The document defines and explains key concepts related to polynomial functions. It states that a real polynomial P(x) of degree n is an expression of the form P(x)=p0+p1x+p2x2+...+pn-1xn-1+pxn, where pn≠0 and n≥0 is an integer. It then provides definitions and examples for important polynomial terms like coefficients, degree, leading term, roots, and zeros.
This document contains a mathematics exam with 30 multiple choice questions covering various topics in calculus, linear algebra, differential equations, real analysis, and probability. The exam has 75 total marks and is divided into sections on vectors and matrices, differential equations, real analysis, and probability. It provides the questions, response options, and asks test takers to select the correct option(s) and write them in the answer booklet.
The document derives the normal probability density function from basic assumptions. It assumes that errors in perpendicular directions are independent, large errors are less likely than small errors, and the distribution is not dependent on orientation. This leads to a differential equation that can only be satisfied by an exponential function, giving the normal distribution. The values of the coefficients are determined by requiring the total area under the curve to be 1 and that the variance equals 1/k. This fully specifies the normal probability density function.
The document discusses unprovability in mathematics and computer science, noting that there are true statements that cannot be proven within certain axiomatic systems due to their inherent incompleteness, as shown by Gödel's incompleteness theorems, and that there are computational problems for which no algorithm can provide a solution, as proven by Turing. The document also provides information about upcoming problem sets and classes.
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
This document contains a mathematics exam with 30 multiple choice questions covering various topics in calculus, linear algebra, differential equations, real analysis, and probability. The exam has 75 total marks and is divided into sections on vectors and matrices, differential equations, real analysis, and probability. It provides the questions, response options, and asks test takers to select the correct option(s) and write them in the answer booklet.
The document derives the normal probability density function from basic assumptions. It assumes that errors in perpendicular directions are independent, large errors are less likely than small errors, and the distribution is not dependent on orientation. This leads to a differential equation that can only be satisfied by an exponential function, giving the normal distribution. The values of the coefficients are determined by requiring the total area under the curve to be 1 and that the variance equals 1/k. This fully specifies the normal probability density function.
The document discusses unprovability in mathematics and computer science, noting that there are true statements that cannot be proven within certain axiomatic systems due to their inherent incompleteness, as shown by Gödel's incompleteness theorems, and that there are computational problems for which no algorithm can provide a solution, as proven by Turing. The document also provides information about upcoming problem sets and classes.
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
1. The passage provides 33 multiple choice questions from a past UPSEE (Uttar Pradesh State Entrance Examination) mathematics paper from 1999.
2. The questions cover a range of mathematics topics including algebra, trigonometry, calculus, probability, and vectors.
3. Each question has 4 possible answer choices labeled a, b, c, or d and tests the examinee's ability to apply mathematical concepts and reasoning to solve problems.
The document provides information about graphing polynomial functions, including:
1) How to determine the degree, leading coefficient, intercepts, and behavior of a polynomial function graph from its standard and factored forms. Activities are provided to match polynomial functions and determine intercepts.
2) How to use the leading coefficient test to determine if a polynomial graph rises or falls on the left and right sides based on whether the leading coefficient is positive or negative and if the degree is odd or even. Examples analyze the behavior of specific polynomial function graphs.
3) How to sign a table to summarize the intercepts, degree, leading coefficient, and behavior of polynomial function graphs. Students are asked to graph specific functions and
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
CVPR2010: higher order models in computer vision: Part 1, 2zukun
This document discusses tractable higher order models in computer vision using random field models. It introduces Markov random fields (MRFs) and factor graphs as graphical models for computer vision problems. Higher order models that include factors over cliques of more than two variables can model problems more accurately but are generally intractable. The document discusses various inference techniques for higher order models such as relaxation, message passing, and decomposition methods. It provides examples of how higher order and global models can be used in problems like segmentation, stereo matching, reconstruction, and denoising.
The document provides an overview of calculating limits. It begins with announcements about assignments and exams for a Calculus I course. The outline then previews the key topics to be covered, including the concept of a limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The bulk of the document explains the error-tolerance game approach to defining a limit and works through examples of basic limits. It also establishes four limit laws for arithmetic operations: addition of limits, subtraction of limits, scaling of limits, and multiplication of limits.
This document discusses natural number objects (NNOs) in the categorical logic framework of Dialectica categories. It begins by motivating the use of linear logic and Dialectica categories to study recursion and iteration. It then provides background on NNOs, Dialectica categories, and their structure. The document considers defining NNOs in Dialectica categories using either the cartesian or tensor structure. It presents a trivial NNO that can be formed using the cartesian structure. The goal is to investigate linear recursion and iteration through the study of NNOs in Dialectica categories.
The average value of a function f(x) over an interval (a,b) can be approximated as:
f(x) = (f(x1) + f(x2) + ... + f(xn))/n, where x1, x2, ..., xn are values in the interval.
The Fourier coefficients for a periodic function f(x) are:
a0 = (1/π) ∫ f(x) dx
an = (2/π) ∫ f(x) cos(nx) dx
bn = (2/π) ∫ f(x) sin(nx) dx
The Fourier series expansion of
The document summarizes key aspects of parabolas as conic sections:
1) A parabola is defined as the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
2) The standard form of the equation of a parabola is y=ax^2, where the vertex is at the origin, the focus is on the y-axis, and the directrix is the x-axis.
3) Examples are worked through to find the equation, focus, directrix, and other properties of parabolas given information like the vertex or standard form equation.
The document summarizes the complex form of Fourier series. It states that after substituting sine and cosine terms into the Fourier series formula, the complex form involves a summation of terms with coefficients multiplied by exponential terms with integer multiples of i and x. It provides the formulas for calculating the coefficients c0, c1, c2, etc. and gives an example function defined over an interval to demonstrate the complex form.
The document defines algebraic expressions and discusses various algebraic operations such as addition, subtraction, multiplication, division, and factorization of algebraic expressions. It provides examples to illustrate each operation. Factorization is described as expressing a complicated polynomial as the product of simpler polynomial factors. Common factoring techniques are mentioned, including factoring the difference of squares and factoring trinomials.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
This document provides an overview of functions and function notation that will be used in Calculus. It defines a function as an equation where each input yields a single output. Examples demonstrate determining if equations are functions and evaluating functions using function notation. The key concepts of domain and range of a function are explained. The document concludes by finding the domains of various functions involving fractions, radicals, and inequalities.
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
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.
We discuss the ideas of monotonicity (increasing or decreasing) and concavity (up or down) of a function. Because of the Mean Value Theorem, we can determine these characteristics using derivatives.
This document contains an unsolved mathematics paper from 1983 consisting of 25 multiple choice questions testing concepts in algebra, geometry, trigonometry, and calculus. The paper is divided into three sections - single answer multiple choice questions, true/false statements, and fill in the blank questions. Sample questions include solving equations, finding roots, determining geometric properties of figures, evaluating integrals and derivatives, and identifying monotonic behavior of functions.
This document covers exponential functions. [1] Exponential functions can be expressed in the form y = bx, where b > 0 and b ≠ 1. They are used to model growth and decay situations. [2] The key properties of exponential graphs are that they have a domain of all real numbers, range of positive real numbers, no x-intercepts, a y-intercept of (0,1), and are either increasing or decreasing based on the value of b. [3] Various b values result in faster or slower growth, with larger b values growing faster and b between 0 and 1 decreasing.
11 X1 T03 01 inequations and inequalities (2010)Nigel Simmons
The document discusses solving inequalities involving quadratic and rational expressions. For quadratic inequalities, it explains how to factorize the expression and then determine the solutions by identifying what range of values satisfy the inequality. It notes that solutions are typically in the form of -6 < x < 1 or x < -4 or x > 1. For rational inequalities, it outlines the steps of finding where the denominator is zero, solving the equality, plotting points on a number line, and then testing different values of the variable to identify the range that satisfies the inequality.
11X1 T14 03 arithmetic & geometric means (2011)Nigel Simmons
The document defines and provides formulas for the arithmetic mean and geometric mean. The arithmetic mean is calculated by summing all values and dividing by the total number of values. The geometric mean is calculated by multiplying all values together and taking the nth root of the product, where n is the number of values. An example is provided to find the arithmetic mean and geometric mean of the values 4 and 25.
1. The passage provides 33 multiple choice questions from a past UPSEE (Uttar Pradesh State Entrance Examination) mathematics paper from 1999.
2. The questions cover a range of mathematics topics including algebra, trigonometry, calculus, probability, and vectors.
3. Each question has 4 possible answer choices labeled a, b, c, or d and tests the examinee's ability to apply mathematical concepts and reasoning to solve problems.
The document provides information about graphing polynomial functions, including:
1) How to determine the degree, leading coefficient, intercepts, and behavior of a polynomial function graph from its standard and factored forms. Activities are provided to match polynomial functions and determine intercepts.
2) How to use the leading coefficient test to determine if a polynomial graph rises or falls on the left and right sides based on whether the leading coefficient is positive or negative and if the degree is odd or even. Examples analyze the behavior of specific polynomial function graphs.
3) How to sign a table to summarize the intercepts, degree, leading coefficient, and behavior of polynomial function graphs. Students are asked to graph specific functions and
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
CVPR2010: higher order models in computer vision: Part 1, 2zukun
This document discusses tractable higher order models in computer vision using random field models. It introduces Markov random fields (MRFs) and factor graphs as graphical models for computer vision problems. Higher order models that include factors over cliques of more than two variables can model problems more accurately but are generally intractable. The document discusses various inference techniques for higher order models such as relaxation, message passing, and decomposition methods. It provides examples of how higher order and global models can be used in problems like segmentation, stereo matching, reconstruction, and denoising.
The document provides an overview of calculating limits. It begins with announcements about assignments and exams for a Calculus I course. The outline then previews the key topics to be covered, including the concept of a limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The bulk of the document explains the error-tolerance game approach to defining a limit and works through examples of basic limits. It also establishes four limit laws for arithmetic operations: addition of limits, subtraction of limits, scaling of limits, and multiplication of limits.
This document discusses natural number objects (NNOs) in the categorical logic framework of Dialectica categories. It begins by motivating the use of linear logic and Dialectica categories to study recursion and iteration. It then provides background on NNOs, Dialectica categories, and their structure. The document considers defining NNOs in Dialectica categories using either the cartesian or tensor structure. It presents a trivial NNO that can be formed using the cartesian structure. The goal is to investigate linear recursion and iteration through the study of NNOs in Dialectica categories.
The average value of a function f(x) over an interval (a,b) can be approximated as:
f(x) = (f(x1) + f(x2) + ... + f(xn))/n, where x1, x2, ..., xn are values in the interval.
The Fourier coefficients for a periodic function f(x) are:
a0 = (1/π) ∫ f(x) dx
an = (2/π) ∫ f(x) cos(nx) dx
bn = (2/π) ∫ f(x) sin(nx) dx
The Fourier series expansion of
The document summarizes key aspects of parabolas as conic sections:
1) A parabola is defined as the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
2) The standard form of the equation of a parabola is y=ax^2, where the vertex is at the origin, the focus is on the y-axis, and the directrix is the x-axis.
3) Examples are worked through to find the equation, focus, directrix, and other properties of parabolas given information like the vertex or standard form equation.
The document summarizes the complex form of Fourier series. It states that after substituting sine and cosine terms into the Fourier series formula, the complex form involves a summation of terms with coefficients multiplied by exponential terms with integer multiples of i and x. It provides the formulas for calculating the coefficients c0, c1, c2, etc. and gives an example function defined over an interval to demonstrate the complex form.
The document defines algebraic expressions and discusses various algebraic operations such as addition, subtraction, multiplication, division, and factorization of algebraic expressions. It provides examples to illustrate each operation. Factorization is described as expressing a complicated polynomial as the product of simpler polynomial factors. Common factoring techniques are mentioned, including factoring the difference of squares and factoring trinomials.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
This document provides an overview of functions and function notation that will be used in Calculus. It defines a function as an equation where each input yields a single output. Examples demonstrate determining if equations are functions and evaluating functions using function notation. The key concepts of domain and range of a function are explained. The document concludes by finding the domains of various functions involving fractions, radicals, and inequalities.
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
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.
We discuss the ideas of monotonicity (increasing or decreasing) and concavity (up or down) of a function. Because of the Mean Value Theorem, we can determine these characteristics using derivatives.
This document contains an unsolved mathematics paper from 1983 consisting of 25 multiple choice questions testing concepts in algebra, geometry, trigonometry, and calculus. The paper is divided into three sections - single answer multiple choice questions, true/false statements, and fill in the blank questions. Sample questions include solving equations, finding roots, determining geometric properties of figures, evaluating integrals and derivatives, and identifying monotonic behavior of functions.
This document covers exponential functions. [1] Exponential functions can be expressed in the form y = bx, where b > 0 and b ≠ 1. They are used to model growth and decay situations. [2] The key properties of exponential graphs are that they have a domain of all real numbers, range of positive real numbers, no x-intercepts, a y-intercept of (0,1), and are either increasing or decreasing based on the value of b. [3] Various b values result in faster or slower growth, with larger b values growing faster and b between 0 and 1 decreasing.
11 X1 T03 01 inequations and inequalities (2010)Nigel Simmons
The document discusses solving inequalities involving quadratic and rational expressions. For quadratic inequalities, it explains how to factorize the expression and then determine the solutions by identifying what range of values satisfy the inequality. It notes that solutions are typically in the form of -6 < x < 1 or x < -4 or x > 1. For rational inequalities, it outlines the steps of finding where the denominator is zero, solving the equality, plotting points on a number line, and then testing different values of the variable to identify the range that satisfies the inequality.
11X1 T14 03 arithmetic & geometric means (2011)Nigel Simmons
The document defines and provides formulas for the arithmetic mean and geometric mean. The arithmetic mean is calculated by summing all values and dividing by the total number of values. The geometric mean is calculated by multiplying all values together and taking the nth root of the product, where n is the number of values. An example is provided to find the arithmetic mean and geometric mean of the values 4 and 25.
The document discusses index laws and meanings in algebra. It covers:
- Adding and subtracting like terms, and inability to add unlike terms
- Index laws for multiplication and division of terms with exponents
- Meanings of exponents as they relate to fractions, roots, and powers
- Examples of expanding and simplifying expressions using index laws
- Solutions to exercises involving application of index laws
The document provides instructions for simplifying algebraic fractions. It states that one should always factorize the expression first before cancelling terms. Several worked examples are provided that show the steps to (1) create a common denominator, (2) identify the difference between the old and new denominators, and (3) multiply the numerator by this difference when factorizing.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
11X1 T10 07 sum & product of roots (2011)Nigel Simmons
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
The document discusses several angle theorems related to circles:
1) Opposite angles of a cyclic quadrilateral are supplementary.
2) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
3) Angles subtended at the circumference by the same or equal arcs are equal. Proofs are provided for each theorem using properties of angles at the center or circumference of a circle.
The document discusses how to find the original curve (primitive function) given the derivative (tangent line equation). It states that if the derivative is f'(x)=xn, then the primitive function is f(x)=(x^(n+1))/(n+1)+c. It provides examples such as if f'(x)=3x^4, then f(x)=3x^5/5+c. It also shows how to find the equation of a curve given its gradient function and a point it passes through.
12X1 T08 02 general binomial expansions (2011)Nigel Simmons
The document discusses relationships in Pascal's triangle. It shows that the binomial coefficient nCk can be expressed as n-1Ck-1 + n-1Ck. It also shows that Pascal's triangle is symmetrical, with nCk = nCn-k for 1 ≤ k ≤ n-1.
A particle undergoing projectile motion obeys two equations of motion: its horizontal acceleration is 0 and its vertical acceleration is -g. The particle has initial conditions when t=0 of an initial horizontal velocity of vcosθ and initial vertical velocity of vsinθ, where v is the initial speed and θ is the launch angle. For example, this can be used to model a ball thrown with an initial speed of 25 m/s at an angle of tan-1(3/4) to the ground to determine properties of its motion.
The document discusses several theorems related to tangents of circles:
1) The angle between a tangent line and radius drawn to the point of contact is 90 degrees.
2) Equal tangents can be drawn from any external point to a circle, and the line joining the point to the center is an axis of symmetry.
3) If two circles share a common tangent, the centers and point of contact must be collinear.
Mathematical induction has the following steps:
1) Prove that the statement is true for the base case (usually n=1).
2) Assume the statement is true for some integer k.
3) Using the assumption from step 2, prove the statement is true for k+1.
4) By proving the statement true for n=1 and showing that if it is true for k then it is true for k+1, the statement is true for all positive integers n.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
11 x1 t11 02 parabola as a locus (2012)Nigel Simmons
The document describes the geometric definition of a parabola. A parabola is defined as the locus of a point where the distance from a fixed point (the focus) is equal to the distance from a fixed line (the directrix). As a point moves along the parabola, its distance from the focus is always equal to its distance from the directrix. The document provides examples of finding the focus, directrix, and focal length given the standard equation of a parabola.
The document defines polynomial functions. A polynomial P(x) of degree n is an expression of the form:
P(x) = p0 + p1x + p2x2 + ... + pn-1xn-1 + pnxn
Where pn ≠ 0. The degree n must be an integer greater than or equal to 0. The coefficients are p0, p1, p2...pn. The degree is the highest power of x. A polynomial equation is P(x) = 0 and the roots are the solutions. The x-intercepts of the graph are the zeros. Examples are provided to demonstrate polynomials and determining the degree and whether a polynomial is monic.
1) The document discusses polynomials, including adding, subtracting, multiplying, and dividing polynomials. It also covers finding the roots and zeros of polynomials.
2) Partial fraction decomposition is explained for denominators that are linear factors, repeated linear factors, quadratic factors, and repeated quadratic factors.
3) Key concepts covered include the remainder and factor theorems, zeros of polynomials, and converting improper rational expressions to proper rational expressions using long division and partial fraction decomposition.
1) The document defines different types of polynomials including linear, quadratic, and cubic polynomials. It gives examples of each type.
2) Key information about polynomials includes that the degree refers to the highest power of the variable, and that a polynomial's zeros are the values where it equals 0.
3) Properties of polynomial zeros are discussed, such as that a linear polynomial has 1 zero, a quadratic polynomial has up to 2 zeros, and a cubic polynomial has up to 3 zeros. Relations between coefficients and zeros are also presented.
This document provides information about a Core Mathematics C3 exam taken by Edexcel students. It includes instructions for students taking the exam, information about materials allowed and provided, and 8 questions testing various calculus, geometry, and trigonometry concepts. The exam is 1 hour and 30 minutes long and contains a total of 75 marks across the 8 questions. Students are advised to show their working clearly and label answers to parts of questions.
This document discusses polynomial functions and how to graph them. It defines a polynomial as a sum of terms with non-negative integer exponents. Polynomial graphs are smooth curves that may be lines, parabolas, or higher-order curves. To graph a polynomial, one determines the end behavior from the leading term, finds the x-intercepts by setting the polynomial equal to 0, and uses intercepts and test points to plot the graph over intervals. Multiplicity of roots affects whether the graph crosses or is tangent to the x-axis at those points.
This presentation discusses polynomials and their key properties. It defines what polynomials are, including their terms, degrees, and different types (constant, linear, quadratic, cubic). It explains the relationship between the zeros of a polynomial and its coefficients. Specifically, it states that the sum of the zeros is equal to the negative of the coefficient of x^1, and the product of the zeros is equal to the constant term. It also discusses how to find the zeros of a polynomial by setting it equal to 0 and factoring. Examples are provided to illustrate various polynomial concepts and properties.
The document discusses curve sketching of polynomial functions. It explains that the appearance of a polynomial graph depends on whether the exponent is odd or even. It also notes that the maximum number of roots is equal to the degree of the polynomial function. The steps for sketching a polynomial graph are outlined as: 1) Find the y-intercept, 2) Find all roots, 3) Determine the sign over intervals defined by roots, and 4) Sketch the graph.
Let Pn(x) be the Legendre polynomial of degree n. Then the generating function for Pn(x) is given by:
∞
1
Pn(x)tn = √
n=0
1 − 2xt + t2
Differentiating both sides with respect to t, we get:
∞
∑nPn(x)tn-1 = -xt(1 − 2xt + t2)-1/2 + (1 − 2xt + t2)-3/2
n=1
Multiplying both sides by (1 − 2xt + t2)1/2, we get:
∞
∑
This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.
This document contains 15 multiple choice and free response questions about sinusoidal functions and graphs. It tests concepts like identifying amplitude, period, phase shift, and writing equations to represent sinusoidal graphs in terms of sine and cosine. The questions progress from identifying properties of given graphs and equations to sketching graphs, writing equations to represent graphs, and applying concepts to word problems involving real-world sinusoidal situations.
This document contains 15 multiple choice and free response questions about sinusoidal functions and their graphs. Key concepts covered include:
- Identifying amplitude, period, and phase shift from graphs of sinusoidal functions
- Writing equations to represent sinusoidal graphs in terms of sine and cosine
- Sketching transformed sinusoidal graphs (shifts, stretches, reflections)
- Finding amplitude, period, phase shift, and vertical/horizontal shifts from equations
- Relating sinusoidal equations to their real world applications like a roller coaster track
This document discusses various methods for finding the zeros or roots of polynomial functions, including factoring, factor theorem, synthetic division, and using the principle that every polynomial of degree n has n zeros. It provides examples of finding the zeros of polynomials by factorization, using a given zero to find other zeros through synthetic division, and identifying which numbers are zeros of various polynomials. Exercises are included for students to practice finding remaining zeros given one zero and identifying polynomial factors.
Rational Zeros and Decarte's Rule of Signsswartzje
The Rational Zero Theorem provides a method to determine all possible rational zeros of a polynomial function. It states that if p/q is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient. Descartes' Rule of Signs can be used to determine the maximum number of positive and negative real zeros by counting the variations in sign of the polynomial function and its substitution of -x. It provides bounds on the number of positive and negative real zeros that are either the number of variations in sign or less by an even integer. The example demonstrates applying these methods to determine all 16 possible rational zeros and the bounds of 0 positive and either 3 or 1 negative real zeros for the given polynomial.
This document provides information about polynomials including definitions, types, terms, and relationships between coefficients and zeros. It begins with acknowledging those who helped create the presentation. It then defines a polynomial as an expression with variable terms raised to whole number powers. The main types discussed are linear, quadratic, and cubic polynomials. Linear polynomials have one zero while quadratics have two zeros and cubics have three. Relationships are defined between the zeros and coefficients. Graphs of linear and quadratic polynomials are presented. The division algorithm for polynomials is also explained.
1) A radical function is of the form y = f(x) = ax + b, where changing a and b affects the graph.
2) Graphing shows that if a > 0 the graph increases, if a < 0 the graph decreases, larger a makes the graph steeper, and closer to 0 makes the graph flatter.
3) The value of b is the y-intercept, and the domain is all x ≥ 0 while the range is all y above or below b depending on if the graph increases or decreases.
1) The document discusses rational power functions and provides the standard frequencies of musical notes starting from A above middle C.
2) It gives an expression for calculating the frequency of any note n notes above A as 440*(2^(1/12))^n.
3) Using this expression, it calculates the frequency of the G note that is two notes below A.
The document defines different types of polynomials and their key properties. It discusses linear, quadratic, and cubic polynomials, and defines them based on their highest degree term. It also covers the degree of a polynomial, zeros of polynomials, and the relationship between the zeros and coefficients of quadratic and cubic polynomials. Finally, it discusses the division algorithm for polynomials.
This document contains practice problems and solutions for combining functions. It includes:
1. Multiple choice questions about compositions of functions.
2. Explicit equations for compositions and composite functions using given functions f(x), g(x), h(x), and k(x).
3. Graphing composite functions and determining their domains.
4. Evaluating composite functions for given values of x.
5. Writing composite functions as sums or compositions of simpler functions.
The document provides an overview of topics covered in Grade 9 math term 1, including:
1) Different types of numbers and their properties such as natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
2) Representation of real numbers on the number line and their decimal expansions.
3) Polynomials, including their classification based on terms and degrees. Properties such as zeros of polynomials and dividing one polynomial by another are discussed.
4) Factorization of polynomials using algebraic identities.
The document provides information about polynomials including:
1) A polynomial is an expression constructed from variables and constants using operations of addition, subtraction, multiplication, and exponents.
2) The degree of a polynomial refers to the exponent of its highest term. For example, a quadratic polynomial is degree 2 and a cubic is degree 3.
3) The zeros of a polynomial are the values that make the polynomial equal to 0. Finding the zeros involves solving the polynomial equation.
4) The relationships between the zeros and coefficients of a polynomial can be used to find unknown coefficients or zeros.
Similar to 11X1 T15 01 polynomial definitions (2011) (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
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
2. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
3. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
4. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
n 0 and is an integer
5. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
n 0 and is an integer
coefficients: p0 , p1 , p2 , , pn
6. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
n 0 and is an integer
coefficients: p0 , p1 , p2 , , pn
index (exponent): the powers of the pronumerals.
7. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
n 0 and is an integer
coefficients: p0 , p1 , p2 , , pn
index (exponent): the powers of the pronumerals.
degree (order): the highest index of the polynomial. The
polynomial is called “polynomial of degree n”
8. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
n 0 and is an integer
coefficients: p0 , p1 , p2 , , pn
index (exponent): the powers of the pronumerals.
degree (order): the highest index of the polynomial. The
polynomial is called “polynomial of degree n”
n
leading term: pn x
9. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
n 0 and is an integer
coefficients: p0 , p1 , p2 , , pn
index (exponent): the powers of the pronumerals.
degree (order): the highest index of the polynomial. The
polynomial is called “polynomial of degree n”
n
leading term: pn x
leading coefficient: pn
10. Polynomial Functions
A real polynomial P(x) of degree n is an expression of the form;
P x p0 p1 x p2 x 2 pn1 x n1 pn x n
where : pn 0
n 0 and is an integer
coefficients: p0 , p1 , p2 , , pn
index (exponent): the powers of the pronumerals.
degree (order): the highest index of the polynomial. The
polynomial is called “polynomial of degree n”
n
leading term: pn x
leading coefficient: pn
monic polynomial: leading coefficient is equal to one.
12. P(x) = 0: polynomial equation
y = P(x): polynomial function
13. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
14. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
15. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
4
b) 2
x 3
x2 3
c)
4
d) 7
16. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
b) 2
x 3
x2 3
c)
4
d) 7
17. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3
c)
4
d) 7
18. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3 1 2 3
c) YES, x
4 4 4
d) 7
19. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3 1 2 3
c) YES, x
4 4 4
d) 7 YES, 7x 0
20. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3 1 2 3
c) YES, x
4 4 4
d) 7 YES, 7x 0
(ii) Determine whether P( x) x 3 8 x 1 7 x 11 2 x 2 1 4 x 2 3 is
monic and state its degree.
21. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3 1 2 3
c) YES, x
4 4 4
d) 7 YES, 7x 0
(ii) Determine whether P( x) x 3 8 x 1 7 x 11 2 x 2 1 4 x 2 3 is
monic and state its degree.
P( x) 8 x 4 x3 7 x 11 8 x 4 6 x 2 4 x 2 3
22. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3 1 2 3
c) YES, x
4 4 4
d) 7 YES, 7x 0
(ii) Determine whether P( x) x 3 8 x 1 7 x 11 2 x 2 1 4 x 2 3 is
monic and state its degree.
P( x) 8 x 4 x3 7 x 11 8 x 4 6 x 2 4 x 2 3
x3 2 x 2 7 x 8
23. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3 1 2 3
c) YES, x
4 4 4
d) 7 YES, 7x 0
(ii) Determine whether P( x) x 3 8 x 1 7 x 11 2 x 2 1 4 x 2 3 is
monic and state its degree.
P( x) 8 x 4 x3 7 x 11 8 x 4 6 x 2 4 x 2 3
x3 2 x 2 7 x 8 monic, degree = 3
24. P(x) = 0: polynomial equation
y = P(x): polynomial function
roots: solutions to the polynomial equation P(x) = 0
zeros: the values of x that make polynomial P(x) zero. i.e. the x
intercepts of the graph of the polynomial.
e.g. (i) Which of the following are polynomials?
1
a) 5 x 3 7 x 2
2
NO, can’t have fraction as a power
4
NO, can’t have negative as a power 4 x 3
1
b) 2 2
x 3
x2 3 1 2 3
c) YES, x Exercise 4A; 1, 2acehi, 3bdf,
4 4 4 6bdf, 7, 9d, 10ad, 13
0
d) 7 YES, 7x
(ii) Determine whether P( x) x 3 8 x 1 7 x 11 2 x 2 1 4 x 2 3 is
monic and state its degree.
P( x) 8 x 4 x3 7 x 11 8 x 4 6 x 2 4 x 2 3
x3 2 x 2 7 x 8 monic, degree = 3