The document is a math worksheet containing calculus problems involving functions. It includes 21 problems involving operations on functions such as composition, inversion and transformations of function graphs. The problems involve determining expressions for composed functions, inverses, graphs of related functions obtained through transformations of an original function graph. The document also provides answers to the problems.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function g∘f. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
The document discusses inverse functions. It defines a one-to-one function as a function where the horizontal line test shows that every horizontal line intersects the graph at most one point. This ensures that each input is mapped to a single output. An inverse function undoes the original function - if f(x) is the original function, its inverse f^-1(x) satisfies f^-1(f(x)) = x.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
- The document discusses linear approximations and Newton's method for finding roots of functions.
- It provides examples of using the linear approximation L(x) = f(x0) + f'(x0)(x - x0) to estimate function values and find roots.
- Newton's method is introduced as xi+1 = xi - f(xi)/f'(xi) to iteratively find better approximations of roots.
- Several examples are worked through step-by-step to demonstrate both linear approximations and Newton's method.
This document provides examples and exercises on determining composite functions from given functions. It includes:
- Examples of determining possible functions f and g for composite functions like y = (x + 4)2 and y = √x + 5.
- A table sketching and finding domains and ranges for various composite functions like y = f(f(x)) and y = f(g(x)) given functions f(x) and g(x).
- Exercises to determine composite functions f(g(x)) and possible functions f, g, and h for more complex functions like y = x2 - 6x + 5.
- Questions about restrictions on variables and domains for composite functions
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition.
(2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function g∘f. The domain and range of the composite functions are discussed.
(3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
The document discusses inverse functions. It defines a one-to-one function as a function where the horizontal line test shows that every horizontal line intersects the graph at most one point. This ensures that each input is mapped to a single output. An inverse function undoes the original function - if f(x) is the original function, its inverse f^-1(x) satisfies f^-1(f(x)) = x.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
- The document discusses linear approximations and Newton's method for finding roots of functions.
- It provides examples of using the linear approximation L(x) = f(x0) + f'(x0)(x - x0) to estimate function values and find roots.
- Newton's method is introduced as xi+1 = xi - f(xi)/f'(xi) to iteratively find better approximations of roots.
- Several examples are worked through step-by-step to demonstrate both linear approximations and Newton's method.
This document provides examples and exercises on determining composite functions from given functions. It includes:
- Examples of determining possible functions f and g for composite functions like y = (x + 4)2 and y = √x + 5.
- A table sketching and finding domains and ranges for various composite functions like y = f(f(x)) and y = f(g(x)) given functions f(x) and g(x).
- Exercises to determine composite functions f(g(x)) and possible functions f, g, and h for more complex functions like y = x2 - 6x + 5.
- Questions about restrictions on variables and domains for composite functions
The document summarizes key concepts about composition functions:
1. Composition functions are not commutative.
2. They are associative.
3. Identity functions leave other functions unchanged when composed.
The document also provides examples of determining the component functions of composition functions and finding inverse functions.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.
The document discusses combining functions through addition, subtraction, multiplication, and division. Some key points:
- Functions can be combined using the same rules as algebraic expressions, such as f(x) + g(x) = (f + g)(x).
- An example demonstrates combining two functions f(x) and g(x) through addition, subtraction, multiplication, and division.
- Composite functions are discussed where the output of one function acts as the input for another function, written as f(g(x)).
- Several examples demonstrate evaluating composite functions by first evaluating the inner function and then the outer function.
- It is shown that f(g(x)) is not
The document provides exercises on composition of functions. It gives the definitions of various functions f(x) and g(x) and asks to calculate f(x)+g(x), f(x)-g(x), f(x)*g(x), fog(x), gof(x), fof(x), and gog(x) for different functions f(x) and g(x). It provides 30 problems to calculate the composition of the given functions through addition, subtraction, multiplication and composition of functions.
Profº. Marcelo Santos Chaves - Cálculo I (Limites e Continuidades) - Exercíci...MarcelloSantosChaves
1. The document discusses limits and continuities. It provides solutions to calculating the limits of 6 different functions as x approaches certain values.
2. The solutions involve algebraic manipulations such as factoring, simplifying, and applying limit properties. Various limit results are obtained such as 1, -6, 0.
3. The techniques demonstrated include making substitutions to simplify indeterminate forms, factoring, and taking limits of rational functions as the variables approach certain values.
This document provides sample questions and answers related to various mathematics topics, including functions, quadratic equations/functions, simultaneous equations, indices and logarithms, and coordinate geometry.
The questions are multiple choice or short answer questions assessing skills like solving equations, graphing functions, finding inverse functions, and working with logarithmic and exponential expressions.
The answers section provides fully worked out solutions to each question in a clear format. This document appears to be a reference for teachers or students to use for exam preparation or review, with the goal of assessing conceptual understanding of core algebra and geometry topics.
1. The function f(x) is defined as 2 + x^2 and g(x) is defined as 1 + x^2.
2. It is given that f(g(x)) = 3 + 2(g(x) - 1) + (g(x) - 1)^2.
3. Substituting g(x) = 1 + x^2 into the equation for f(g(x)) yields f(x) = 2 + x^2.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
This document contains solutions to checkpoint questions about transforming graphs of functions. It includes examples of translating graphs by shifting them horizontally and vertically based on changes to the x and y variables in the function. It also contains an example of reflecting a graph across the x-axis. The questions require sketching the transformed graphs on grids and writing the equations of the transformed functions based on the given transformations.
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
This document contains a midterm exam for an engineering mathematics course. It consists of 4 problems:
1. Finding the general solution to two linear ODEs.
2. Finding the complementary and particular solutions for a given non-homogeneous linear ODE, and using them to solve an IVP.
3. Repeating steps from problem 2 for another given non-homogeneous linear ODE.
4. Modeling and solving an ODE describing the motion of a damped spring-mass system subject to an external force.
This document provides solutions to review problems involving combining functions through addition, subtraction, multiplication, division, and composition. Some key examples include:
- Sketching the graphs of f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x) given the graphs of f(x) and g(x)
- Writing explicit equations for combinations of functions and determining their domains and ranges
- Evaluating composite functions like f(g(x)) and g(f(x)) given definitions of f(x) and g(x)
- Determining if two functions are inverses using their compositions
The document contains 15 multiple choice questions about functions. The questions cover topics such as exponential decay functions, quadratic functions, maximum and minimum values of functions, and function definitions and properties.
1) The student solved several integral evaluation problems and derivative problems.
2) They sketched the region bounded by two curves and found its area.
3) Several functions were analyzed, including finding their derivatives, extrema, concavity, asymptotes and sketching their graphs.
4) Some proofs and word problems involving applications of calculus like radioactive decay were also addressed.
This document discusses linear equations and curve fitting. It provides 18 examples of using a linear system to solve for the coefficients of linear, quadratic, and cubic polynomials that fit given data points. It also provides examples of using a linear system to solve for the coefficients of circle and central conic equations that fit given points. The linear systems are set up and solved, providing the resulting equations that fit the data in each example.
Este documento descreve uma oficina sobre o uso educativo de blogs para professores. A oficina ocorrerá em dois dias com 8 horas de duração e tem como objetivo principal apresentar como blogs podem ser aplicados no ambiente educacional, permitindo a divulgação de trabalhos de sala de aula e a interação entre professores e alunos.
El documento describe un estudio sobre el efecto de diferentes dosis de fósforo en el crecimiento de plantas de maíz. El estudio incluyó la medición de parámetros como la altura, ancho de hojas y peso de las plantas tratadas con diferentes niveles de fósforo, con el objetivo de observar las diferencias en el desarrollo del cultivo. Los resultados se presentaron en gráficos de crecimiento y peso para cada tratamiento.
Este documento discute o projeto Esfera, uma iniciativa para melhorar a qualidade da resposta humanitária a desastres. O projeto fornece normas mínimas, indicadores e orientações para setores como água, saneamento, segurança alimentar e abrigo. O documento descreve a evolução do projeto desde 1997 e como ele visa tornar os princípios humanitários em ações concretas para ajudar as populações afetadas por desastres.
O documento apresenta uma lista de locais ao redor do mundo, incluindo Veneza, Canadá, Tailândia, Amazônia, Grécia, Alpes Suíços, Mar Negro, Panamá, Mar Vermelho, Rio de Janeiro e Paris, destacando a beleza do planeta mesmo à noite. A música Il mondo, de Sérgio Endrigo, é apresentada por Renato Cardoso em um site sobre Bauru.
The document summarizes key concepts about composition functions:
1. Composition functions are not commutative.
2. They are associative.
3. Identity functions leave other functions unchanged when composed.
The document also provides examples of determining the component functions of composition functions and finding inverse functions.
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.
The document discusses combining functions through addition, subtraction, multiplication, and division. Some key points:
- Functions can be combined using the same rules as algebraic expressions, such as f(x) + g(x) = (f + g)(x).
- An example demonstrates combining two functions f(x) and g(x) through addition, subtraction, multiplication, and division.
- Composite functions are discussed where the output of one function acts as the input for another function, written as f(g(x)).
- Several examples demonstrate evaluating composite functions by first evaluating the inner function and then the outer function.
- It is shown that f(g(x)) is not
The document provides exercises on composition of functions. It gives the definitions of various functions f(x) and g(x) and asks to calculate f(x)+g(x), f(x)-g(x), f(x)*g(x), fog(x), gof(x), fof(x), and gog(x) for different functions f(x) and g(x). It provides 30 problems to calculate the composition of the given functions through addition, subtraction, multiplication and composition of functions.
Profº. Marcelo Santos Chaves - Cálculo I (Limites e Continuidades) - Exercíci...MarcelloSantosChaves
1. The document discusses limits and continuities. It provides solutions to calculating the limits of 6 different functions as x approaches certain values.
2. The solutions involve algebraic manipulations such as factoring, simplifying, and applying limit properties. Various limit results are obtained such as 1, -6, 0.
3. The techniques demonstrated include making substitutions to simplify indeterminate forms, factoring, and taking limits of rational functions as the variables approach certain values.
This document provides sample questions and answers related to various mathematics topics, including functions, quadratic equations/functions, simultaneous equations, indices and logarithms, and coordinate geometry.
The questions are multiple choice or short answer questions assessing skills like solving equations, graphing functions, finding inverse functions, and working with logarithmic and exponential expressions.
The answers section provides fully worked out solutions to each question in a clear format. This document appears to be a reference for teachers or students to use for exam preparation or review, with the goal of assessing conceptual understanding of core algebra and geometry topics.
1. The function f(x) is defined as 2 + x^2 and g(x) is defined as 1 + x^2.
2. It is given that f(g(x)) = 3 + 2(g(x) - 1) + (g(x) - 1)^2.
3. Substituting g(x) = 1 + x^2 into the equation for f(g(x)) yields f(x) = 2 + x^2.
1. The document defines relations and functions. It provides examples of relations including r1, r2, r3, r4, and r5.
2. Functions are defined as mappings from a domain A to a range B. Examples of one-to-one, many-to-one, and onto functions are given.
3. Different types of functions are described including constant, linear, quadratic, polynomial, rational, absolute value, step, and periodic functions. Examples are provided for each type.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
This document contains solutions to checkpoint questions about transforming graphs of functions. It includes examples of translating graphs by shifting them horizontally and vertically based on changes to the x and y variables in the function. It also contains an example of reflecting a graph across the x-axis. The questions require sketching the transformed graphs on grids and writing the equations of the transformed functions based on the given transformations.
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
This document contains a midterm exam for an engineering mathematics course. It consists of 4 problems:
1. Finding the general solution to two linear ODEs.
2. Finding the complementary and particular solutions for a given non-homogeneous linear ODE, and using them to solve an IVP.
3. Repeating steps from problem 2 for another given non-homogeneous linear ODE.
4. Modeling and solving an ODE describing the motion of a damped spring-mass system subject to an external force.
This document provides solutions to review problems involving combining functions through addition, subtraction, multiplication, division, and composition. Some key examples include:
- Sketching the graphs of f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x) given the graphs of f(x) and g(x)
- Writing explicit equations for combinations of functions and determining their domains and ranges
- Evaluating composite functions like f(g(x)) and g(f(x)) given definitions of f(x) and g(x)
- Determining if two functions are inverses using their compositions
The document contains 15 multiple choice questions about functions. The questions cover topics such as exponential decay functions, quadratic functions, maximum and minimum values of functions, and function definitions and properties.
1) The student solved several integral evaluation problems and derivative problems.
2) They sketched the region bounded by two curves and found its area.
3) Several functions were analyzed, including finding their derivatives, extrema, concavity, asymptotes and sketching their graphs.
4) Some proofs and word problems involving applications of calculus like radioactive decay were also addressed.
This document discusses linear equations and curve fitting. It provides 18 examples of using a linear system to solve for the coefficients of linear, quadratic, and cubic polynomials that fit given data points. It also provides examples of using a linear system to solve for the coefficients of circle and central conic equations that fit given points. The linear systems are set up and solved, providing the resulting equations that fit the data in each example.
Este documento descreve uma oficina sobre o uso educativo de blogs para professores. A oficina ocorrerá em dois dias com 8 horas de duração e tem como objetivo principal apresentar como blogs podem ser aplicados no ambiente educacional, permitindo a divulgação de trabalhos de sala de aula e a interação entre professores e alunos.
El documento describe un estudio sobre el efecto de diferentes dosis de fósforo en el crecimiento de plantas de maíz. El estudio incluyó la medición de parámetros como la altura, ancho de hojas y peso de las plantas tratadas con diferentes niveles de fósforo, con el objetivo de observar las diferencias en el desarrollo del cultivo. Los resultados se presentaron en gráficos de crecimiento y peso para cada tratamiento.
Este documento discute o projeto Esfera, uma iniciativa para melhorar a qualidade da resposta humanitária a desastres. O projeto fornece normas mínimas, indicadores e orientações para setores como água, saneamento, segurança alimentar e abrigo. O documento descreve a evolução do projeto desde 1997 e como ele visa tornar os princípios humanitários em ações concretas para ajudar as populações afetadas por desastres.
O documento apresenta uma lista de locais ao redor do mundo, incluindo Veneza, Canadá, Tailândia, Amazônia, Grécia, Alpes Suíços, Mar Negro, Panamá, Mar Vermelho, Rio de Janeiro e Paris, destacando a beleza do planeta mesmo à noite. A música Il mondo, de Sérgio Endrigo, é apresentada por Renato Cardoso em um site sobre Bauru.
O documento discute a exploração e sofrimento de crianças em diversas partes do mundo, incluindo trabalho infantil, desnutrição, tráfico humano e guerras. Enfatiza a importância de promover o desenvolvimento infantil e políticas sociais compassivas para assegurar um futuro melhor às próximas gerações.
The document discusses how to create a website that generates leads. It recommends optimizing the website for search engines through techniques like search engine optimization (SEO). Key steps include choosing relevant keywords, optimizing page titles and descriptions, and avoiding duplicate or low-quality content that could hurt SEO rankings. The #1 mistake most websites make is failing to effectively capture visitor information through forms or other methods. With the right online strategy, a website can become a lead generation machine for businesses.
La Unión Europea ha acordado un embargo petrolero contra Rusia en respuesta a la invasión de Ucrania. El embargo prohibirá las importaciones marítimas de petróleo ruso a la UE y pondrá fin a las entregas a través de oleoductos dentro de seis meses. Esta medida forma parte de un sexto paquete de sanciones de la UE destinadas a aumentar la presión económica sobre Moscú y privar al Kremlin de fondos para financiar su guerra.
1) La digestión comienza en la boca y continúa a través del esófago hasta el estómago.
2) En el estómago, los alimentos son reducidos durante 1-6 horas.
3) Los nutrientes son luego absorbidos en el intestino delgado y transportados a través de la circulación sanguínea para alimentar las células del cuerpo, mientras que los residuos no absorbidos pasan al intestino grueso para ser eliminados.
Este documento contém mensagens de alunos da 6a série para suas mães expressando gratidão e afeição. As mensagens descrevem as mães como fontes de amor, alegria, luz e confiança que tornam a vida dos filhos melhor. A professora agradece aos alunos por existirem.
This resume summarizes Deepik A Grover's objective, education, work experience, projects, internships, achievements, and extracurricular activities. Deepik seeks an opportunity to contribute professionally while being innovative. She has an MBA in finance and IT from Bharati Vidyapeeth University and worked as a customer service associate for Fidelity Worldwide Investments since 2013, handling client queries and complaints. She completed internships at the National Informatics Centre and MMTC Limited. Her achievements include honors from marketing and case study competitions.
OCHOA_C, El modelo Lee-Carter para estimar y pronosticar mortalidad, Una apli...Carlos Andr Ochoa
Este documento presenta dos aplicaciones del modelo Lee-Carter para estimar y pronosticar tasas de mortalidad. En la primera aplicación, se estima un modelo Lee-Carter para la dinámica de mortalidad en Colombia entre 1951-1999 y se generan proyecciones hasta 2011. En la segunda aplicación, se analiza la mortalidad académica de estudiantes de pregrado en la Universidad Nacional de Colombia entre 1989-2006 y se realizan proyecciones hasta 2011. Los resultados sugieren una disminución en las tasas de mortalidad para Colombia y una mayor deserción estudiantil en los primer
The document discusses techniques for sketching graphs of functions, including:
- Using the increasing/decreasing test to determine if a function is increasing or decreasing based on the sign of the derivative
- Using the concavity test to determine if a graph is concave up or down based on the second derivative
- A checklist for completely graphing a function, including finding critical points, inflection points, asymptotes, and putting together the information about monotonicity and concavity.
The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.
The document discusses functions and their properties. It defines a function as a rule that assigns exactly one output value to each input value in its domain. Functions can be represented graphically, numerically in a table, or with an algebraic rule. The domain of a function is the set of input values, while the range is the set of output values. Basic operations like addition, subtraction, multiplication and division can be performed on functions in the same way as real numbers. Composition of functions is defined as evaluating one function using the output of another as the input.
Exponential and logarithm functions are important in both theory and practice. They examine the graphs of exponential functions f(x)=ax where a>0 and logarithm functions f(x)=loga(x) where a>0. It is important to practice these functions so their properties become intuitive. Key properties include exponential functions where a>1 increase rapidly for positive x and 0<a<1 increase for decreasing negative x, and both pass through (0,1). The natural logarithm function f(x)=ln(x) is particularly important.
The document discusses partial differentiation and its applications. It covers functions of two variables, first and second partial derivatives, and applications including the Cobb-Douglas production function and finding marginal productivity from a production function. Examples are provided to demonstrate calculating partial derivatives of various functions and applying partial derivatives in contexts like production analysis.
The document discusses linear approximations of functions. It provides examples of determining:
1) The value of a function f(x) at a point x0 and the derivative f'(x0)
2) The linear approximation L(x) = f(x0) + f'(x0)(x - x0)
3) Using L(x) to estimate the value of f(x) near x0
The document discusses linear approximations of functions. It provides examples of determining:
1) The value of a function f(x) at a point x0 and the derivative f'(x0)
2) The linear approximation L(x) = f(x0) + f'(x0)(x - x0)
3) Using L(x) to estimate the value of f(x) near x0
This document summarizes the steps taken to solve several calculus problems:
1) For part (a), the function f(x) is factorized as ex multiplied by (x2 - 5x - 8) which is then further factorized as (2x - 5).
2) In part (b), the function f(x) is factorized as e^x^2-x multiplied by (x2 - x) which is then further factorized as 2x - 1.
3) Part (c) defines the function f(x) as the natural log of (x*e^x - 3) which is simplified to x/(e^x - 3).
This document provides questions about finding the equations of tangents and normals to various polynomial functions at given points. It contains 8 parts with multiple questions each about finding the equations of tangents to polynomial curves and normals to polynomial curves at specified points using differentiation.
1. The document provides examples of calculating derivatives of various functions using the power, constant multiple, sum and difference, and chain rules of differentiation.
2. Examples include finding the derivatives of polynomials, rational functions, radical functions, and compositions of functions.
3. The four main rules of differentiation are outlined and then applied to specific examples to calculate the derivative of each function.
This document provides examples and exercises on inverse functions. It first shows an example of finding the inverse of the function f(x) = 2x-3/(x+2). It gives the steps to solve for x in terms of y and obtain the inverse function f^-1(x) = -2x-3/(x-2). It then asks the reader to verify that f(f^-1(x)) = x. The exercises provide 20 functions and ask the reader to find their inverses and verify the inverses are correct. It also gives graphs of 8 functions and asks the reader to determine properties of the inverse graphs, including their domains and ranges and any fixed or end points.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
This document covers key concepts in calculus including:
- Computing derivatives using notation such as f'(a) and dy/dx
- Relationships between differentiability and continuity
- Using derivatives to find horizontal tangents, max/min points, and inflection points
- Applying derivative rules such as the sum and constant multiple rules
- Examples of computing derivatives of various functions and determining differentiability
A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
1. The document discusses function notation, evaluating functions, and finding the domain from the equation of a function.
2. It provides examples of different types of function notation including f(x)=x^2+1 and {(x,y)|y=x^2+1}.
3. Evaluating a function involves substitution, and the document gives an example of evaluating f(x)=x^2+3x-1 when x=-2.
4. To find the domain from the equation, one looks at what type of equation it is such as linear, absolute value, quadratic, radical, fractional, or other. The domain is the set of inputs that can be substituted into the
The document provides information about functions. It defines a function as a relation where each input has exactly one output. A function has a domain, which is the set of all legal inputs, and a range, which is the set of all possible outputs. Notation for a function is y=f(x), where y depends on x. Examples are provided of evaluating functions at different inputs and finding the domains of functions. Piecewise functions are also introduced.
This document provides information on key concepts related to derivatives including:
1. Critical numbers and how to find them using the first derivative test
2. How the first derivative relates to intervals of increasing and decreasing functions
3. How to determine local maxima and minima using the first derivative test
4. How to find absolute maxima and minima on a closed interval
5. How to determine concavity using the second derivative test and identify inflection points. Worked examples are provided to demonstrate each concept.
This document provides a summary of precalculus concepts including:
1. Functions and their graphs including function definitions, transformations, combinations, and compositions of functions.
2. Trigonometry including trigonometric functions, graphs of trigonometric functions, and trigonometric identities.
3. Graphs of second-degree equations including circles, parabolas, ellipses, and hyperbolas.
The document contains examples and explanations of key precalculus topics to serve as a review for a Math 131 course. It covers essential functions like polynomials, rational functions, and transcendental functions. It also discusses trigonometric functions and their graphs along with transformations of functions.
This document provides an overview of functions, limits, and continuity. It defines key concepts such as domain and range of functions, and examples of standard real functions. It also covers even and odd functions, and how to calculate limits, including left and right hand limits. Methods for evaluating algebraic limits using substitution, factorization, and rationalization are presented. The objectives are to understand functions, domains, ranges, and how to evaluate limits of functions.
O documento apresenta um resumo sobre lógica de programação ministrado por Andrei Bastos na UFES em 2014, abordando conceitos básicos como objetivos do curso, bibliografia, conceitos de algoritmo, formas de representação como fluxograma e variáveis.
1) O vértice E do paralelepípedo pertence à reta r de equação z2yx −=−= . As coordenadas de E são determinadas como (10,12,12).
2) Sabendo que o vértice B tem coordenadas (0,1,1) e que A(0,0,1) são vértices consecutivos de um quadrado no plano 01z2yx: =−+−α , as coordenadas dos outros dois vértices são determinadas como (0,2,1) e (0,3,3).
3) Uma equação do plan
Este documento apresenta os principais conceitos de vetores e geometria analítica em três frases:
1) Introduz o sistema cartesiano ortogonal no R3 para definir as coordenadas de um ponto e discute a noção de segmentos orientados equipolentes.
2) Define vetor como uma classe de equivalência de segmentos orientados equipolentes e apresenta propriedades da adição e multiplicação de vetores.
3) Aborda dependência e independência linear de vetores, combinações lineares, bases e coordenadas de vetores em relação a bases.
O documento descreve os sistemas de coordenadas cartesianas e polares. Explica como representar pontos no plano usando cada sistema e fornece as equações de transformação entre os sistemas. Também apresenta as equações polares das principais cônicas - circunferência, elipse, hipérbole e parábola - em termos da distância polar ρ e do ângulo polar θ.
O documento discute as seções cônicas, curvas planas obtidas da interseção de um plano com um cone de revolução. Apresenta breve histórico sobre o estudo destas curvas desde a Grécia Antiga, destacando contribuições de Arquimedes, Apolônio, Galileu e Newton. Em seguida, define e apresenta as equações das principais seções cônicas: elipse, hipérbole, parábola e circunferência.
O documento apresenta fórmulas e conceitos para calcular distâncias e ângulos entre objetos vetoriais como pontos, retas e planos no espaço tridimensional. Inclui definições de distância entre dois pontos, um ponto e uma reta, um ponto e um plano, entre duas retas, dois planos e uma reta e um plano. Também apresenta fórmulas para calcular ângulos entre vetores, retas, planos e entre uma reta e um plano. Exemplos ilustram o cálculo destas grandezas.
O documento descreve vários conceitos relacionados a planos em geometria analítica, incluindo:
1) Definições de plano, equação vetorial e casos particulares de planos;
2) Métodos para representar planos através de equações paramétricas, geral e segmentária;
3) Cálculo do vetor normal a um plano e relações entre os vetores normais de planos.
O documento define e discute conceitos básicos sobre retas em geometria analítica, incluindo: (1) a equação vetorial de uma reta que contém um ponto e tem direção de um vetor, (2) as diferentes formas de escrever a equação de uma reta, como equações paramétricas e simétricas, (3) a condição para três pontos serem alinhados e (4) as posições relativas entre duas retas, como paralelas, concorrentes e reversas.
1) O documento apresenta os conceitos de produto escalar e produto vetorial entre vetores no espaço R3.
2) O produto escalar é definido como o número real θ⋅⋅=⋅ cos|v||u|vu, onde θ é o ângulo entre os vetores u e v. Já o produto vetorial é definido como um vetor.
3) São apresentadas propriedades e interpretações geométricas desses produtos, como a relação entre o módulo do produto escalar e a projeção de um vetor na direção do
O documento discute conceitos fundamentais de dependência linear e bases de vetores. Ele define combinação linear, vetores linearmente independentes (LI) e dependentes (LD), e apresenta teoremas relacionados a essas noções. O documento também discute o que é uma base de vetores e apresenta exemplos de bases nos espaços R2 e R3.
O documento descreve representações geométricas de vetores no plano e no espaço. No plano, vetores são representados por pares ordenados de números reais e divididos em quadrantes. No espaço, vetores são representados por ternas de números reais e divididos em oitantes. O documento também apresenta operações com vetores e conceitos como cossenos diretores e projeções.
O documento introduz os conceitos fundamentais de vetores e operações com vetores. Apresenta a definição formal de vetor como uma classe de equipolência de segmentos orientados e define as noções de módulo, direção e sentido de um vetor. Descreve as principais operações com vetores - adição, subtração e multiplicação por escalar - utilizando os métodos da poligonal e do paralelogramo.
Este capítulo apresenta as transformações geométricas de translação e rotação de eixos no plano cartesiano R2. A translação translada o sistema de coordenadas para uma nova origem, enquanto a rotação gira os eixos em torno da origem por um ângulo θ. As equações de translação e rotação são fornecidas, e exemplos ilustram como aplicá-las para reduzir equações de cônicas à forma mais simples.
Este documento apresenta definições e teoremas sobre isomorfismo de espaços vetoriais. [1] Define transformação linear bijetora como isomorfismo e apresenta propriedades como a existência de inversa e isomorfismo entre espaços da mesma dimensão. [2] Aplica os conceitos em exemplos de verificação de isomorfismo e determinação da transformação inversa.
O documento resume os principais conceitos de transformação linear entre espaços vetoriais, incluindo: (1) Definição formal de transformação linear; (2) Propriedades das transformações lineares, como núcleo e imagem; (3) Operações com transformações lineares, como adição, subtração e produto escalar.
O documento discute representações matriciais de transformações lineares. Define-se a matriz de uma transformação linear como sendo formada pelas coordenadas dos vetores da imagem de uma base em relação a outra base. Mostra-se que esta matriz representa completamente a transformação e que propriedades algébricas desta são refletidas na matriz, como inversibilidade. Exemplos ilustram os conceitos.
Este documento apresenta três frases:
1) Define matriz mudança de base como representando as coordenadas de vetores de uma base em relação a outra.
2) Explica que a matriz mudança de base relaciona as coordenadas dos vetores de uma base quando escritos como combinação linear dos vetores da outra base.
3) Apresenta três teoremas sobre propriedades das matrizes mudança de base e exemplos ilustrando seu cálculo e aplicação.
1. O documento discute conceitos fundamentais de álgebra linear como base, dimensão e coordenadas de vetores. É apresentada a definição formal de base e exemplos para R3.
2. São listadas as bases canônicas dos principais espaços vetoriais como Rn, M(2x2) e Pn. É explicado o Teorema da Invariância e o processo para obter uma base de um subespaço.
3. Os conceitos de dimensão, subespaços e suas propriedades são definidos. São mostrados teoremas e proposições
[1] A combinação linear é uma soma ponderada de vetores, onde os pesos são escalares. Um vetor é combinação linear de outros se puder ser escrito dessa forma. [2] O subespaço gerado por um conjunto de vetores S é o conjunto de todos os vetores que podem ser escritos como combinação linear dos vetores de S. [3] Vetores são linearmente independentes se a única solução para sua combinação linear ser nula é quando todos os escalares são nulos.
Este documento apresenta os conceitos fundamentais de álgebra linear, incluindo:
1) A definição de corpo, que é um conjunto com operações de adição e multiplicação que satisfazem certas propriedades. Exemplos de corpos incluem os números complexos e reais.
2) A definição de espaço vetorial, que é um conjunto com operações de adição vetorial e produto escalar satisfazendo propriedades específicas. Exemplos incluem Rn.
3) A definição de subespaço vetorial, que é um subconjunto
1. Lista 2 de C´lculo I
a 2010-2 3
UFES - Universidade Federal do Esp´
ırito Santo Opera¸˜es com fun¸˜es
co co
DMAT - Departamento de Matem´tica
a
Fun¸˜o composta
ca
LISTA 2 - 2010-2 Transforma¸˜es em gr´ficos
co a
1
1. Se f (x) = 3x2 + 2 e g(x) = , determine:
3x + 2
(a) (f + g)(x) (c) (f · g)(x) g
(e) (x)
f f
(b) (f (x))−1 (d) (x) (f) (f ◦ g)(x)
g
3−x
2. Seja f (x) = . Determine:
x
(a) f x2 − (f (x))2 1 1 (c) (f ◦ f )(x)
(b) f −
x f (x)
1
−x , x < 0 , x<0 (a) (f ◦ g)(x)
3. Dadas f (x) = e g(x) = √x , determine:
x2 , x ≥ 0 x , x≥0 (b) (g ◦ f )(x)
Nos exerc´
ıcios 4. a 11., a partir do gr´fico da fun¸˜o y = f (x) dado abaixo, esboce o gr´fico da
a ca a
fun¸˜o dada.
ca
y = f(x)
4. y = f (|x|) 8. y = f (x + 2) 4
2
9. y = f (x) + 3
5. y = |f (x)| –8 –6 –4 –2 0 2 4 6 8 10 12 14
x
f (x) + |f (x)| –2
10. y = –4
6. y = f (−x) 2
f (x) − |f (x)|
7. y = −f (x) 11. y =
2
Esboce os gr´ficos das fun¸˜es dos exerc´
a co ıcios 12. a 17.
12. f (x) = 8 − 3 x
2 −1 14. f (x) = |x − 1|3 16. f (x) = (|x| − 1)3
√
13. f (x) = 2|x| − 6 15. f (x) = 1 + 3 1 − x 17. f (x) = x2 − 4|x| + 3
Nos exerc´
ıcios 18. a 21. determime o dom´
ınio, a imagem e esboce o gr´fico da fun¸˜o dada.
a ca
18. f (x) = 3 sen (2πx) x 1
20. f (x) = sen − , 0 ≤ x ≤ 4π
2 2
x 1 π
19. f (x) = tan 21. f (x) = sec x −
2 2 3
2. Lista 2 de C´lculo I
a 2010-2 4
RESPOSTAS
9x3 + 6x2 + 6x + 5 3 2
1. (a) y = ,x = − (d) y = 9x3 + 6x2 + 6x + 4, x = −
3x + 2 2 3
1 1 2
(b) y = 2 (e) y = 3 ,x = −
3x + 2 9x + 6x2 + 6x + 4 3
3x2 + 2 2 18x2 + 24x + 11 2
(c) y = ,x = − (f) y = ,x = −
3x + 2 3 9x2 + 12x + 4 3
6x − 2x2 − 6 2
9x − 3x − 3 4x − 3
2. (a) y = (b) y = (c) y =
x2 3−x 3−x
√
1 −x , x < 0
− , x<0 (b) (g ◦ f )(x) =
3. (a) (f ◦ g)(x) = x x , x≥0
x , x≥0
4. 7. 10.
y y y
4 4 4
2 2 2
x x x
–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14
–2 –2 –2
–4 –4 –4
5. 8. 11.
y y
y
4 4
4
2 2
2
x x
–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 x
–2 –2 –14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14
–2
–4 –4
–4
6. 9.
y y
4 6
2 4
x 2
–14 –10 –8 –6 –4 –2 2 4 6 8 10 12 14
–2 x
–14 –10 –8 –6 –4 –2 0 2 4 6 8 10 12 14
–4 –2
12. 14. 16.
y y y
8 2 2
6
1
4 1
x
2
x –4 –3 –2 –1 1 2 3 4
10 20
x
–2 2 4 –1
–2
13. 15. 17.
y
8
y y
2
6 4
4 x
0 2
2 –4 –2 2 4 6
x x
–10 –8 –6 –4 –2 0 2 4 6 8 10 –2 –6 –4 –2 2 4 6
18. 20.
y
3
2
2
1
–6 –4 –2 0 2 4 x6
0 x
5 10
–2
19. 21.
y y
10 10
5
x x
–10 10 –15 –10 –5 5 10 15
–5
–10 –10