This document discusses stationary points (SPs) and how to identify and classify them. It explains that SPs occur when the derivative of a function is equal to 0. It provides examples of finding SPs by taking the derivative, setting it equal to 0, and solving for x. It also introduces the concept of using a nature table to determine whether a SP is a maximum or minimum turning point by examining the sign of the derivative just before and after the SP. The document demonstrates this process on several examples, finding the SPs and using the nature table to classify them.
A function is a relation where each element of the domain is paired with exactly one element of the range. The key characteristics are that there is one and only one output for each input. Some examples are provided to demonstrate determining if a relation represents a function based on this definition. The vertical line test is also introduced as a visual way to determine if a graph depicts a function. Finally, examples are given of evaluating functions for given inputs using function notation.
The document discusses the Fundamental Theorem of Algebra, which establishes that any polynomial of degree n has n complex roots, counting multiplicity. It also states that if a polynomial has real coefficients, then its complex roots must occur in conjugate pairs. The proof of this second part is then shown. It involves using properties of complex conjugates, such as (az)* = a(z*) if a is real, to show that if z is a root, then its conjugate z* is also a root.
The document discusses different methods for finding the point of intersection between two lines, including:
1) Using simultaneous equations by setting the two line equations equal to each other and solving.
2) Using substitution by replacing the y-value in one equation with the expression for y in the other equation.
3) Using the "y=y" tactic by setting the y-expressions in the two equations equal to each other and solving for x and y.
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
Hyperbolic functions are useful in mathematics and physics. The main hyperbolic functions are the hyperbolic sine, cosine, and tangent. Hyperbolic curves include the catenary curve, which describes the shape of a hanging chain and is modeled by the hyperbolic cosine function. Hyperbolic functions are applied in areas like physics, differential equations, and special relativity.
This document discusses stationary points (SPs) and how to identify and classify them. It explains that SPs occur when the derivative of a function is equal to 0. It provides examples of finding SPs by taking the derivative, setting it equal to 0, and solving for x. It also introduces the concept of using a nature table to determine whether a SP is a maximum or minimum turning point by examining the sign of the derivative just before and after the SP. The document demonstrates this process on several examples, finding the SPs and using the nature table to classify them.
A function is a relation where each element of the domain is paired with exactly one element of the range. The key characteristics are that there is one and only one output for each input. Some examples are provided to demonstrate determining if a relation represents a function based on this definition. The vertical line test is also introduced as a visual way to determine if a graph depicts a function. Finally, examples are given of evaluating functions for given inputs using function notation.
The document discusses the Fundamental Theorem of Algebra, which establishes that any polynomial of degree n has n complex roots, counting multiplicity. It also states that if a polynomial has real coefficients, then its complex roots must occur in conjugate pairs. The proof of this second part is then shown. It involves using properties of complex conjugates, such as (az)* = a(z*) if a is real, to show that if z is a root, then its conjugate z* is also a root.
The document discusses different methods for finding the point of intersection between two lines, including:
1) Using simultaneous equations by setting the two line equations equal to each other and solving.
2) Using substitution by replacing the y-value in one equation with the expression for y in the other equation.
3) Using the "y=y" tactic by setting the y-expressions in the two equations equal to each other and solving for x and y.
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
Hyperbolic functions are useful in mathematics and physics. The main hyperbolic functions are the hyperbolic sine, cosine, and tangent. Hyperbolic curves include the catenary curve, which describes the shape of a hanging chain and is modeled by the hyperbolic cosine function. Hyperbolic functions are applied in areas like physics, differential equations, and special relativity.
This is the slides of the UCLA School of Engineering Matlab workshop on Matlab graphics.
Learning Matlab graphics by examples:
- In 2 hours, you will be able to create publication-quality plots.
- Starts from the basic 2D line plots to more advanced 3D plots.
- You will also learn some advanced topics like fine-tuning the appearance of your figure and the concept of handles.
- You will be able to create amazing animations: we use 2D wave equation and Lorentz attractor as examples.
This document provides instructions and examples for simplifying radical expressions. It defines a radical as a square root expression. It then provides 5 problems with step-by-step explanations and solutions for simplifying radical expressions by finding perfect squares under the radical signs. The problems cover simplifying radicals of variables, combining like radicals, and simplifying fractional radicals.
This document provides formulas and rules for taking derivatives and integrals of common functions including polynomials, trigonometric functions, inverse trigonometric functions, exponentials, logarithms, and others. It also describes techniques for evaluating integrals using substitution, integration by parts, trigonometric substitutions, partial fractions, and splitting products and quotients of trigonometric functions.
This document discusses inverse functions. It begins by defining one-to-one functions and inverse functions. A function f is one-to-one if it passes the horizontal line test. The inverse function f^-1 has the domain and range swapped and satisfies the equation f(x) = y if and only if f^-1(y) = x. Examples are provided of finding inverse trigonometric, hyperbolic, and other functions. The document concludes with exercises involving evaluating functions, finding inverse functions, and using function composition to determine if functions are inverses.
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
This document is an introduction to combinatorics presented by A.B. Benedict Balbuena from the University of the Philippines. It discusses fundamental combinatorics concepts like the addition rule, product rule, and inclusion-exclusion principle. Examples of counting problems are provided to illustrate how to use these rules to calculate the number of possible outcomes in situations involving sets, permutations, and combinations.
This document provides an overview of functions and continuity. It begins with essential questions about determining if functions are one-to-one and/or onto, and determining if functions are discrete or continuous. The document then defines key vocabulary terms related to functions, including one-to-one functions, onto functions, discrete relations, continuous relations, and more. It provides examples to demonstrate these concepts, such as evaluating functions, graphing equations, and determining if a relation represents a function.
This document provides a formula booklet covering topics in mathematics. It contains 24 sections with over 100 formulas related to topics like straight lines, circles, parabolas, ellipses, hyperbolas, limits, differentiation, integration, equations, sequences, series, vectors, and more. For each topic, relevant geometric definitions and properties are stated along with the key formulas.
As três frases são:
Este documento fornece fórmulas para calcular perímetros e áreas de várias figuras geométricas planas, incluindo quadrados, retângulos, losangos, trapézios e triângulos. Fornece as fórmulas para calcular o perímetro de cada figura somando as medidas de seus lados. Também fornece as fórmulas para calcular a área de cada figura usando medidas como base, altura, diagonal e metade da altura.
This is the slides of the UCLA School of Engineering Matlab workshop on Matlab graphics.
Learning Matlab graphics by examples:
- In 2 hours, you will be able to create publication-quality plots.
- Starts from the basic 2D line plots to more advanced 3D plots.
- You will also learn some advanced topics like fine-tuning the appearance of your figure and the concept of handles.
- You will be able to create amazing animations: we use 2D wave equation and Lorentz attractor as examples.
This document provides instructions and examples for simplifying radical expressions. It defines a radical as a square root expression. It then provides 5 problems with step-by-step explanations and solutions for simplifying radical expressions by finding perfect squares under the radical signs. The problems cover simplifying radicals of variables, combining like radicals, and simplifying fractional radicals.
This document provides formulas and rules for taking derivatives and integrals of common functions including polynomials, trigonometric functions, inverse trigonometric functions, exponentials, logarithms, and others. It also describes techniques for evaluating integrals using substitution, integration by parts, trigonometric substitutions, partial fractions, and splitting products and quotients of trigonometric functions.
This document discusses inverse functions. It begins by defining one-to-one functions and inverse functions. A function f is one-to-one if it passes the horizontal line test. The inverse function f^-1 has the domain and range swapped and satisfies the equation f(x) = y if and only if f^-1(y) = x. Examples are provided of finding inverse trigonometric, hyperbolic, and other functions. The document concludes with exercises involving evaluating functions, finding inverse functions, and using function composition to determine if functions are inverses.
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
This document is an introduction to combinatorics presented by A.B. Benedict Balbuena from the University of the Philippines. It discusses fundamental combinatorics concepts like the addition rule, product rule, and inclusion-exclusion principle. Examples of counting problems are provided to illustrate how to use these rules to calculate the number of possible outcomes in situations involving sets, permutations, and combinations.
This document provides an overview of functions and continuity. It begins with essential questions about determining if functions are one-to-one and/or onto, and determining if functions are discrete or continuous. The document then defines key vocabulary terms related to functions, including one-to-one functions, onto functions, discrete relations, continuous relations, and more. It provides examples to demonstrate these concepts, such as evaluating functions, graphing equations, and determining if a relation represents a function.
This document provides a formula booklet covering topics in mathematics. It contains 24 sections with over 100 formulas related to topics like straight lines, circles, parabolas, ellipses, hyperbolas, limits, differentiation, integration, equations, sequences, series, vectors, and more. For each topic, relevant geometric definitions and properties are stated along with the key formulas.
As três frases são:
Este documento fornece fórmulas para calcular perímetros e áreas de várias figuras geométricas planas, incluindo quadrados, retângulos, losangos, trapézios e triângulos. Fornece as fórmulas para calcular o perímetro de cada figura somando as medidas de seus lados. Também fornece as fórmulas para calcular a área de cada figura usando medidas como base, altura, diagonal e metade da altura.