.
This document provides a calculus cheat sheet covering key topics in limits, derivatives, and integrals. It defines limits, including one-sided limits and limits at infinity. Properties of limits are listed. Derivatives are defined and basic rules like the power, constant multiple, sum, difference, and chain rules are covered. Common derivatives are provided. Higher order derivatives and the second derivative are defined. Evaluation techniques like L'Hospital's rule, polynomials at infinity, and piecewise functions are summarized.
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
BBMP1103 - Sept 2011 exam workshop - part 8Richard Ng
This document summarizes steps to solve constrained optimization problems using Lagrange multipliers. It provides an example of finding the minimum value of the function f(x,y)=5x^2-6y^2-xy subject to the constraint x+2y=24. The steps are: [1] Express the constraint as g(x,y)=0, [2] Form the Lagrange function F(x,y,λ)=f(x,y)-λg(x,y), [3] Take partial derivatives and set equal to 0, [4] Solve the system of equations for a minimum of (6,9). Additional practice problems and questions are also presented.
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
1) Euler-Bernoulli bending theory and Timoshenko beam theory describe the stresses and deflections of beams under bending loads.
2) Euler-Bernoulli theory assumes a beam's cross-section remains plane and perpendicular to the neutral axis during bending. Timoshenko theory accounts for shear deformation.
3) Both theories relate the bending moment M and shear force V to the beam's deflection w and its derivatives, allowing calculation of stresses, forces, and deflections for given beam geometries and loads.
The document provides an overview of constrained optimization using Lagrange multipliers. It begins with motivational examples of constrained optimization problems and then introduces the method of Lagrange multipliers, which involves setting up equations involving the functions to optimize and constrain and a Lagrange multiplier. Examples are worked through to demonstrate solving these systems of equations to find critical points. Caution is advised about dividing equations where one side could be zero. A contour plot example visually depicts the constrained critical points.
The document describes the process of integration by partial fractions. It explains that when the degree of the numerator is greater than or equal to the denominator, division is performed. Otherwise, the denominator is factored. For each linear factor, the numerator is written as a sum of terms divided by that factor. For multiple linear factors, the numerator is written as a sum of terms divided by powers of that factor. Examples are provided to demonstrate these steps.
This document contains 20 multiple integral exercises with solutions. Some of the exercises involve calculating double integrals over specified regions, while others involve setting up approximations of double integrals using Riemann sums. Exercise 19 involves sketching solid regions in 3D space and Exercise 20 involves sketching surfaces defined by z=f(x,y).
The document discusses support vector machines (SVM) for classification. It begins by introducing the concepts of maximum margin hyperplane and soft margin. It then formulates the SVM optimization problem to find the maximum margin hyperplane using Lagrange multipliers. The optimization problem is solved using Kuhn-Tucker conditions to obtain the dual formulation only in terms of the support vectors. Kernel tricks are introduced to handle non-linear decision boundaries. The formulation is extended to allow for misclassification errors by introducing slack variables ξ and a penalty parameter C.
Profº. Marcelo Santos Chaves - Cálculo I (Limites e Continuidades) - Exercíci...MarcelloSantosChaves
1. The document discusses limits and continuities. It provides solutions to calculating the limits of 6 different functions as x approaches certain values.
2. The solutions involve algebraic manipulations such as factoring, simplifying, and applying limit properties. Various limit results are obtained such as 1, -6, 0.
3. The techniques demonstrated include making substitutions to simplify indeterminate forms, factoring, and taking limits of rational functions as the variables approach certain values.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
Edhole School provides best Information about Schools in India, Delhi, Noida, Gurgaon. Here you will get about the school, contact, career, etc. Edhole Provides best study material for school students."
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps for graphing each type of combination function, which include separately graphing the individual functions and then using properties of the combination to determine points and features of the combined graph. An example is worked through for each type of combination function.
The document summarizes key concepts from Lesson 28 on Lagrange multipliers, including:
1) Restating the method of Lagrange multipliers and providing justifications through elimination, graphical, and symbolic approaches.
2) Discussing second order conditions for constrained optimization problems, noting the importance of compact feasibility sets.
3) Providing the theorem on Lagrange multipliers and examples of its application to problems with more than two variables or multiple constraints.
The document discusses various techniques for constructing shadows and lighting effects in 3D computer graphics, including using projection matrices to generate shadow polygons and accounting for factors like light source positioning, radial intensity attenuation, and surface reflectance properties. It also examines methods for animating camera movement and introducing texture mapping to surfaces.
Lesson 13: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is notes for a calculus class covering derivatives of exponential and logarithmic functions. It includes:
- Announcements about upcoming review sessions and an exam on sections 1.1-2.5.
- An outline of topics to be covered, including derivatives of the natural exponential function, natural logarithm function, other exponentials/logarithms, and logarithmic differentiation.
- Definitions and properties of exponential functions, the natural number e, and logarithmic functions.
- Examples of graphs of various exponential and logarithmic functions.
- Derivatives of exponential functions and proofs involving limits.
Quantum fields on the de sitter spacetime - Ion CotaescuSEENET-MTP
This document summarizes research on defining quantum fields on de Sitter spacetime. It discusses how external symmetries allow defining conserved observables like an energy operator, despite doubts in the literature. New quantum modes were obtained for scalar, Dirac, and vector fields on de Sitter spacetime using this energy operator. The paper reviews defining fields in local frames where spin is well-defined, and how isometries give rise to conserved quantities through external symmetry transformations that involve gauge transformations and diffeomorphisms. Generators of field representations are constructed from orbital and spin parts related to Killing vectors and structure functions.
The document discusses generating functions. It defines a generating function G(z) as a power series representation of a sequence <an> = a0, a1, a2, ... . Properties of generating functions include that differentiating or multiplying generating functions results in new generating functions, and that generating functions can reveal relationships between sequences.
The document discusses Laplace's equation, which describes steady state situations in physics. It provides examples of using separation of variables to solve Laplace's equation in Cartesian and polar coordinates. These include problems in electrostatics, stress analysis, and heat transfer. The key points are that Laplace's equation has no time dependence and describes steady state distributions, and separation of variables is a common technique for solving problems described by this equation in different coordinate systems.
This document discusses feature extraction in computer vision systems. It focuses on edge and corner detection methods. Edge detection aims to locate boundaries between objects and background in images. Common approaches discussed include Sobel and Canny edge detectors, which apply first and second derivative filters to detect edges. Corner detection aims to find stable points of interest across images for tracking objects. It involves computing the eigenvalues of a matrix formed from the image gradient to identify corners.
The document provides an overview of techniques for solving different types of ordinary differential equations (ODEs):
1. It describes prototypes and solution methods for various types of first-order ODEs, including separable, exact, homogeneous, Bernoulli, and linear.
2. It discusses techniques for solving second and higher-order linear ODEs with constant or Cauchy-Euler coefficients, including the auxiliary equation and using the variation of parameters or undetermined coefficients methods.
3. It mentions series solutions centered around x=0.
4. For homogeneous systems of ODEs, it outlines converting between system and matrix forms, finding eigenvalues and eigenvectors, and using the eigenstructure to solve
The document outlines methods for graphing functions that involve addition, subtraction, multiplication, and division of other functions. It provides steps such as graphing the individual functions separately first before combining them based on the operation. Examples are given to illustrate each method, including identifying points where the individual functions are equal to 0 or 1 and investigating asymptotes.
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
1) Euler-Bernoulli bending theory and Timoshenko beam theory describe the stresses and deflections of beams under bending loads.
2) Euler-Bernoulli theory assumes a beam's cross-section remains plane and perpendicular to the neutral axis during bending. Timoshenko theory accounts for shear deformation.
3) Both theories relate the bending moment M and shear force V to the beam's deflection w and its derivatives, allowing calculation of stresses, forces, and deflections for given beam geometries and loads.
The document provides an overview of constrained optimization using Lagrange multipliers. It begins with motivational examples of constrained optimization problems and then introduces the method of Lagrange multipliers, which involves setting up equations involving the functions to optimize and constrain and a Lagrange multiplier. Examples are worked through to demonstrate solving these systems of equations to find critical points. Caution is advised about dividing equations where one side could be zero. A contour plot example visually depicts the constrained critical points.
The document describes the process of integration by partial fractions. It explains that when the degree of the numerator is greater than or equal to the denominator, division is performed. Otherwise, the denominator is factored. For each linear factor, the numerator is written as a sum of terms divided by that factor. For multiple linear factors, the numerator is written as a sum of terms divided by powers of that factor. Examples are provided to demonstrate these steps.
This document contains 20 multiple integral exercises with solutions. Some of the exercises involve calculating double integrals over specified regions, while others involve setting up approximations of double integrals using Riemann sums. Exercise 19 involves sketching solid regions in 3D space and Exercise 20 involves sketching surfaces defined by z=f(x,y).
The document discusses support vector machines (SVM) for classification. It begins by introducing the concepts of maximum margin hyperplane and soft margin. It then formulates the SVM optimization problem to find the maximum margin hyperplane using Lagrange multipliers. The optimization problem is solved using Kuhn-Tucker conditions to obtain the dual formulation only in terms of the support vectors. Kernel tricks are introduced to handle non-linear decision boundaries. The formulation is extended to allow for misclassification errors by introducing slack variables ξ and a penalty parameter C.
Profº. Marcelo Santos Chaves - Cálculo I (Limites e Continuidades) - Exercíci...MarcelloSantosChaves
1. The document discusses limits and continuities. It provides solutions to calculating the limits of 6 different functions as x approaches certain values.
2. The solutions involve algebraic manipulations such as factoring, simplifying, and applying limit properties. Various limit results are obtained such as 1, -6, 0.
3. The techniques demonstrated include making substitutions to simplify indeterminate forms, factoring, and taking limits of rational functions as the variables approach certain values.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
Likelihood is sometimes difficult to compute because of the complexity of the model. Approximate Bayesian computation (ABC) makes it easy to sample parameters generating approximation of observed data.
Edhole School provides best Information about Schools in India, Delhi, Noida, Gurgaon. Here you will get about the school, contact, career, etc. Edhole Provides best study material for school students."
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps for graphing each type of combination function, which include separately graphing the individual functions and then using properties of the combination to determine points and features of the combined graph. An example is worked through for each type of combination function.
The document summarizes key concepts from Lesson 28 on Lagrange multipliers, including:
1) Restating the method of Lagrange multipliers and providing justifications through elimination, graphical, and symbolic approaches.
2) Discussing second order conditions for constrained optimization problems, noting the importance of compact feasibility sets.
3) Providing the theorem on Lagrange multipliers and examples of its application to problems with more than two variables or multiple constraints.
The document discusses various techniques for constructing shadows and lighting effects in 3D computer graphics, including using projection matrices to generate shadow polygons and accounting for factors like light source positioning, radial intensity attenuation, and surface reflectance properties. It also examines methods for animating camera movement and introducing texture mapping to surfaces.
Lesson 13: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is notes for a calculus class covering derivatives of exponential and logarithmic functions. It includes:
- Announcements about upcoming review sessions and an exam on sections 1.1-2.5.
- An outline of topics to be covered, including derivatives of the natural exponential function, natural logarithm function, other exponentials/logarithms, and logarithmic differentiation.
- Definitions and properties of exponential functions, the natural number e, and logarithmic functions.
- Examples of graphs of various exponential and logarithmic functions.
- Derivatives of exponential functions and proofs involving limits.
Quantum fields on the de sitter spacetime - Ion CotaescuSEENET-MTP
This document summarizes research on defining quantum fields on de Sitter spacetime. It discusses how external symmetries allow defining conserved observables like an energy operator, despite doubts in the literature. New quantum modes were obtained for scalar, Dirac, and vector fields on de Sitter spacetime using this energy operator. The paper reviews defining fields in local frames where spin is well-defined, and how isometries give rise to conserved quantities through external symmetry transformations that involve gauge transformations and diffeomorphisms. Generators of field representations are constructed from orbital and spin parts related to Killing vectors and structure functions.
The document discusses generating functions. It defines a generating function G(z) as a power series representation of a sequence <an> = a0, a1, a2, ... . Properties of generating functions include that differentiating or multiplying generating functions results in new generating functions, and that generating functions can reveal relationships between sequences.
The document discusses Laplace's equation, which describes steady state situations in physics. It provides examples of using separation of variables to solve Laplace's equation in Cartesian and polar coordinates. These include problems in electrostatics, stress analysis, and heat transfer. The key points are that Laplace's equation has no time dependence and describes steady state distributions, and separation of variables is a common technique for solving problems described by this equation in different coordinate systems.
This document discusses feature extraction in computer vision systems. It focuses on edge and corner detection methods. Edge detection aims to locate boundaries between objects and background in images. Common approaches discussed include Sobel and Canny edge detectors, which apply first and second derivative filters to detect edges. Corner detection aims to find stable points of interest across images for tracking objects. It involves computing the eigenvalues of a matrix formed from the image gradient to identify corners.
The document provides an overview of techniques for solving different types of ordinary differential equations (ODEs):
1. It describes prototypes and solution methods for various types of first-order ODEs, including separable, exact, homogeneous, Bernoulli, and linear.
2. It discusses techniques for solving second and higher-order linear ODEs with constant or Cauchy-Euler coefficients, including the auxiliary equation and using the variation of parameters or undetermined coefficients methods.
3. It mentions series solutions centered around x=0.
4. For homogeneous systems of ODEs, it outlines converting between system and matrix forms, finding eigenvalues and eigenvectors, and using the eigenstructure to solve
The document outlines methods for graphing functions that involve addition, subtraction, multiplication, and division of other functions. It provides steps such as graphing the individual functions separately first before combining them based on the operation. Examples are given to illustrate each method, including identifying points where the individual functions are equal to 0 or 1 and investigating asymptotes.
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps such as graphing the individual functions separately first before combining using ordinates, examining the sign of products, and identifying asymptotes. An example for each operation is worked through step-by-step.
This document discusses transformations of functions. It defines various types of transformations including vertical and horizontal stretches and shifts, reflections, and periodic transformations. It provides examples of functions and their transformations. It also discusses even and odd functions. The key points are that transformations can stretch, shrink, shift, or reflect the graph of a function and that even functions are symmetric about the y-axis while odd functions are symmetric about the origin.
The document is notes for a lesson on tangent planes. It provides definitions of tangent lines and planes, formulas for finding equations of tangent lines and planes, and examples of applying these concepts. Specifically, it defines that the tangent plane to a function z=f(x,y) through the point (x0,y0,z0) has normal vector (f1(x0,y0), f2(x0,y0),-1) and equation f1(x0,y0)(x-x0) + f2(x0,y0)(y-y0) - (z-z0) = 0 or z = f(x0,y0) +
The document discusses the cross product of vectors in R3. It begins by defining the cross product as a vector z that is orthogonal to two given vectors x and y. It then shows that z can be uniquely defined as z = (x2y3 - x3y2, x3y1 - x1y3, x1y2 - x2y1). Several properties of the cross product are then discussed, including that it is anticommutative and relates to the area of the parallelogram formed by x and y. The cross product allows computing volumes of parallelepipeds in R3 and relates to both the dot product and scalar triple product of vectors.
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 contains solutions to checkpoint questions about transforming graphs of functions. It includes examples of translating graphs by shifting them horizontally and vertically based on changes to the x and y variables in the function. It also contains an example of reflecting a graph across the x-axis. The questions require sketching the transformed graphs on grids and writing the equations of the transformed functions based on the given transformations.
1. The document provides examples and explanations of concepts in solid geometry including the three dimensional coordinate system, distance formula in three space, and equations for planes, spheres, cylinders, quadric surfaces, and their graphs.
2. Key solid geometry concepts covered include plotting points in three dimensions, finding distances between points and distances from a point to a plane, midpoint formulas, and standard and general equations for planes, spheres, cylinders, ellipsoids, hyperboloids, and paraboloids.
3. Examples are given for graphing equations of a plane, sphere, circular cylinder, parabolic cylinder, and their relation to the standard equations.
This document contains a math lab exercise on graphing rational functions with 4 questions. Question 1 has students predict holes in graphs of rational functions. Question 2 has students predict vertical asymptotes. Question 3 has students predict which graphs have horizontal asymptotes. The document provides step-by-step solutions and explanations for each question.
1. The document provides instructions for sketching exponential and logarithmic graphs. Students are asked to sketch various graphs of exponential and logarithmic functions, noting the shape, asymptotes, and intercepts.
2. Students are asked to describe how changing components of exponential and logarithmic equations, such as coefficients and signs, affects the graphs.
3. The objectives are for students to be able to sketch exponential and logarithmic graphs accurately, including asymptotes and intercepts, and draw straight lines to solve equations.
This document discusses rational functions, which are defined as the ratio of two polynomials. It provides examples of specific rational functions and examines their key properties including vertical and horizontal asymptotes. It discusses how to predict asymptotes from the polynomials and emphasizes the importance of graphing to verify predictions. Guidelines are provided for accurately graphing rational functions on graph paper including showing intercepts, extrema, asymptotes, holes, and using proper scaling.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
C:\Documents And Settings\Smapl\My Documents\Sttj 2010\P& P Berkesan 2010...zabidah awang
Linear programming involves solving problems related to linear inequalities and equations. There are four main steps: (1) write the linear equations and inequalities that describe the situation, (2) graph the equations and shade the feasible region, (3) determine and graph the objective function, (4) determine the optimum value of the objective function graphically. Key skills include identifying and shading feasible regions, writing linear equations and inequalities, and solving problems with integer values of variables within given constraints.
Lesson 26: The Fundamental Theorem of Calculus (Section 4 version)Matthew Leingang
The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.
This document provides examples and explanations of transformations of trigonometric functions including phase shifts and vertical/horizontal shifts. It discusses how to write alternative equations for shifted trig functions by following patterns of + and - signs. Examples are provided comparing graphs of original and transformed trig functions to illustrate various shifts. The document also discusses concepts of phase relationships between voltage and current in AC circuits including being in phase, out of phase, and maximum inductance occurring when voltage leads current by a phase of π/2 radians. Homework problems from p. 282 #1-20 are assigned.
Ch 6.1 & 6.2 Slope of Parallel and Perpendicular Linesmdicken
This document discusses slope, parallel lines, and perpendicular lines. It defines slope as the steepness of a graph line and explains that two points are needed to calculate slope using the rise over run formula. Parallel lines have equal slopes and never intersect, while perpendicular lines have slopes that are negative reciprocals and intersect at 90 degrees. Examples are provided to demonstrate calculating slope and determining if lines are parallel, perpendicular, or neither.
Here are the key steps to find the instantaneous rate of change using a graphing calculator:
1. Graph the function over the appropriate domain.
2. Use the arrow keys to move the cursor to the point where you want to find the instantaneous rate of change.
3. Press the TRACE button and select the tangent option.
4. The calculator will display the slope of the tangent line, which is the instantaneous rate of change at that point.
5. For example, if finding the IROC at x=1 for the function f(x) = x3, you would:
a) Graph f(x) = x3
b) Use arrows to move cursor
Wedderburn College Newsletter 28th March 2012coburgmaths
The document provides information about upcoming parent-teacher-student interviews at Wedderburn College in Victoria, Australia. It discusses that the interviews will take place on March 29th in Wedderburn and March 28th in Inglewood to discuss student learning. It also mentions several other events happening at the end of the term like a casual day fundraiser and working bees planned for the start of next term.
2011 Whole School Action Plan Curriculum PD Presentationcoburgmaths
This document outlines a curriculum professional development session focused on developing a whole school action plan. Key points discussed include:
- The session focused on improving understanding of the e5 framework and identifying different types of questioning.
- A whole school action plan was proposed to improve student learning outcomes through a collaborative approach focusing on areas like learning intentions, feedback, and higher-order questioning.
- Data was presented indicating the need for improvement, and professional learning teams would be formed to design inquiries around addressing this need through collaborative learning and feedback cycles.
This document describes the features of an online platform called Ultranet, including a personalized home page for all users, an express page for personal space, a collaborative space for student learning and work, a community space, a professional development space for teachers, and a content storage and search area called "My content".
This document describes the features of an online platform called Ultranet, including a personalized home page for all users, an express page for personal space, a collaborative space for student learning and work, a community space, a professional development space for teachers, and a content storage and search area called "My content".
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 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 boosts blood flow, releases endorphins, and promotes changes in the brain which help regulate emotions and stress levels.
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 boost feelings of calmness and well-being.
This document discusses a multi-modal approach to teaching mathematics that involves exploring, engaging, explaining, elaborating and evaluating using different modes such as numbers, words, stories, real-life examples, symbols, diagrams, and thinking mathematically. It focuses on using calculation, communication, application, manipulation, visualization and strategies to teach mathematical concepts in a technological age.
This document discusses using a multi-modal think board approach to teach mathematics. It involves using concrete manipulatives, diagrams, illustrations, pictures, charts and graphs to develop students' mental models and understanding of abstract mathematical concepts. Students learn by doing physical activities, making demonstrations, and using manipulatives like paper, cubes and tiles. Diagrams vary in abstraction from literal pictures to representations using dots. They convey ideas vividly. The multi-modal think board can be used for lesson planning, assessment, integrating technology, and reflecting on student and teacher learning.
The document discusses using a multi-modal think board approach to teaching mathematics. It describes the six mathematical modes of thinking - number, word, diagram, symbol, real thing, and story. Examples are provided of how to differentiate mathematics instruction for students using open-ended questions within these six modes. The goal is to engage students in thinking and working mathematically in a variety of ways.
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 document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help alleviate symptoms of mental illness and boost overall mental well-being.
The document discusses linear equations and graphing lines. It covers plotting points, calculating slope, writing equations in slope-intercept form, and graphing lines by making a table or using the slope and y-intercept. Methods are presented for finding the equation of a line given two points, the slope and a point, or from a graph.
The document discusses various types of transformations that can be performed on the graph of the function f(x)=logax. It explains that dilations on the x-axis are shown by coefficients in front of x in the equation, dilations on the y-axis by coefficients in front of the log term. Translations are shown by adding or subtracting terms from x or at the end of the equation for x-axis or y-axis translations respectively. Reflections are shown by changing the sign in front of terms or dividing the entire equation by 1. It also discusses finding degrees from radians and vice versa, finding intercepts of log equations, and determining the composition of two functions.
This document provides instructions on how to solve various types of equations, including:
- Linear equations by rearranging to isolate the variable
- Equations using the null factor law to find zeros
- Exponential and logarithmic equations by equating exponents or rearranging into logarithmic form
- Logarithmic functions using properties of logarithms
- Using the "solve" command to solve equations on a calculator
- Circular function equations by rearranging to isolate the variable
It also provides examples and exercises to practice these techniques.
Unlocking the Secrets of IPTV App Development_ A Comprehensive Guide.pdfWHMCS Smarters
With IPTV apps, you can access and stream live TV, on-demand movies, series, and other content you like online. Viewers have more flexibility and customization of content to watch. To develop the best IPTV app that functions, you must combine creative problem-solving skills and technical knowledge. This post will look into the details of IPTV app development, so keep reading to learn more.
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Morgan Freeman is Jimi Hendrix: Unveiling the Intriguing Hypothesisgreendigital
In celebrity mysteries and urban legends. Few narratives capture the imagination as the hypothesis that Morgan Freeman is Jimi Hendrix. This fascinating theory posits that the iconic actor and the legendary guitarist are, in fact, the same person. While this might seem like a far-fetched notion at first glance. a deeper exploration reveals a rich tapestry of coincidences, speculative connections. and a surprising alignment of life events fueling this captivating hypothesis.
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Introduction to the Hypothesis: Morgan Freeman is Jimi Hendrix
The idea that Morgan Freeman is Jimi Hendrix stems from a mix of historical anomalies, physical resemblances. and a penchant for myth-making that surrounds celebrities. While Jimi Hendrix's official death in 1970 is well-documented. some theorists suggest that Hendrix did not die but instead reinvented himself as Morgan Freeman. a man who would become one of Hollywood's most revered actors. This article aims to delve into the various aspects of this hypothesis. examining its origins, the supporting arguments. and the cultural impact of such a theory.
The Genesis of the Theory
Early Life Parallels
The hypothesis that Morgan Freeman is Jimi Hendrix begins by comparing their early lives. Jimi Hendrix, born Johnny Allen Hendrix in Seattle, Washington, on November 27, 1942. and Morgan Freeman, born on June 1, 1937, in Memphis, Tennessee, have lived very different lives. But, proponents of the theory suggest that the five-year age difference is negligible and point to Freeman's late start in his acting career as evidence of a life lived before under a different identity.
The Disappearance and Reappearance
Jimi Hendrix's death in 1970 at the age of 27 is a well-documented event. But, theorists argue that Hendrix's death staged. and he reemerged as Morgan Freeman. They highlight Freeman's rise to prominence in the early 1970s. coinciding with Hendrix's supposed death. Freeman's first significant acting role came in 1971 on the children's television show "The Electric Company," a mere year after Hendrix's passing.
Physical Resemblances
Facial Structure and Features
One of the most compelling arguments for the hypothesis that Morgan Freeman is Jimi Hendrix lies in the physical resemblance between the two men. Analyzing photographs, proponents point out similarities in facial structure. particularly the cheekbones and jawline. Both men have a distinctive gap between their front teeth. which is rare and often highlighted as a critical point of similarity.
Voice and Mannerisms
Supporters of the theory also draw attention to the similarities in their voices. Jimi Hendrix known for his smooth, distinctive speaking voice. which, according to some, resembles Morgan Freeman's iconic, deep, and soothing voice. Additionally, both men share certain mannerisms. such as their calm demeanor and eloquent speech patterns.
Artistic Parallels
Musical and Acting Talents
Jimi Hendrix was regarded as one of t
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The advent of Over-The-Top (OTT) players has brought a seismic shift in the television industry, transforming how we consume media. These digital platforms, which deliver content directly over the internet, have outpaced traditional cable and satellite television, offering unparalleled convenience, variety, and personalization. Here’s an in-depth look at how OTT players are revolutionizing the TV viewing experience.
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Introduction
Leonardo DiCaprio is synonymous with Hollywood stardom and acclaimed performances. has a unique connection with one of America's most beloved sports events—the Super Bowl. The "Leonardo DiCaprio Super Bowl" phenomenon combines the worlds of cinema and sports. drawing attention from fans of both domains. This article delves into the multifaceted relationship between DiCaprio and the Super Bowl. exploring his appearances at the event, His involvement in Super Bowl advertisements. and his cultural impact that bridges the gap between these two massive entertainment industries.
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Leonardo DiCaprio: The Hollywood Icon
Early Life and Career Beginnings
Leonardo Wilhelm DiCaprio was born in Los Angeles, California, on November 11, 1974. His journey to stardom began at a young age with roles in television commercials and educational programs. DiCaprio's breakthrough came with his portrayal of Luke Brower in the sitcom "Growing Pains" and later as Tobias Wolff in "This Boy's Life" (1993). where he starred alongside Robert De Niro.
Rise to Stardom
DiCaprio's career skyrocketed with his performance in "What's Eating Gilbert Grape" (1993). earning him his first Academy Award nomination. He continued to gain acclaim with roles in "Romeo + Juliet" (1996) and "Titanic" (1997). the latter of which cemented his status as a global superstar. Over the years, DiCaprio has showcased his versatility in films like "The Aviator" (2004). "Start" (2010), and "The Revenant" (2015), for which he finally won an Academy Award for Best Actor.
Environmental Activism
Beyond his film career, DiCaprio is also renowned for his environmental activism. He established the Leonardo DiCaprio Foundation in 1998, focusing on global conservation efforts. His commitment to ecological issues often intersects with his public appearances. including those related to the Super Bowl.
The Super Bowl: An American Institution
History and Significance
The Super Bowl is the National Football League (NFL) championship game. is one of the most-watched sporting events in the world. First played in 1967, the Super Bowl has evolved into a cultural phenomenon. featuring high-profile halftime shows, memorable advertisements, and significant media coverage. The event attracts a diverse audience, from avid sports fans to casual viewers. making it a prime platform for celebrities to appear.
Entertainment and Advertisements
The Super Bowl is not only about football but also about entertainment. The halftime show features performances by some of the biggest names in the music industry. while the commercials are often as anticipated as the game itself. Companies invest millions in Super Bowl ads. creating iconic and sometimes controversial commercials that capture public attention.
Leonardo DiCaprio's Super Bowl Appearances
A Celebrity Among the Fans
Leonardo DiCaprio's presence at the Super Bowl has noted several times. As a high-profile celebrity. DiCaprio attracts
SERV is the ideal spot for savory food, refreshing beverages, and exciting entertainment. Each visit promises an unforgettable experience with daily promotions, live music, and engaging games such as pickleball. Offering five distinct food concepts inspired by popular street food, as well as coffee and dessert options, there's something to satisfy every taste. For more information visit our website: https://servfun.com/
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Tom Cruise Daughter: An Insight into the Life of Suri Cruisegreendigital
Tom Cruise is a name that resonates with global audiences for his iconic roles in blockbuster films and his dynamic presence in Hollywood. But, beyond his illustrious career, Tom Cruise's personal life. especially his relationship with his daughter has been a subject of public fascination and media scrutiny. This article delves deep into the life of Tom Cruise daughter, Suri Cruise. Exploring her upbringing, the influence of her parents, and her current life.
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Introduction: The Fame Surrounding Tom Cruise Daughter
Suri Cruise, the daughter of Tom Cruise and Katie Holmes, has been in the public eye since her birth on April 18, 2006. Thanks to the media's relentless coverage, the world watched her grow up. As the daughter of one of Hollywood's most renowned actors. Suri has had a unique upbringing marked by privilege and scrutiny. This article aims to provide a comprehensive overview of Suri Cruise's life. Her relationship with her parents, and her journey so far.
Early Life of Tom Cruise Daughter
Birth and Immediate Fame
Suri Cruise was born in Santa Monica, California. and from the moment she came into the world, she was thrust into the limelight. Her parents, Tom Cruise and Katie Holmes. Were one of Hollywood's most talked-about couples at the time. The birth of their daughter was a anticipated event. and Suri's first public appearance in Vanity Fair magazine set the tone for her life in the public eye.
The Impact of Celebrity Parents
Having celebrity parents like Tom Cruise and Katie Holmes comes with its own set of challenges and privileges. Suri Cruise's early life marked by a whirlwind of media attention. paparazzi, and public interest. Despite the constant spotlight. Her parents tried to provide her with an upbringing that was as normal as possible.
The Influence of Tom Cruise and Katie Holmes
Tom Cruise's Parenting Style
Tom Cruise known for his dedication and passion in both his professional and personal life. As a father, Cruise has described as loving and protective. His involvement in the Church of Scientology, but, has been a point of contention and has influenced his relationship with Suri. Cruise's commitment to Scientology has reported to be a significant factor in his and Holmes' divorce and his limited public interactions with Suri.
Katie Holmes' Role in Suri's Life
Katie Holmes has been Suri's primary caregiver since her separation from Tom Cruise in 2012. Holmes has provided a stable and grounded environment for her daughter. She moved to New York City with Suri to start a new chapter in their lives away from the intense scrutiny of Hollywood.
Suri Cruise: Growing Up in the Spotlight
Media Attention and Public Interest
From stylish outfits to everyday activities. Suri Cruise has been a favorite subject for tabloids and entertainment news. The constant media attention has shaped her childhood. Despite this, Suri has managed to maintain a level of normalcy, thanks to her mother's efforts.
1. y = cos x
e.g. y = (x + 3)2 (x − 1)2
1
Copyright itute.com 2006
0 π 2π
Free download & print from www.itute.com –3 0 1
Do not reproduce by other means
–1
Mathematical Methods 3,4 e.g. y = (x + 2 )3 (x − 1) y = tan x
Summary sheets
Distance between two points
–2 0 1 0 π 2π
= (x2 − x1 )2 + ( y2 − y1 )2
x +x y +y
Mid-point = 1 2 , 1 2 e.g. y = (x + 2)4
2 2 Modulus functions
Parallel lines, m1 = m2 x, x ≥ 0
y= x =
Perpendicular lines, − x , x < 0
1 –2 0
m1m2 = −1 or m2 = − Transformations of y = f (x )
m1
Examples of power functions: (1) Vertical dilation (dilation away from the
Graphs of polynomial functions in x-axis, dilation parallel to the y-axis) by
1
factorised form: y = x −1 y = factor k. y = kf (x )
Quadratics e.g. y = (x + 1)(x − 3) x
(2) Horizontal dilation (dilation away from
0 the y-axis, dilation parallel to the x-axis) by
1
factor . y = f (nx )
n
–1 0 3
(3) Reflection in the x-axis. y = − f (x )
y = x −2 (4) Reflection in the y-axis. y = f (− x )
e.g. y = (x − 3)2
1 (5) Vertical translation (translation parallel
y = 2
to the y-axis) by c units.
x
y = f (x ) ± c , + up, – down.
0
0 3 1
(6) Horizontal translation (translation
Cubics e.g. y = 3(x + 1)(x − 1)(x − 2 ) y= x2
parallel to the x-axis) by b units.
y = f (x ± b ) , + left, – right.
(y = x ) *Always carry out translations last in
0 sketching graphs.
Example 1 Sketch y = − 2(x − 1) + 2
Exponential functions:
-1 0 1 2 2
y = a x where a = 2, e,10
0 1 2
e.g. y = (x + 1)2 (x − 1)
10x ex 2x
–1 0 1 Example 2 Sketch y = 2 1 − x .
Rewrite as y = 2 − (x − 1) .
1 2
asymptotic 0
e.g. y = (x + 1) 3
0 1
Logarithmic functions: Relations and functions:
–1 0 y = log a x where a = 2, e,10 A relation is a set of ordered pairs (points).
If no two ordered pairs have the same first
2x element, then the relation is a function.
Quartics e.g. y = (x + 3)(x + 1)(x − 1)(x − 2 ) ex *Use the vertical line test to determine
whether a relation is a function.
10x *Use the horizontal line test to determine
–3 –1 0 1 2 0 1 whether a function is one-to-one or many-to-
asymptotic one.
*The inverse of a relation is given by its
reflection in the line y = x .
*The inverse of a one-to-one function is a
e.g. y = (x + 3)2 (x − 1)(x − 2 ) Trigonometric functions: function and is denoted by f −1 . The inverse
y = sin x
of a many-to-one function is not a function
1
and therefore cannot be called inverse
–3 0 1 2 0 π 2π function, and f −1 cannot be used to denote
the inverse.
–1
2. Factorisation of polynomials: Quadratic formula: Index laws:
( )
(1) Check for common factors first. 2
Solutions of ax + bx + c = 0 are am n
(2) Difference of two squares, a ma n = a m+ n , = am−n , am = a mn
an
( ) − 3 = (x − 3)(x + 3)
2
2 − b ± b − 4ac
e.g. x 4 − 9 = x 2 2 2 2 x= . 1 1
(ab )n = a nbn , = a−n , am =
( 3 )(x + 3 )(x + 3)
2a
= x− 2 n −m
Graphs of transformed trig. functions a a
(3) Trinomials, by trial and error, π 1 1
e.g. y = −2 cos 3 x − + 1 , rewrite a 0 = 1, a 2 =na
e.g. 2 x 2 − x − 1 = (2 x + 1)(x − 1) 2 = a,a n
(4) Difference of two cubes, e.g. π
( )
Logarithm laws:
equation as y = −2 cos 3 x − + 1 .
x3 − y 3 = (x − y ) x 2 + xy + y 2 6 a
log a + log b = log ab, log a − log b = log
3 b
(5) Sum of two cubes, e.g. 8 + a = The graph is obtained by reflecting it in the
(
23 + a3 = (2 + a ) 4 − 2a + a 2 ) x-axis, dilating it vertically so that its
amplitude becomes 2, dilating it horizontally log ab = b log a, log
1
b
= − log b, log a a = 1
(6) Grouping two and two, 2π
e.g. x3 + 3x 2 + 3 x + 1 = x3 + 1 + 3 x 2 + 3 x( ) ( ) so that its period becomes
3
, translating log1 = 0, log 0 = undef , log(neg ) = undef
(
= (x + 1) x − x + 1 + 3 x(x + 1)
2
) upwards by 1 and right by
π
.
Change of base:
= (x + 1)(x ) 6 log b x
2
− x + 1 + 3x log a x = ,
log b a
= (x + 1)(x+ 2 x + 1 = (x + 1)3
2
) 3 log e 7
(7) Grouping three and one, e.g. log 2 7 = = 2.8 .
log e 2
e.g. x 2 − 2 x − y 2 + 1 π 5π
0 Exponential equations:
( )
= x 2 − 2 x + 1 − y 2 = (x − 1)2 − y 2 –1
6 6
e.g. 2e3 x = 5, e3 x = 2.5 , 3x = loge 2.5 ,
= (x − 1 − y )(x − 1 + y ) 1
(8) Completing the square, e.g. x= loge 2.5
3
1 1
2 2 Solving trig. equations
x2 + x −1 = x2 + x + − −1 e.g. 2e 2 x − 3e x − 2 = 0 ,
2 2 3
2
2
e.g. Solve sin 2 x =
2
, 0 ≤ x ≤ 2π .
( ) − 3(e )− 2 = 0 ,
2 ex
2 x
= x2
+x+
1
−
4
5
4
=x +
1
2
−
2
5
∴ 0 ≤ 2 x ≤ 4π ,
π 2π π 2π
(2e + 1)(e − 2) = 0 , since 2e
x x x
+1 ≠ 0 ,
2x = , , + 2π , + 2π x x
∴ e − 2 = 0 , e = 2 , x = loge 2 .
3 3 3 3
1 5
x + 1 + 5
= x +
− π π 7π 4π
2 2 2 2
∴x = , , , . Equations involving log:
6 3 6 3 e.g. loge (1 − 2 x ) + 1 = 0 , loge (1 − 2 x ) = −1 ,
(9) Factor theorem, x x
e.g. sin = 3 cos , 0 ≤ x ≤ 2π . 1 1
e.g. P(x ) = x3 − 3x 2 + 3 x − 1 2 2 1 − 2 x = e −1 , 2 x = 1 − e −1 , x = 1 − .
2 e
P(− 1) = (− 1)3 − 3(− 1)2 + 3(− 1) − 1 ≠ 0 sin
x
e.g. log10 (x − 1) = 1 − log10 (2 x − 1)
x 2 = 3 , tan x = 3 ,
P(1) = 13 − 3(1)2 + 3(1) − 1 = 0 0 ≤ ≤π,
x log10 (x − 1) + log10 (2 x − 1) = 1
2 2
∴ (x − 1) is a factor. cos
2 log10 (x − 1)(2 x − 1) = 1 , (x − 1)(2 x − 1) = 10 ,
Long division: x π 2π
∴ = , ∴x = . 2 x 2 − 3x − 9 = 0 , (2 x + 3)(x − 3) = 0 ,
x2 − 2x + 1 2 3 3 3
x − 1)x3 − 3 x 2 + 3 x − 1 x = − , 3 . 3 is the only solution because
2
(
− x3 − x 2 ) Exact values for trig. functions:
x=−
3
makes the log equation undefined.
− 2 x 2 + 3x xo x sin x cos x tan x 2
(
− − 2x2 + 2 x ) 0
30
0
π/6
0
1/2
1
√3/2
0
1/√3 Equation of inverse:
x −1 Interchange x and y in the equation to obtain
45 π/4 1/√2 1/√2 1
− (x − 1) 60 π/3 √3/2 1/2 √3
the equation of the inverse. If possible
express y in terms of x.
0 90 π/2 1 0 undef
(
∴ P(x ) = (x − 1) x 2 − 2 x + 1 = (x − 1)3 ) 120 2π/3 √3/2 –1/2 –√3
e.g. y = 2(x − 1)2 + 1 , x = 2( y − 1)2 + 1 ,
x −1
135 3π/4 1/√2 –1/√2 –1 2( y − 1)2 = x − 1 , ( y − 1)2 = ,
Remainder theorem: 150 5π/6 1/2 –√3/2 –1/√3 2
e.g. when P (x ) = x3 − 3 x 2 + 3 x − 1 is 180 π 0 –1 0 x −1
210 7π/6 –1/2 –√3/2 1/√3 y=± +1 .
divided by x + 2 , the remainder is 2
225 5π/4 –1/√2 –1/√2 1
P (− 2) = (− 2)3 − 3(− 2)2 + 3(− 2) − 1 = −11 240 4π/3 –√3/2 –1/2 √3 e.g. y = −
2
+4, x = −
2
+4 ,
When it is divided by 2 x − 3 , the remainder x −1 y −1
270 3π/2 –1 0 undef
3 1 300 5π/3 –√3/2 1/2 –√3 2 2
is P = . x−4= − , y −1 = − ,
2 8 315 7π/4 –1/√2 1/√2 –1 y −1 x−4
330 11π/6 –1/2 √3/2 –1/√3 2
y=− +1 .
360 2π 0 1 0 x−4
3. e.g. y = −2e x −1 + 1 , x = −2e y −1 + 1 , Differentiation rules: The approx. change in a function is
The product rule: For the multiplication of = f (a + h ) − f (a ) = hf ′(a ) ,
1− x
2e y −1 = 1 − x , e y −1 = , two functions, y = u (x )v(x ) , e.g. e.g. find the approx. change in cos x when x
2
y = x 2 sin 2 x , let u = x 2 , v = sin 2 x , π
1− x 1− x changes from to 1.6. Let f (x ) = cos x ,
y − 1 = loge , y = log e +1. dy du dv 2
2 2 =v +u π
dx dx dx then f ′(x ) = − sin x . Let a = , then
e.g. y = − loge (1 − 2 x ) − 1 , ( )
= (sin 2 x )(2 x ) + x 2 (2 cos 2 x )
π
2
π
x = − loge (1 − 2 y ) − 1 , = 2 x(sin 2 x + x cos 2 x ) f ′(a ) = − sin = −1 and h = 1.6 − = 0.03
2 2
The quotient rule: For the division of
loge (1 − 2 y ) = −(x + 1) , 1 − 2 y = e − ( x +1) u (x ) log e x Change in cos x = hf ′(a ) = 0.03×− 1 = −0.03
functions, y = , e.g. y = ,
1
2
(
2 y = 1 − e − ( x +1) , y = 1 − e − ( x +1) .) v(x ) x Rate of change:
dy
dx
is the rate of change of
du dv
The binomial theorem: v −u dx
dy dx dx
= y with respect to x. v = , velocity is the
e.g. Expand (2 x − 1)4 dx v2 dt
= 4C0 (2 x )4 (− 1)0 + 4C1 (2 x )3 (− 1)1 rate of change of position x with respect to
(x ) 1 − (loge x )(1)
dv
+ 4C2 (2 x )2 (− 1)2 + 4C3 (2 x )1 (− 1)3 x 1 − log e x time t. a = , acceleration a is the rate of
= 2 = . dt
x x2
+ 4C4 (2 x )0 (− 1)4 = ...... The chain rule: For composite functions,
change of velocity v with respect to t.
e.g. Find the coefficient of x2 in the
y = f (u ( x) ) , e.g. y = e cos x . Average rate of change: Given y = f (x ) ,
expansion of (2 x − 3)5 .
dy dy du when x = a , y = f (a ) , when x = b ,
Let u = cos x , y = eu , = ×
The required term is 5C3 (2 x )2 (− 3)3 dx du dx y = f (b ) , the average rate of change of y
( )
= 10 4 x 2 (− 27 ) = −1080 x 2 . ( )( sin x) = −e
= eu − cos x
sin x . with respect to x =
∆y
=
f (b ) − f (a )
.
∴ the coefficient of x2 is –1080. dy ∆x b−a
Finding stationary points: Let = 0 and
dx
Differentiation rules: solve for x and then y, the coordinates of the Deducing the graph of gradient function
dy from the graph of a function
y = f (x ) = f ' (x ) stationary point.
f(x)
dx Nature of stationary point at x = a :
•
ax n anx n −1 Local Local Inflection
•
max. min. point
a(x + c )n an(x + c )n −1 x<a dy dy dy
0 x
>0 <0 > 0 , (< 0)
a(bx + c )n abn(bx + c )n −1 dx dx dx
x=a dy dy dy
a sin x a cos x =0 =0 =0 o
dx dx dx
a sin (x + c ) a cos(x + c ) x>a dy dy dy f’(x)
a sin (bx + c ) ab cos(bx + c ) <0 >0 > 0 , (< 0)
dx dx dx o o
a cos x −a sin x Equation of tangent and normal at x = a :
•
a cos(x + c ) − a sin (x + c ) 1) Find the y coordinate if it is not given. 0 o x
dy
a cos(bx + c ) − ab sin (bx + c ) 2) Gradient of tangent mT = at x = a . •
dx
a tan x a sec 2 x 3) Use y − y1 = mT (x − x1 ) to find equation Deducing the graph of function from the
a tan (x + c ) a sec (x + c )
2 of tangent. graph of anti-derivative function
a tan (bx + c )
1
ab sec 2 (bx + c ) 4) Find gradient of normal mN = − . ∫ f(x)dx+ c
mT
x x
ae ae 5) Use y − y1 = m N (x − x1 ) to find equation •
x+c
ae ae x + c of the normal. •
Linear approximation: 0 x
ae bx + c abe bx + c To find the approx. value of a function, use
a log e x a f (a + h ) ≈ f (a ) + hf ′(a ) , e.g. find the
x approx. value of 25.1 . Let f (x ) = x ,
o
a log e bx a 1
x then f ′(x ) = . Let a = 25 and h = 0.1 , f(x)
2 x o o
a log e (x + c ) a
then f (a + h ) = 25.1 , f (a ) = 25 = 5 ,
x+c 0 o • x
1
a log e b(x + c ) a f ′(a ) = = 0.1 .
2 25 •
x+c
∴ 25.1 ≈ 5 + 0.1 × 0.1 = 5.01
a log e (bx + c ) ab
bx + c
4. Anti-differentiation (indefinite integrals): Estimate area by left (or right) rectangles Graphics calculator :
Pr ( X = a ) = binompdf (n, p, a )
f (x )
∫ f (x)dx Left Right Pr ( X ≤ a ) = binomcdf (n, p, a )
ax n for n ≠ −1 a n +1
x a b a b Pr ( X < a ) = binomcdf (n, p, a − 1)
n +1 Area between two curves: Pr ( X ≥ a ) = 1 − binomcdf (n, p, a − 1)
a (x + c )n , n ≠ −1 a
(x + c )n +1 y = g (x ) Pr ( X > a ) = 1 − binomcdf (n, p, a )
n +1 y = f (x ) Pr (a ≤ X ≤ b ) = binomcdf (n, p, b )
a (bx + c )n , n ≠ −1 a
(bx + c )n +1 a 0 b −binomcdf (n, p, a − 1)
(n + 1)b Probability density functions f (x ) for
a a log e x , x > 0 x ∈ [a, b] . y y = f (x )
x a log e (− x ) , x < 0 Firstly find the x-coordinates of the
a a log e (x + c ) intersecting points, a, b, then evaluate
b a c b x
x+c
a a
log e (bx + c )
A=
∫ [ f (x) − g (x)]dx . Always the function
a
For f (x ) to be a probability density
above minus the function below. function, f (x ) > 0 and
bx + c b b
For three intersecting points:
ae x ae x Pr (a < X < b ) =
∫ f (x)dx = 1.
a
ae x + c ae x + c y = f (x ) c b
ae bx + c a bx + c
e
y = g (x ) Pr ( X < c ) =
∫ f (x)dx , Pr(X > c) = ∫ f (x)dx
a c
b a b 0 c Normal distributions are continuous prob.
a sin x − a cos x distributions. The graph of a normal dist. has
b c
a sin (x + c ) − a cos(x + c )
∫ [ f (x) − g (x)]dx ∫ [g (x) − f (x )]dx
a bell shape and the area under the graph
A= +
represents probability. Total area = 1.
a sin (bx + c )
a b
a
− cos(bx + c )
b
Discrete probability distributions: ( )
N 1 µ1 , σ 2 , N 2 µ 2 , σ 2 . ( )
In general, in the form of a table,
a cos x a sin x x x1 x2 x3 ...... 1 2 µ1 < µ 2
a cos(x + c ) a sin (x + c )
Pr ( X = x ) p1 p2 p3 ......
a cos(bx + c ) a 0 µ1 µ2 X
sin (bx + c ) p1 , p2 , p3 ,... have values from 0 to 1 and
b
p1 + p2 + p3 + ... = 1 . ( 2
)
N1 µ , σ 1 , N 2 µ , σ 2 ( 2
).
Definite integrals: µ = E ( X ) = x1 p1 + x2 p2 + x3 p3 + ... 1 σ1 < σ 2
π π
Var ( X ) = x1 p1 + x2 p2 + x3 p3 + ... − µ 2
2 2 2 2
π π 2
e.g.
∫
0
2 cos x − dx = sin x −
3 3 0 σ = sd ( X ) = Var ( X )
0 X
π π π If random variable Y = aX + b ,
= sin − − sin 0 − The standard normal distribution:
2 3 3 E (Y ) = aE ( X ) + b , Var (Y ) = a 2 × Var ( X ) has µ = 0 and σ = 1 . N (0,1)
and sd (Y ) = a × sd ( X ) .
π π 1+ 3
= sin − sin − = . 95% probability interval : (µ − 2δ , µ + 2δ ) µ σ 2
6 3 2
Pr ( A ∩ B )
Properties of definite integrals: Conditional prob: Pr A B = ( ) Pr (B )
.
b b
1)
∫ kf (x)dx = k ∫ f (x)dx
a a
Binomial distributions are examples of
discrete prob. distributions. Sampling with
0 Z
b b b Graphics calculator: Finding probability,
2) [ f (x ) ± g (x )]dx = f (x )dx ± g (x )dx
∫ ∫ ∫
replacement has a binomial distribution.
a a a Number of trials = n. In a single trial, prob. Pr ( X < a ) = normalcdf (− E 99, a, µ , σ )
b c b of success = p, prob. of failure = q = 1- p. Pr ( X > a ) = normalcdf (a, E 99, µ , σ )
∫ a ∫ ∫
3) f (x )dx = f (x )dx + f (x )dx ,
a c
The random variable X is the number of
successes in the n trials. The binomial dist.
Pr (a < X < b ) = normalcdf (a, b, µ , σ )
b a Finding quantile, e.g. given Pr ( X < x ) = 0.7
is Pr ( X = x )= n C x p x q n− x , x = 0,1,2,... with
∫ ∫
where a < c < b . 4) f (x )dx = − f (x )dx
a b x = invNorn(0.7, µ , σ ) .
b a a µ = np and σ = npq = np(1 − p ) . Given Pr ( X > x ) = 0.7 , then
∫ a ∫ b∫
4) f (x )dx = − f (x )dx , 5) f (x )dx = 0.
a
** Effects of increasing n on the graph of a Pr ( X < x ) = 1 − 0.7 = 0.3 and
Area ‘under’ curve: binomial distribution. (1) more points x = invNorm(0.3, µ , σ ) .
(2) lower probability for each x value
(3) becoming symmetrical , bell shape. X −µ
b To find µ and/or σ, use Z = to
y = f (x ) A=
∫ f (x)dx
a
** Effects of changing p on the graph of a
binomial distribution. (1) bell shape when
σ
convert X to Z first, e.g. find µ given σ = 2
a 0 b p = 0.5 (2) positively skewed if p < 0.5 4−µ
y = f (x ) and Pr ( X < 4) = 0.8 . Pr Z < = 0.8 ,
(3) negatively skewed if p > 0.5 2
a c 0 b p = 0.5 p < 0.5 p > 0.5 4−µ
c b
∴ = invNorm(0.8) = 0.8416 ,
2
∫
A = − f (x )dx +
a ∫ f (x)dx
c ∴ µ = 2.3168 .