1) The document discusses the concept of limits in mathematics. A limit describes the value a function approaches as the input value gets closer to a certain point, without reaching it.
2) A limit is needed because functions are often not defined at certain points, but their values can still be determined by approaching those points.
3) To find the limit of a function f(x) as x approaches a value c, we determine if the left-hand limit and right-hand limit both exist and are equal. If so, that common value is the limit. If not, the limit does not exist.
1) The document discusses limit theorems and trigonometric function limits. It introduces basic limit theorems including limits of constants, sums, products, quotients, and composite functions.
2) Examples are provided to illustrate evaluating limits using the limit theorems, including limits approaching positive/negative infinity.
3) The document also discusses evaluating limits of rational functions by factorizing the numerator and denominator into their highest-order terms.
1) The document discusses the continuity of functions at a point and over an interval.
2) A function f is defined to be continuous at a point c if the limit of f(x) as x approaches c exists and is equal to f(c).
3) For a function to be continuous over an interval (a,b), it must be continuous at every point c within the interval.
This document discusses the concept of limits in mathematics. It defines a limit as the value a function approaches as the input values get closer to a certain point, without actually reaching it. For a limit of a function f(x) as x approaches c to exist, the left-hand limit and right-hand limit must both exist and be equal to the same number L. A limit may not exist if the left and right sides are different, or if the values oscillate or approach infinity as x approaches c. The document also formally defines a limit using epsilon-delta notation.
This document discusses limits and how to calculate them. It defines a limit as a number a function approaches as the input value approaches a certain number. It provides examples of using graphs and tables on a calculator to find limits, and discusses direct substitution and the replacement theorem for evaluating limits. Special cases like one-sided limits, limits of polynomials, and limits involving radicals are also covered.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
The document discusses limits, continuity, and related concepts. Some key points:
1) It defines the concept of a limit and explains how to evaluate one-sided and two-sided limits. A limit exists only if the left and right-sided limits are equal.
2) Continuity is defined as a function being defined at a point, and the limit existing and being equal to the function value. Functions like tan(x) are only continuous where the denominator is not 0.
3) Theorems are presented for evaluating limits of polynomials, sums, products, quotients of continuous functions, and the squeeze theorem. Piecewise functions may or may not be continuous depending on behavior at points of discontin
1) The document discusses the concept of limits in mathematics. A limit describes the value a function approaches as the input value gets closer to a certain point, without reaching it.
2) A limit is needed because functions are often not defined at certain points, but their values can still be determined by approaching those points.
3) To find the limit of a function f(x) as x approaches a value c, we determine if the left-hand limit and right-hand limit both exist and are equal. If so, that common value is the limit. If not, the limit does not exist.
1) The document discusses limit theorems and trigonometric function limits. It introduces basic limit theorems including limits of constants, sums, products, quotients, and composite functions.
2) Examples are provided to illustrate evaluating limits using the limit theorems, including limits approaching positive/negative infinity.
3) The document also discusses evaluating limits of rational functions by factorizing the numerator and denominator into their highest-order terms.
1) The document discusses the continuity of functions at a point and over an interval.
2) A function f is defined to be continuous at a point c if the limit of f(x) as x approaches c exists and is equal to f(c).
3) For a function to be continuous over an interval (a,b), it must be continuous at every point c within the interval.
This document discusses the concept of limits in mathematics. It defines a limit as the value a function approaches as the input values get closer to a certain point, without actually reaching it. For a limit of a function f(x) as x approaches c to exist, the left-hand limit and right-hand limit must both exist and be equal to the same number L. A limit may not exist if the left and right sides are different, or if the values oscillate or approach infinity as x approaches c. The document also formally defines a limit using epsilon-delta notation.
This document discusses limits and how to calculate them. It defines a limit as a number a function approaches as the input value approaches a certain number. It provides examples of using graphs and tables on a calculator to find limits, and discusses direct substitution and the replacement theorem for evaluating limits. Special cases like one-sided limits, limits of polynomials, and limits involving radicals are also covered.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
The document discusses limits, continuity, and related concepts. Some key points:
1) It defines the concept of a limit and explains how to evaluate one-sided and two-sided limits. A limit exists only if the left and right-sided limits are equal.
2) Continuity is defined as a function being defined at a point, and the limit existing and being equal to the function value. Functions like tan(x) are only continuous where the denominator is not 0.
3) Theorems are presented for evaluating limits of polynomials, sums, products, quotients of continuous functions, and the squeeze theorem. Piecewise functions may or may not be continuous depending on behavior at points of discontin
The document discusses limits and continuity of functions. It provides examples of computing one-sided limits, limits at points of discontinuity, and limits involving algebraic, trigonometric, exponential and logarithmic functions. The key rules for limits include the properties of limits, the sandwich theorem, and limits of compositions of functions. Continuity of functions is defined as a function having a limit equal to its value at a point. Polynomials, trigonometric functions and exponentials are shown to be continuous everywhere they are defined.
This document provides an overview of key concepts related to limits and continuity, including:
1) Defining what a limit means both graphically and algebraically as the input value gets closer to a given number without reaching it.
2) Explaining how to find limits through direct substitution when possible, or by simplifying rational functions.
3) Introducing one-sided limits and infinite limits.
4) Detailing how limits can involve multiple variables.
5) Defining continuity as having no holes, jumps, or vertical asymptotes at a given point, and how to determine continuity algebraically for different function types like polynomials, rational functions, and piecewise functions.
The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.
Limits and Continuity - Intuitive Approach part 1FellowBuddy.com
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
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The document discusses key concepts related to limits, continuity, and differentiation. It defines what it means for a variable x to approach a finite number a or infinity, and provides the formal definitions of one-sided limits and two-sided limits. It also discusses indeterminate forms when limits take on forms like 0/0, infinity/infinity, or infinity - infinity. The document outlines several properties of limits, including limits of even and odd functions. It distinguishes between the limit of a function as x approaches a, denoted limx→af(x), versus the function value at that point, f(a). Finally, it states standard theorems about limits, such as the sum and product of two functions whose limits exist
This document discusses functions, limits, and continuity. It begins by defining functions, domains, ranges, and some standard real functions like constant, identity, modulus, and greatest integer functions. It then covers limits of functions including one-sided limits and properties of limits. Examples are provided to illustrate evaluating limits using substitution and factorization methods. The overall objectives are to understand functions, domains, ranges, limits of functions and methods to evaluate limits.
This document is an 18-page review on limits, continuity, and the definition of the derivative in calculus. It begins with formal definitions of the derivative of a function, the derivative at a point, and continuity. It then provides examples and practice problems related to evaluating limits, including as x approaches infinity or a number, and limits related to continuity and derivatives. The document concludes with several free response questions involving analyzing functions for continuity and differentiability over an interval. In summary, this review covers key calculus concepts of limits, continuity, and the definition of the derivative through formal definitions, examples, and practice problems.
This document discusses topics in partial differentiation including:
1) The geometrical meaning of partial derivatives as the slope of the tangent line to a surface.
2) Finding the equation of the tangent plane and normal line to a surface.
3) Taylor's theorem and Maclaurin's theorem for functions with two variables, which can be used to approximate functions and calculate errors.
The document discusses different types of limits in calculus and analysis. A limit describes the behavior of a function near a particular input value. There are several types of limits including one-sided limits as x approaches a from the left or right, direct substitution limits where the value is simply substituted, and limits involving techniques like factoring, rationalization, as x approaches infinity, trigonometric functions, or the number e.
The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
This document discusses rules for taking derivatives of various functions including:
1. The derivative of a constant function is 0.
2. The power rule states that the derivative of x^n is nx^{n-1}.
3. Higher derivatives can be found by taking additional derivatives, and the nth derivative is written as f^(n).
It also covers the product rule, quotient rule, and applying rules to polynomials and exponential functions.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others constant. It provides examples of computing first and second partial derivatives.
2) Implicit differentiation is introduced as a way to find partial derivatives of functions defined implicitly rather than explicitly. The chain rule is also discussed.
3) Methods are presented for finding partial derivatives of functions of two or three variables, including using implicit differentiation and the chain rule. Examples are provided to illustrate these concepts.
This document defines and explains partial derivatives. It begins by defining a partial derivative as the rate of change of a function with respect to one variable, holding other variables fixed. It then covers notation, calculating partial derivatives, interpreting them geometrically and as rates of change, higher derivatives, and applications to partial differential equations.
This document discusses various topics relating to partial derivatives including Leibniz notation, Clairaut's theorem, the chain rule, and directional derivatives. It provides two example problems - finding the partial derivative fxy for a given function, and finding the directional derivative of another function in a given direction. Sources for further information on partial derivatives are also listed.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
This document discusses partial derivatives of functions with multiple variables. It defines partial derivatives as derivatives of a function where all but one variable is held constant. For a function z=f(x,y), the partial derivatives with respect to x and y are defined. Higher order partial derivatives and partial derivatives of functions with more than two variables are also introduced. Examples are provided to demonstrate calculating first and second order partial derivatives.
This document provides definitions and examples of various types of numbers and functions. It discusses:
- Number sets including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- Types of intervals such as closed, open, and semi-open/semi-closed.
- Definitions of a function, including domain, co-domain, and range. Methods of representing functions include mapping, algebraic, and ordered pairs.
- Classification of functions as algebraic vs. transcendental, even vs. odd, explicit vs. implicit, continuous vs. discontinuous, and increasing vs. decreasing.
- Properties of even and odd functions are also discussed.
The document discusses limits and continuity of functions. It provides examples of computing one-sided limits, limits at points of discontinuity, and limits involving algebraic, trigonometric, exponential and logarithmic functions. The key rules for limits include the properties of limits, the sandwich theorem, and limits of compositions of functions. Continuity of functions is defined as a function having a limit equal to its value at a point. Polynomials, trigonometric functions and exponentials are shown to be continuous everywhere they are defined.
This document provides an overview of key concepts related to limits and continuity, including:
1) Defining what a limit means both graphically and algebraically as the input value gets closer to a given number without reaching it.
2) Explaining how to find limits through direct substitution when possible, or by simplifying rational functions.
3) Introducing one-sided limits and infinite limits.
4) Detailing how limits can involve multiple variables.
5) Defining continuity as having no holes, jumps, or vertical asymptotes at a given point, and how to determine continuity algebraically for different function types like polynomials, rational functions, and piecewise functions.
The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.
Limits and Continuity - Intuitive Approach part 1FellowBuddy.com
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
www.anuragtyagiclasses.com
The document discusses key concepts related to limits, continuity, and differentiation. It defines what it means for a variable x to approach a finite number a or infinity, and provides the formal definitions of one-sided limits and two-sided limits. It also discusses indeterminate forms when limits take on forms like 0/0, infinity/infinity, or infinity - infinity. The document outlines several properties of limits, including limits of even and odd functions. It distinguishes between the limit of a function as x approaches a, denoted limx→af(x), versus the function value at that point, f(a). Finally, it states standard theorems about limits, such as the sum and product of two functions whose limits exist
This document discusses functions, limits, and continuity. It begins by defining functions, domains, ranges, and some standard real functions like constant, identity, modulus, and greatest integer functions. It then covers limits of functions including one-sided limits and properties of limits. Examples are provided to illustrate evaluating limits using substitution and factorization methods. The overall objectives are to understand functions, domains, ranges, limits of functions and methods to evaluate limits.
This document is an 18-page review on limits, continuity, and the definition of the derivative in calculus. It begins with formal definitions of the derivative of a function, the derivative at a point, and continuity. It then provides examples and practice problems related to evaluating limits, including as x approaches infinity or a number, and limits related to continuity and derivatives. The document concludes with several free response questions involving analyzing functions for continuity and differentiability over an interval. In summary, this review covers key calculus concepts of limits, continuity, and the definition of the derivative through formal definitions, examples, and practice problems.
This document discusses topics in partial differentiation including:
1) The geometrical meaning of partial derivatives as the slope of the tangent line to a surface.
2) Finding the equation of the tangent plane and normal line to a surface.
3) Taylor's theorem and Maclaurin's theorem for functions with two variables, which can be used to approximate functions and calculate errors.
The document discusses different types of limits in calculus and analysis. A limit describes the behavior of a function near a particular input value. There are several types of limits including one-sided limits as x approaches a from the left or right, direct substitution limits where the value is simply substituted, and limits involving techniques like factoring, rationalization, as x approaches infinity, trigonometric functions, or the number e.
The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
This document discusses rules for taking derivatives of various functions including:
1. The derivative of a constant function is 0.
2. The power rule states that the derivative of x^n is nx^{n-1}.
3. Higher derivatives can be found by taking additional derivatives, and the nth derivative is written as f^(n).
It also covers the product rule, quotient rule, and applying rules to polynomials and exponential functions.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others constant. It provides examples of computing first and second partial derivatives.
2) Implicit differentiation is introduced as a way to find partial derivatives of functions defined implicitly rather than explicitly. The chain rule is also discussed.
3) Methods are presented for finding partial derivatives of functions of two or three variables, including using implicit differentiation and the chain rule. Examples are provided to illustrate these concepts.
This document defines and explains partial derivatives. It begins by defining a partial derivative as the rate of change of a function with respect to one variable, holding other variables fixed. It then covers notation, calculating partial derivatives, interpreting them geometrically and as rates of change, higher derivatives, and applications to partial differential equations.
This document discusses various topics relating to partial derivatives including Leibniz notation, Clairaut's theorem, the chain rule, and directional derivatives. It provides two example problems - finding the partial derivative fxy for a given function, and finding the directional derivative of another function in a given direction. Sources for further information on partial derivatives are also listed.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
This document discusses partial derivatives of functions with multiple variables. It defines partial derivatives as derivatives of a function where all but one variable is held constant. For a function z=f(x,y), the partial derivatives with respect to x and y are defined. Higher order partial derivatives and partial derivatives of functions with more than two variables are also introduced. Examples are provided to demonstrate calculating first and second order partial derivatives.
This document provides definitions and examples of various types of numbers and functions. It discusses:
- Number sets including natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- Types of intervals such as closed, open, and semi-open/semi-closed.
- Definitions of a function, including domain, co-domain, and range. Methods of representing functions include mapping, algebraic, and ordered pairs.
- Classification of functions as algebraic vs. transcendental, even vs. odd, explicit vs. implicit, continuous vs. discontinuous, and increasing vs. decreasing.
- Properties of even and odd functions are also discussed.
The document defines and provides examples of functions. Some key points:
- A function is a correspondence between two sets such that each element in the first set corresponds to a unique element in the second set.
- Examples of functions include relationships between a company's profits and production levels, a bacteria culture's size over time, and learning time for a word list versus word list length.
- The domain of a function is the set of input values, and the range is the set of output values.
- Functions can be represented algebraically, with tables of values, or with graphs. Operations can be performed on functions.
- Examples of functions include constant, identity, linear, quadratic, rational
The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.
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.
1. The document discusses various mathematical functions including absolute value, floor/ceiling functions, factorials, modular arithmetic, exponential functions, logarithmic functions, and polynomials.
2. It defines key properties of functions such as one-to-one, onto, bijective, and inverse functions. It also covers function composition and important sequences like geometric and arithmetic progressions.
3. The document provides formulas for summing sequences and common summations like the sum of the first n natural numbers and the sum of the first n odd integers.
Mauricio opened a bank account with $20 and deposits $10 each week. His account balance can be modeled as a linear function f(x) = 20 + 10x, where x is the number of weeks and f(x) is the balance in dollars. The function shows that after 0 weeks the balance is $20, after 1 week it is $30, after 2 weeks $40, and so on, increasing by $10 each week.
This document defines functions and discusses key concepts related to functions including:
- A function relates each element of its domain to a unique element of its range.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, equations, and graphs.
- The domain of a function is the set of possible inputs, and the range is the set of possible outputs.
The document is about quadratic polynomial functions and contains the following information:
1. It discusses investigating relationships between numbers expressed in tables to represent them in the Cartesian plane, identifying patterns and creating conjectures to generalize and algebraically express this generalization, recognizing when this representation is a quadratic polynomial function of the type y = ax^2.
2. It provides examples of converting algebraic representations of quadratic polynomial functions into geometric representations in the Cartesian plane, distinguishing cases in which one variable is directly proportional to the square of the other, using or not using software or dynamic algebra and geometry applications.
3. It discusses characterizing the coefficients of quadratic functions, constructing their graphs in the Cartesian plane, and determining their zeros (
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
The document discusses various concepts related to functions and graphs:
1) It defines what a function is and provides examples of different types of functions such as identity, constant, polynomial, and rational functions.
2) It explains the terminology used in functions such as domain, co-domain, pre-image, and image.
3) It discusses the properties of one-to-one, onto, and bijective functions and provides examples of each. The concepts of inverse and composition of functions are also introduced.
4) Useful mathematical functions like floor, ceiling, and round are defined. Concepts related to limits, continuity, differentiability of functions and their graphs are explained briefly.
5
The document discusses key concepts in set theory and functions, including:
- Sets can contain numbers, elements, and be represented using curly brackets.
- Venn diagrams use overlapping circles to show logical connections between sets.
- A function has a domain (input) and range (output), where each input is mapped to a unique output.
- Composite functions combine other functions by substituting one into another.
- Inverse functions reverse the input and output of a function if it exists.
- Common functions that can be graphed include linear, quadratic, trigonometric, cubic, exponential and logarithmic functions.
Higher Maths 121 Sets And Functions 1205778086374356 2Niccole Taylor
The document discusses key concepts in set theory and functions, including:
- Sets can contain numbers, elements, and be represented using curly brackets.
- Venn diagrams use overlapping circles to show logical connections between sets.
- A function has a domain (input) and range (output), where each input is mapped to a unique output.
- Composite functions combine other functions by substituting one into another.
- Inverse functions reverse the input and output of a function if it exists.
- Common functions that can be graphed include linear, quadratic, trigonometric, cubic, exponential and logarithmic functions.
The document defines functions and discusses:
- Functions are sets of ordered pairs with each first element paired to a unique second element.
- Functions can be one-to-one or many-to-one.
- Functions are represented in set notation, tabular form, as equations, and as graphs.
- The domain of a function is the set of first elements and the range is the set of second elements.
1) The document discusses various topics in mathematics including sets, functions, composite functions, exponential and logarithmic graphs, and graph transformations.
2) It provides definitions and examples of sets, functions, and how to represent functions using formulas, arrow diagrams, and graphs. Composite functions are defined as functions of other functions.
3) The document explains how to graph exponential and logarithmic functions and describes the key features of these graphs. It also discusses how different transformations can move a graph in various ways, such as reflecting it or stretching/squashing it.
The document discusses relations and functions. A relation pairs elements from two sets, while a function pairs each element from the first set (domain) to only one element in the second set (range). An example is provided of a relation that is not a function because it pairs one element from the domain to two elements in the range. Functions can be represented numerically, algebraically, graphically, or verbally. Evaluating functions involves substituting values into the function formula. A function's domain consists of the valid inputs, while its range consists of the possible outputs.
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 discusses functions and their graphs. It defines piecewise functions, absolute value functions, and greatest integer functions. It provides examples of evaluating and sketching graphs of these types of functions. The objectives are to sketch graphs of functions, determine domains and ranges from graphs, identify functions from relations using graphs, define and evaluate piecewise, absolute value, and greatest integer functions. Exercises are provided to apply these concepts.
Dokumen tersebut membahas berbagai jenis bangunan air untuk irigasi seperti gorong-gorong, syphon, talang, bangunan terjun tegak dan miring. Memberikan penjelasan tentang parameter perencanaan seperti kehilangan energi akibat gesekan, peralihan, dan belokan serta contoh perhitungan dimensi talang.
1. Gorong-gorong dan siphon adalah bangunan persilangan yang mengalirkan air di bawah struktur lain seperti jalan.
2. Perencanaan bangunan persilangan mempertimbangkan kehilangan energi akibat gesekan, peralihan, dan belokan.
3. Kecepatan aliran harus dihitung dengan tepat agar air dapat mengalir melalui siphon.
Dokumen tersebut membahas tiga jenis bangunan air yaitu Cipoletti, pintu sorong, dan balok sekat. Cipoletti digunakan untuk mengukur debit, sedangkan pintu sorong dan balok sekat digunakan untuk mengatur tinggi muka air. Dokumen ini juga menjelaskan rumus debit untuk ketiga bangunan air tersebut beserta contoh perhitungannya.
Dokumen tersebut membahas tentang kebutuhan air untuk irigasi dan pertanian, termasuk faktor-faktor yang mempengaruhi kebutuhan air, cara perhitungannya, serta contoh perhitungan kebutuhan air untuk tanaman padi di beberapa musim tanam.
Dokumen tersebut membahas tentang perencanaan bangunan talang untuk jaringan irigasi, termasuk parameter perencanaan seperti kehilangan energi, dimensi, kemiringan, dan penulangan beton."
Kuliah dilaksanakan di lapangan Embung Bengawan Kota Tarakan membahas dua jenis bangunan pengatur muka air yaitu pintu skot balok dan pintu sorong. Pintu sorong terdiri dari rangkat pintu, daun pintu, stang ulir untuk membuka dan menutup daun pintu. Pintu skot balok terbuat dari susunan balok-balok kayu yang dapat diatur tingginya untuk mengontrol debit masuk. Perhitungan hidrolis untuk menentuk
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ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
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Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
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detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
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train and test our model. The results of our experiments show that our CNN-LSTM method is much better
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3. FUNGSI
Fungsi f adalah suatu aturan yang memetakan/mengawankan setiap
𝑥 ∈ 𝐴 (anggota dari himpunan A) dengan tepat “satu” 𝑦 ∈ 𝐵 (anggota
himpunan B).
Dapat dituliskan :
𝑦 = 𝑓(𝑥)
Beberapa istilah pada fungsi :
Himpunan A merupakan daerah asal (domain) 𝐷𝑓
Himpunan B (kodomain) yang merupakan derah kawan,
sedangkan himpunan semua anggota B yang memiliki pasangan
disebut daerah hasil dari fungsi atau range 𝑅𝑓.
4. FUNGSI
Fungsi tidak membolehkan objek dalam daerah asal
dipasangkan lebih dari satu pada daerah hasil.
Notasi Fungsi :
Untuk memberi nama Fungsi
digunakanan Sebuah huruf tunggal
f (atau g atau F )
maka f(x) DIBACA
“f dari x” ATAU “f pada x “
Bukan Fungsi
5. Fungsi
Bukan fungsi, sebab ada elemen A yang
mempunyai 2 kawan.
Bukan fungsi, sebab ada elemen A yang
tidak mempunyai kawan.
A B
FUNGSI
6. Notasi fungsi Untuk menyatakan bahwa fungsi f mengawankan anggota-anggota
himpunan A terhadap anggota-anggota B,
f : A B
f : x 2x dibaca f mengawankan x terhadap 2x.
f : x x2+3x+5 dibaca f mengawankan x terhadap x2+3x+5.
Rumus fungsi
f(x)=2x
f(x)=x2+3x+5
f(x) = y
x disebut variabel independent, dan y disebut variabel dependent.
7. Perhatikan gambar di atas
Himpunan A = {1,2,3,4} dan Himpunan B = {1,2,3,5,6,7}, suatu fungsi
yang memetakan 𝑓: 𝐴 → 𝐵 ditentukan oleh f(x) = 2x – 1, maka daerah
hasil atau range dari himpunan di atas dapat dinyatakan dalam :
2(1) – 1 = 1 ; 2(2) – 1 = 3 ; 2(3) – 1 = 5 ; 2(4) – 1 = 7
𝑅𝑓 = {1,3,5,7}
1
2
3
4
1
2
3
5
6
7
A B
DOMAIN DAN RANGE
8. DOMAIN DAN RANGE
Tentukan domain dan range dari 𝑓 𝑥 = 𝑥 − 2
Fungsi di atas tidak terdefinisi (tidak memberikan nilai real) jika x-
2<0 atau x<2
Artinya, fungsi f terdefinisi jika x≥2
Dengan demikian, 𝐷𝑓 = [2, ∞)= 𝑥 𝑥 ≥ 2, 𝑥 ∈ 𝑅
Untuk 𝑥 ≥ 2, diperoleh 𝑥 − 2 ≥ 0
Sehingga 𝑓 𝑥 = 𝑥 − 2 ≥ 0
Dengan demikian, 𝑅𝑓 = [0, ∞)= 𝑥 𝑥 ≥ 0, 𝑥 ∈ 𝑅
9. Contoh :
Jika 𝑓 𝑥 = 𝑥3
− 4, maka
Untuk x = 2,
f (2) = (2)3-4 = …
Untuk x = -1,
f (-1)= (-1)3-4 = …
Aturan
Daerah Asal
Daerah Hasil
10. Latihan Soal
Untuk f(x) = x2 – 2x, cari dan sederhanakan:
a. f(4)
b. f(4 + h)
c. f(4 + h) – f(4)
11. JENIS FUNGSI:
Jenis Fungsi :
Fungsi konstan: f(x) = C,
Fungsi linear : f(x) = ax + b
Fungsi kuadrat : f(x) = ax2 +bx + c
Fungsi eksponensial : f(x) = ex
Fungsi logaritma : f(x) = log x
12. FUNGSI KOMPOSISI
Diberikan fungsi f(x) dan g(x), komposisi fungsi
antara f(x) dan g(x) ditulis (f 𝜊 g)(x) = f(g(x))
Domain dari (f 𝜊 g)(x) adalah himpunan semua
bilangan x dengan domain g(x) sehingga g(x) di
dalam Df
14. GRAFIK FUNGSI
Cara menggambar grafik fungsi yang baik
adalah dengan membuat tabel nilai-nilai
sehingga diperoleh pasangan nilai dari
peubah fungsi yang mewakili suatu titik.
Untuk menggambar garis lurus diperlukan
dua titik, untuk menggambar fungsi kuadrat
minimal dibutuhkan tiga titik.Misal, gambar
grafik fungsi f(x)=x+1 sebagai berikut.
15. Fungsi Konstan
Definisi
f : x C dengan C konstan disebut
fungsi konstan (tetap). Fungsi f
mengawankan setiap bilangan real
dengan C.
Contoh 4.6
Fungsi f(x) = 3
16. Fungsi Linear
Definisi
Fungsi pada bilangan real
yang didefinisikan f(x) = ax +
b, a dan b konstan dengan a ≠
0
Contoh 4.7
Gambarlah grafik fungsi y = 2x + 3
17. Dengan menentukan titik-titik potong dengan sumbu-x dan sumbu-y
y = 2x + 3
Titik potong grafik dengan sumbu-x :
y = 0 : 0 = 2x + 3
-2x = 3
x = -
3
2
sehingga titik potong grafik dengan sumbu x adalah −
3
2
, 0
Titik potong grafik dengan sumbu-y :
X = 0 : y = 2x + 3
y = 2.0 + 3
y = 0 + 3
y = 3
Sehingga titik potong grafik dengan sumbu-y adalah (0,3)
18.
19.
20. Fungsi Kuadrat
Definisi
Bentuk umum fungsi kuadrat adalah y = ax2+bx+c dengan a, b, c ∈ R dan a≠0
Grafik fungsi kuadrat juga sering disebut fungsi parabola. Jika a > 0, parabola terbuka ke
atas sehingga mempunyai titik balik minimum, dan jika a < 0 parabola terbuka ke bawah
sehingga mempunyai titik balik maksimum.
21.
22.
23.
24. OPERASI FUNGSI
Jika f dan g dua fungsi maka jumlah f + g, selisih f – g, hasil kali fg,
hasil bagi f/g dan perpangkatan fn adalah fungsi-fungsi dengan
daerah asal berupa irisan dari daerah asal f dan daerah asal g, dan
dirumuskan sebagai berikut.
(f + g)(x) = f (x) + g(x)
(f – g)(x) = f (x) – g(x)
(f x g)(x) = f (x) x g(x)
(f / g)(x) = f (x) / g(x) asalkan g(x) ≠ 0
25. TUGAS
1. Untuk f(x) = x2+x dan g(x) = 2(x+1), carilah:
a. (f + g)(2)
b. (f - g)(2)
c. (f /g)(1)
d. (f∘g)(1)
e. (g∘f)(1)