This document describes a module on hyperbolic functions and differentiation. It begins by introducing hyperbolic functions such as sinh, cosh, and tanh, which are analogous to trigonometric functions but correspond to hyperbolas rather than circles. It defines these functions in terms of exponentials and discusses their properties. The document then examines series expressions for the hyperbolic functions and explores their connection to trigonometric functions through complex numbers. It concludes by covering advanced hyperbolic functions, identities, and differentiating hyperbolic functions.
The document discusses the process for finding the eigenvalues of a square matrix. It begins by defining the characteristic equation as det(A - λI) = 0, where A is the matrix and λI subtracts λ from the diagonal. The characteristic polynomial is obtained by computing this determinant. For a 2x2 matrix, it is a quadratic equation that can be factored to find the two eigenvalues. Larger matrices may require numerical methods. The sum of eigenvalues equals the trace, and their product equals the determinant. A matrix will always have n eigenvalues for its size n. An example problem is presented to demonstrate the full process.
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.
The document discusses relations and functions. It defines a relation as a set of ordered pairs, with the domain as the set of first coordinates and the range as the set of second coordinates. A function is a special type of relation where each element of the domain is paired with exactly one element of the range, or no two ordered pairs have the same first coordinate. Examples are provided to illustrate relations, identifying their domains and ranges, and to demonstrate the vertical line test for determining if a relation is a function.
The document discusses relations, functions, domains, ranges, and evaluating functions. A relation is a set of ordered pairs, while a function is a relation where each input is mapped to only one output. To determine if a relation is a function, one can use the vertical line test or create a mapping diagram. The domain of a relation is the set of all inputs, while the range is the set of all outputs. Evaluating a function involves substituting inputs into the function rule to obtain the corresponding outputs.
This document defines relations and functions in mathematics. A relation is a set of ordered pairs where the domain is the set of all x values and the range is the set of all y values. A function assigns each element in the domain (set of x values) to exactly one element in the range (set of y values). Functions are commonly represented by letters like f(x), where f denotes the name of the function and x is the variable. The left side of a function equation tells us the name and variable of the function, not that the function is being multiplied.
1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
A relation is a set of ordered pairs where the domain is the set of all first elements and the range is the set of all second elements. A function is a special type of relation where each element of the domain is mapped to exactly one element of the range. When finding the domain of a function, valid inputs are any values that do not result in division by zero or taking the square root of a negative number.
The document discusses the process for finding the eigenvalues of a square matrix. It begins by defining the characteristic equation as det(A - λI) = 0, where A is the matrix and λI subtracts λ from the diagonal. The characteristic polynomial is obtained by computing this determinant. For a 2x2 matrix, it is a quadratic equation that can be factored to find the two eigenvalues. Larger matrices may require numerical methods. The sum of eigenvalues equals the trace, and their product equals the determinant. A matrix will always have n eigenvalues for its size n. An example problem is presented to demonstrate the full process.
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.
The document discusses relations and functions. It defines a relation as a set of ordered pairs, with the domain as the set of first coordinates and the range as the set of second coordinates. A function is a special type of relation where each element of the domain is paired with exactly one element of the range, or no two ordered pairs have the same first coordinate. Examples are provided to illustrate relations, identifying their domains and ranges, and to demonstrate the vertical line test for determining if a relation is a function.
The document discusses relations, functions, domains, ranges, and evaluating functions. A relation is a set of ordered pairs, while a function is a relation where each input is mapped to only one output. To determine if a relation is a function, one can use the vertical line test or create a mapping diagram. The domain of a relation is the set of all inputs, while the range is the set of all outputs. Evaluating a function involves substituting inputs into the function rule to obtain the corresponding outputs.
This document defines relations and functions in mathematics. A relation is a set of ordered pairs where the domain is the set of all x values and the range is the set of all y values. A function assigns each element in the domain (set of x values) to exactly one element in the range (set of y values). Functions are commonly represented by letters like f(x), where f denotes the name of the function and x is the variable. The left side of a function equation tells us the name and variable of the function, not that the function is being multiplied.
1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
A relation is a set of ordered pairs where the domain is the set of all first elements and the range is the set of all second elements. A function is a special type of relation where each element of the domain is mapped to exactly one element of the range. When finding the domain of a function, valid inputs are any values that do not result in division by zero or taking the square root of a negative number.
The document discusses functions, relations, domains, ranges and using vertical line tests, mappings, tables and graphs to represent and analyze functions. It provides examples of determining if a relation is a function, finding domains and ranges, modeling function rules with tables and graphs using given domains, and solving other function problems.
The document contains 37 multiple choice questions about sets, relations, functions and their properties. Specifically, it contains questions about: determining if a relation is reflexive, symmetric, transitive or a function; evaluating functions at given values; determining the domain and range of functions; determining if functions are one-to-one, onto or both; and other properties of functions like being even, odd or periodic. The answers to all questions are provided at the end.
This document defines key concepts related to relations and functions including ordered pairs, coordinate planes, relations, functions, graphical representations, domain and range, and the vertical line test. It provides examples and explanations of these terms. Ordered pairs represent points on a graph and functions are defined as sets of ordered pairs where no x-value is repeated, while relations allow repeated x-values. The vertical line test can be used to determine if a relation qualifies as a function by checking if a vertical line passes through only one point for each x-value.
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
This document outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
The document provides information about functions and relations. It defines a function as a relation where each x-value is paired with exactly one y-value. To determine if a relation is a function, it describes using the vertical line test, where a relation is a function if a vertical line can only intersect the graph at one point. It gives examples of applying the vertical line test to graphs and determining the domain and range of relations.
1. The document discusses relations and functions, including identifying their domain and range. It provides examples of plotting points on a Cartesian plane and using graphs to represent equations.
2. Functions are defined as relations where each input is mapped to only one output. Several examples are given to demonstrate determining if a relation qualifies as a function using techniques like the vertical line test.
3. The key concepts of domain, range, and using graphs are illustrated through multiple examples of sketching and analyzing relations and functions.
1. The document discusses functions and how to determine if a relation is a function. A function is a relation where each input is mapped to exactly one output.
2. It provides examples of evaluating whether relations are functions and how to identify the domain and range. Functions can be represented by ordered pairs in a set or graphed on a coordinate plane.
3. The vertical line test is introduced as a way to visually check if a relation is a function by seeing if any vertical line passes through more than one point.
The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of x-values and the range is the set of y-values. A function is a relation where each domain value is paired with exactly one range value. The document provides examples of determining if a relation represents a function using the vertical line test and evaluating functions using function notation.
A data structure is a specialized format for organizing, processing, retrieving and storing data. ... For instance, in an object-oriented programming language, the data structure and its associated methods are bound together as part of a class definition.
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
This document provides examples and explanations of functions and relations. It begins by defining a relation as a set of ordered pairs and a function as a relation where each element of the domain has exactly one match in the range. Several examples are given of functions and non-functions. Students are then asked to determine whether additional relations are functions or not and to provide their own examples using concepts like planets, leadership values, and colors.
1) A function is a relationship between two sets of data where each input value in the domain corresponds to exactly one output value in the range.
2) Function notation represents a relationship as f(x) where x is plugged into the original equation. For example, the function f(x)=5x+9 defines the relationship.
3) To determine if a graph represents a function, the vertical line test can be used. If a vertical line can only intersect the graph in one point, it passes the test and is a function. If a line intersects in multiple points, it is not a function.
This document discusses character tables and their application in group theory. It contains the following key points:
1. Character tables describe the transformation properties of molecular coordinates and basis functions under different symmetry operations.
2. Group theory can be used to predict chirality, polarity, hybrid orbitals, molecular orbitals, and vibrational spectra of molecules based on their symmetry properties.
3. Molecular vibrations are classified as infrared (IR) active if the vibration changes the dipole moment and Raman active if it changes the polarizability of the molecule, based on the symmetry of the dipole moment and polarizability.
This document discusses relations and functions for class 12. It defines concepts like cartesian product, relations, domain and codomain, range, types of relations such as reflexive, symmetric, transitive and equivalence relations. It also defines functions, one-to-one and onto functions, bijective functions, composition of functions, and inverse functions. An example is provided to show that a given relation is reflexive but not symmetric or transitive. Another example shows a given function is one-to-one but not onto.
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
This document discusses solving systems of linear equations in three or more variables using row operations, matrices, and their inverses. It covers putting systems into row echelon form through elementary row operations, solving systems using the inverse of the coefficient matrix when it exists, and fitting a quadratic function to data points by setting up the problem as a system of linear equations. Examples are provided to illustrate these concepts and techniques.
The document reviews key concepts about functions including domain, range, and evaluating functions. It provides examples of determining if a relation is a function using mapping diagrams and the vertical line test. It also gives examples of finding the domain and range of functions from graphs and equations. Practice problems are included for students to determine domains, ranges, and evaluate functions.
Bab 4 dan bab 5 algebra and trigonometryCiciPajarakan
This document discusses exponential functions and their properties. Exponential functions have the form f(x) = ax, where a is the base. They model phenomena like population growth, compound interest, and radioactive decay. Exponential functions either increase or decrease rapidly depending on whether the base a is greater than or less than 1. Their graphs have a characteristic shape, with the x-axis as a horizontal asymptote. Logarithmic functions are the inverses of exponential functions and can be used to solve equations involving exponential models.
Important Questions of fourier series with theoretical study Engg. Mathem...Mohammad Imran
This document provides information about Fourier series and even/odd functions from the Jahangirbad Institute of Technology. It defines Fourier series and even/odd functions, lists their properties, and provides 14 important questions about finding Fourier series for various periodic functions over different intervals. The questions aim to find the Fourier series and deduce trigonometric identities.
The document discusses functions, relations, domains, ranges and using vertical line tests, mappings, tables and graphs to represent and analyze functions. It provides examples of determining if a relation is a function, finding domains and ranges, modeling function rules with tables and graphs using given domains, and solving other function problems.
The document contains 37 multiple choice questions about sets, relations, functions and their properties. Specifically, it contains questions about: determining if a relation is reflexive, symmetric, transitive or a function; evaluating functions at given values; determining the domain and range of functions; determining if functions are one-to-one, onto or both; and other properties of functions like being even, odd or periodic. The answers to all questions are provided at the end.
This document defines key concepts related to relations and functions including ordered pairs, coordinate planes, relations, functions, graphical representations, domain and range, and the vertical line test. It provides examples and explanations of these terms. Ordered pairs represent points on a graph and functions are defined as sets of ordered pairs where no x-value is repeated, while relations allow repeated x-values. The vertical line test can be used to determine if a relation qualifies as a function by checking if a vertical line passes through only one point for each x-value.
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
This document outlines a summer course in linear algebra. It covers topics such as sets and operations on sets, relations and functions, polynomial theorems, and exponential and logarithmic equations. The course will teach students how to solve various types of word problems involving linear equations in two variables. It will also cover matrices, including Gaussian elimination and determinants.
The document provides information about functions and relations. It defines a function as a relation where each x-value is paired with exactly one y-value. To determine if a relation is a function, it describes using the vertical line test, where a relation is a function if a vertical line can only intersect the graph at one point. It gives examples of applying the vertical line test to graphs and determining the domain and range of relations.
1. The document discusses relations and functions, including identifying their domain and range. It provides examples of plotting points on a Cartesian plane and using graphs to represent equations.
2. Functions are defined as relations where each input is mapped to only one output. Several examples are given to demonstrate determining if a relation qualifies as a function using techniques like the vertical line test.
3. The key concepts of domain, range, and using graphs are illustrated through multiple examples of sketching and analyzing relations and functions.
1. The document discusses functions and how to determine if a relation is a function. A function is a relation where each input is mapped to exactly one output.
2. It provides examples of evaluating whether relations are functions and how to identify the domain and range. Functions can be represented by ordered pairs in a set or graphed on a coordinate plane.
3. The vertical line test is introduced as a way to visually check if a relation is a function by seeing if any vertical line passes through more than one point.
The document defines relations and functions. A relation is a set of ordered pairs where the domain is the set of x-values and the range is the set of y-values. A function is a relation where each domain value is paired with exactly one range value. The document provides examples of determining if a relation represents a function using the vertical line test and evaluating functions using function notation.
A data structure is a specialized format for organizing, processing, retrieving and storing data. ... For instance, in an object-oriented programming language, the data structure and its associated methods are bound together as part of a class definition.
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
This document provides examples and explanations of functions and relations. It begins by defining a relation as a set of ordered pairs and a function as a relation where each element of the domain has exactly one match in the range. Several examples are given of functions and non-functions. Students are then asked to determine whether additional relations are functions or not and to provide their own examples using concepts like planets, leadership values, and colors.
1) A function is a relationship between two sets of data where each input value in the domain corresponds to exactly one output value in the range.
2) Function notation represents a relationship as f(x) where x is plugged into the original equation. For example, the function f(x)=5x+9 defines the relationship.
3) To determine if a graph represents a function, the vertical line test can be used. If a vertical line can only intersect the graph in one point, it passes the test and is a function. If a line intersects in multiple points, it is not a function.
This document discusses character tables and their application in group theory. It contains the following key points:
1. Character tables describe the transformation properties of molecular coordinates and basis functions under different symmetry operations.
2. Group theory can be used to predict chirality, polarity, hybrid orbitals, molecular orbitals, and vibrational spectra of molecules based on their symmetry properties.
3. Molecular vibrations are classified as infrared (IR) active if the vibration changes the dipole moment and Raman active if it changes the polarizability of the molecule, based on the symmetry of the dipole moment and polarizability.
This document discusses relations and functions for class 12. It defines concepts like cartesian product, relations, domain and codomain, range, types of relations such as reflexive, symmetric, transitive and equivalence relations. It also defines functions, one-to-one and onto functions, bijective functions, composition of functions, and inverse functions. An example is provided to show that a given relation is reflexive but not symmetric or transitive. Another example shows a given function is one-to-one but not onto.
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
This document discusses solving systems of linear equations in three or more variables using row operations, matrices, and their inverses. It covers putting systems into row echelon form through elementary row operations, solving systems using the inverse of the coefficient matrix when it exists, and fitting a quadratic function to data points by setting up the problem as a system of linear equations. Examples are provided to illustrate these concepts and techniques.
The document reviews key concepts about functions including domain, range, and evaluating functions. It provides examples of determining if a relation is a function using mapping diagrams and the vertical line test. It also gives examples of finding the domain and range of functions from graphs and equations. Practice problems are included for students to determine domains, ranges, and evaluate functions.
Bab 4 dan bab 5 algebra and trigonometryCiciPajarakan
This document discusses exponential functions and their properties. Exponential functions have the form f(x) = ax, where a is the base. They model phenomena like population growth, compound interest, and radioactive decay. Exponential functions either increase or decrease rapidly depending on whether the base a is greater than or less than 1. Their graphs have a characteristic shape, with the x-axis as a horizontal asymptote. Logarithmic functions are the inverses of exponential functions and can be used to solve equations involving exponential models.
Important Questions of fourier series with theoretical study Engg. Mathem...Mohammad Imran
This document provides information about Fourier series and even/odd functions from the Jahangirbad Institute of Technology. It defines Fourier series and even/odd functions, lists their properties, and provides 14 important questions about finding Fourier series for various periodic functions over different intervals. The questions aim to find the Fourier series and deduce trigonometric identities.
This document provides an overview of Fourier analysis techniques for communication engineering experiments. It introduces Fourier series as a way to expand periodic signals into a sum of complex exponentials. The Fourier series coefficients represent the contribution of each harmonic frequency. MATLAB will be used to implement Fourier analysis and observe its applications in communication systems. Students are expected to review basic MATLAB commands and complete pre-lab exercises on vector operations and plotting signals before conducting the experiment.
1) The document outlines a course curriculum that covers functions, rational functions, one-to-one functions, exponential functions, and logarithmic functions over 32 hours spread across 8 weeks.
2) It provides the chapter titles and learning objectives for each chapter, along with the topics and hours allocated to each lesson.
3) Key concepts covered include functions as models of real-life situations, representing functions as sets of ordered pairs, tables, graphs, and piecewise functions.
This document discusses inverses of functions. It begins by introducing the concept of inverses and using the story of the Sneetches to provide an example. It then discusses how to find the inverse of a function by switching the x's and y's and solving for y. Several examples are worked through. It emphasizes that a function must be one-to-one to have an inverse and discusses using the horizontal line test to determine if a function is one-to-one. The document concludes by discussing how the composition of a function and its inverse will result in the original input and exploring examples of functions that are inverses of each other.
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docxanhlodge
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
The document outlines the course outline for a General Mathematics course covering several topics:
1. Functions and relations including ordered pairs, arrow diagrams, tables of values, equations, and graphs.
2. Specific function types including constant, linear, quadratic, cubic, and other functions.
3. Evaluating, operating on (adding, subtracting, multiplying, dividing, composing), and finding inverses of functions.
4. Rational, exponential, and logarithmic functions and their properties.
5. Basic business mathematics topics like simple and compound interest.
6. Logic including propositions, truth values, and fallacies.
The document discusses quadratic functions. It begins by defining a quadratic function as a function of the form f(x) = ax2 + bx + c, where the highest exponent is 2. It explains that the graph of a quadratic function is a smooth curve called a parabola. It also notes that you can determine if a function is quadratic based on its equation having x2 as the highest term or if its graph is a parabola. The document provides examples of quadratic functions and non-quadratic functions to illustrate these concepts.
Module 4 exponential and logarithmic functionsdionesioable
This document provides an overview of a module on logarithmic functions. It discusses the definition of logarithmic functions as the inverse of exponential functions, how to graph logarithmic functions by reflecting the graph of the corresponding exponential function across the line y=x, and properties of logarithmic function graphs like their domains, ranges, asymptotes, and behavior. It also covers laws of logarithms and how to solve logarithmic equations. The document is designed to teach students to define logarithmic functions, graph them, use laws of logarithms, and solve simple logarithmic equations.
This document discusses functions and how to determine if a relation is a function. It defines a function as a relation where each element of the domain maps to exactly one element of the range. It provides examples of tables, coordinate points, and graphs, and explains how to use the vertical line test to determine if a graph represents a function. It also discusses how to determine if a relation is a function algebraically by solving for the output variable and checking if there is only one output for each input.
Lab 5 template Lab 5 - Your Name - MAT 275 Lab The M.docxsmile790243
Lab 5 template
%% Lab 5 - Your Name - MAT 275 Lab
% The Mass-Spring System
%% EX 1 10 pts
%A) 1 pts | short comment
%
%B) 2 pts | short comment
%
%C) 1 pts | short comment
%
%D) 1 pts
%E) 2 pts | List the first 3-4 t values either in decimal format or as
%fractions involving pi
%F) 3 pts | comments. | (1 pts for including two distinct graphs, each with y(t) and v(t) plotted)
%% EX 2 10 pts
%A) 5 pts
% add commands to LAB05ex1 to compute and plot E(t). Then use ylim([~,~]) to change the yaxis limits.
% You don't need to include this code but at least one plot of E(t) and a comment must be
% included!
%B) 2 pts | write out main steps here
% first differentiate E(t) with respect to t using the chain rule. Then
% make substitutions using the expression for omega0 and using the
% differential equation
%C) 3 pts | show plot and comment
%% EX 3 10 pts
%A) 3 pts | modify the system of equations in LAB05ex1a
% write the t value and either a) show correponding graph or b) explain given matlab
% commands
%B) 2 pts | write t value and max |V| value; include figure
%note: velocity magnitude is like absolute value!
%C) 3 pts | include 3 figures here + comments.
% use title('text') to attach a title to the figure
%D) 2 pts | What needs to happen (in terms of the characteristic equation)
%in order for there to be no oscillations? Impose a condition on the
%characteristic equation to find the critical c value. Write out main steps
%% EX4 10 pts
% A) 5 pts | include 1 figure and comment
%B) 2 pts
% again find dE/dt using the chain rule and make substitutions based on the
% differential equation. You should reach an expression for dE/dt which is
% in terms of y'
%C) 3 pts | include one figure and comment
Exercise (1):
function LAB05ex1
m = 1; % mass [kg]
k = 9; % spring constant [N/m]
omega0=sqrt(k/m);
y0=0.4; v0=0; % initial conditions
[t,Y]=ode45(@f,[0,10],[y0,v0],[],omega0); % solve for 0<t<10
y=Y(:,1); v=Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,'b+-',t,v,'ro-'); % time series for y and v
grid on;
%------------------------------------------------------
function dYdt= f(t,Y,omega0)
y = Y(1); v= Y(2);
dYdt = [v; -omega0^2*y];
Exercise (1a):
function LAB05ex1a
m = 1; % mass [kg]
k = 9; % spring constant [N/m]
c = 1; % friction coefficient [Ns/m]
omega0 = sqrt(k/m); p = c/(2*m);
y0 = 0.4; v0 = 0; % initial conditions
[t,Y]=ode45(@f,[0,10],[y0,v0],[],omega0,p); % solve for 0<t<10
y=Y(:,1); v=Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,'b+-',t,v,'ro-'); % time series for y and v
grid on
%------------------------------------------------------
function dYdt= f(t,Y,omega0,p)
y = Y(1); v= Y(2);
dYdt = [v; ?? ]; % fill-in dv/dt
More instructions for the l ...
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
These notes were developed by Wilson J. Rugh for a signals and systems course taught at Johns Hopkins University between 2000-2005. The notes cover fundamental concepts in analyzing signals and systems in both the time and frequency domains. Key topics include signal classifications, operations on signals, continuous-time and discrete-time linear time-invariant systems, Fourier series, Fourier transforms, and Laplace transforms. The notes are provided freely for educational purposes only.
These notes were developed by Wilson J. Rugh for a signals and systems course at Johns Hopkins University between 2000-2005. The notes cover traditional concepts in time and frequency domain analysis of signals and systems. The document includes a table of contents outlining the topics covered in the notes, which include signal classes, systems, discrete-time and continuous-time linear time-invariant systems, signal representation using Fourier series and transforms. The notes are provided freely online but use is permitted only for non-commercial educational purposes.
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
Second or fourth-order finite difference operators, which one is most effective?Premier Publishers
This paper presents higher-order finite difference (FD) formulas for the spatial approximation of the time-dependent reaction-diffusion problems with a clear justification through examples, “why fourth-order FD formula is preferred to its second-order counterpart” that has been widely used in literature. As a consequence, methods for the solution of initial and boundary value PDEs, such as the method of lines (MOL), is of broad interest in science and engineering. This procedure begins with discretizing the spatial derivatives in the PDE with algebraic approximations. The key idea of MOL is to replace the spatial derivatives in the PDE with the algebraic approximations. Once this procedure is done, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. In other words, only one independent variable is remaining, the resulting semi-discrete problem has now become a system of coupled ordinary differential equations (ODEs) in time. Thus, we can apply any integration algorithm for the initial value ODEs to compute an approximate numerical solution to the PDE. Analysis of the basic properties of these schemes such as the order of accuracy, convergence, consistency, stability and symmetry are well examined.
This document outlines a course on Calculus and Numerical Methods over two parts. Part one covers calculus topics like functions, graphs, limits, differentiation, integration and differential equations over 7 weeks. Part two covers numerical methods topics like errors, root finding, interpolation, numerical differentiation and integration, and solving ordinary differential equations over 6 weeks. There are three learning outcomes focusing on applying calculus and numerical methods concepts, solving problems using programming, and solving real-life problems. Students will be assessed through tests, assignments, midterms and a final exam testing the different learning outcomes. The course then provides details on the topics and subtopics to be covered in the first part on functions and graphs.