Pertemuan 4
•Download as PPTX, PDF•
0 likes•89 views
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.
Report
Share
Report
Share
Recommended
Pertemuan 4
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.
Pertemuan 3
Materi pertemuan 3
Mata Kuliah Matematika Dasar
Jurusan Teknik Sipil
Fakultas Teknik
Universitas Borneo Tarakan
Fungsi dan Grafik Fungsi
Pertemuan 7
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.
1.5 all notes
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.
Pertemuan 7
Kekontinuan Fungsi
Matematika Dasar
Jurusan Teknik Sipil
Fakultas Teknik
Universitas Borneo Tarakan
Limit and continuity (2)
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.
Section 1-5
The document discusses the concept of limits in calculus, including the formal definition of a limit as the value a function approaches as the input gets arbitrarily close to a given number. It provides examples of evaluating limits graphically and algebraically, such as determining the limit of a function as x approaches 3 is 6, even though the function is undefined at x=3 due to a removable discontinuity. The document also discusses using graphs to identify different types of discontinuities and limits.
Limit of Function And Its Types
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.
Recommended
Pertemuan 4
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.
Pertemuan 3
Materi pertemuan 3
Mata Kuliah Matematika Dasar
Jurusan Teknik Sipil
Fakultas Teknik
Universitas Borneo Tarakan
Fungsi dan Grafik Fungsi
Pertemuan 7
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.
1.5 all notes
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.
Pertemuan 7
Kekontinuan Fungsi
Matematika Dasar
Jurusan Teknik Sipil
Fakultas Teknik
Universitas Borneo Tarakan
Limit and continuity (2)
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.
Section 1-5
The document discusses the concept of limits in calculus, including the formal definition of a limit as the value a function approaches as the input gets arbitrarily close to a given number. It provides examples of evaluating limits graphically and algebraically, such as determining the limit of a function as x approaches 3 is 6, even though the function is undefined at x=3 due to a removable discontinuity. The document also discusses using graphs to identify different types of discontinuities and limits.
Limit of Function And Its Types
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.
Mean Value Theorem | Mathematics
The document provides an overview of the Mean Value Theorem and Rolle's Theorem. It discusses that the Mean Value Theorem states that for any function continuous on a closed interval, there exists a point where the slope of the tangent line equals the slope of the secant line through the endpoints. Rolle's Theorem is a special case where if a function is continuous on a closed interval and differentiable on the open interval, if the function is equal at the endpoints, the derivative at some interior point is zero. Graphical interpretations are also provided to illustrate these theorems.
Mean Value Theorems
- Rolle's theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in (a,b) where the derivative f'(c) = 0.
- The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists a value c in (a,b) such that the average rate of change of f over the interval [a,b] equals the instantaneous rate of change
MEAN VALUE THEOREM
The document discusses the Mean Value Theorem, which states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in (a,b) such that:
f(b) - f(a) = f'(c)(b - a)
In other words, there is at least one point where the slope of the tangent line equals the slope of the secant line between points a and b. The document provides examples and illustrations to demonstrate how to apply the Mean Value Theorem.
DBMS 11 | Design Theory [Normalization 1]
1. Anomalies
2. Normalization
3. Functional dependency
4. FD closure, Attribute closure
5. All candidate key finding
6. Extraneous attribute detection
7. Minimal cover
Abstract algebra ppt
The document defines several key concepts in abstract algebra including groups, abelian groups, subgroups, isomorphic groups, normal subgroups, quotient groups, and homomorphisms. A group is a set with a binary operation satisfying closure, associativity, identity, and inverse properties. An abelian group additionally requires the binary operation be commutative. A subgroup is a subset of a group that also forms a group. Isomorphic groups have the same structure despite notational differences. Normal subgroups allow the formation of quotient groups by considering left and right cosets. A homomorphism between groups preserves the group operation.
4.3 Logarithmic Functions
Solve an exponential equation using base 10 logarithms.
Find the missing part of a logarithm equation.
Use properties of logarithms to solve problems.
Mean value theorem
Md. Mizanur Rahaman presented on the Mean Value Theorem of differential calculus. Some key points:
1) The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that the slope of the tangent line at (c,f(c)) equals the slope of the secant line between (a,f(a)) and (b,f(b)).
2) A restricted form of the theorem was proved by Rolle in 1691 for polynomials, while the modern form was stated and proved by Cauchy in 18
Roll's theorem
This document discusses Rolle's Theorem from calculus. Rolle's Theorem states that if a function f is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), and if f(a) = f(b), then there exists at least one value c in the interval (a,b) where the derivative of f is equal to 0. The document provides an example of applying Rolle's Theorem to show that the derivative of the function f(x) = x^2 - 3x + 2 is equal to 0 at some point between the two x-intercepts of the function.
Lar calc10 ch01_sec4
This document discusses continuity and one-sided limits of functions. It defines continuity at a point and on an open interval. Functions can have removable or nonremovable discontinuities. One-sided limits are used to investigate continuity on closed intervals. The intermediate value theorem guarantees that a continuous function takes on all intermediate values between its values at the endpoints of a closed interval. Examples demonstrate determining continuity and applying properties of continuity and the intermediate value theorem.
Rolles theorem
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
Applied Calculus Chapter 1 polar coordinates and vector
The document discusses polar coordinates and vectors. It introduces parametric equations to describe the motion of a particle in the xy-plane over time. The variable t is called the parameter. Examples are provided to demonstrate forming Cartesian equations by eliminating t from parametric equations and graphing parametric equations by plugging in values of t. The document also discusses standard representation and finding direction numbers of vectors in R3.
4.1 Inverse Functions
The document discusses inverse functions. It defines inverse functions as pairs of one-to-one functions that undo each other. To find the inverse of a function, interchange the x and y variables and solve for y. The graph of an inverse function is a reflection of the original function across the line y = x. Examples are provided to illustrate how to determine if a function is one-to-one, find the inverse function, and relate the graph of a function to its inverse.
3.4 Polynomial Functions and Their Graphs
Identify the end behavior of a function
Use the Intermediate Value Theorem and the Boundedness Theorem to locate zeros of a function.
Hipster maths
This is an introduction to Analytic Combinatorics. I gave this talk as part of the PSS the 9 October 2014 at the University of Bath... needless to say I threw in a couple of hipster jokes.
Rolle's Theorem
Rolle's Theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), where f(a) = f(b), then there exists at least one number c in the open interval (a,b) where the derivative f'(c) is equal to 0. The document provides an example of applying Rolle's Theorem to the polynomial function f(x) = x^2 - 2.5x + 1.5 on the interval [1,2], showing that the derivative is 0 at x = 1.5.
Ejercicios propuestos 2 Estructuras Discretas II Verónica Torres
This document contains solutions to exercises on discrete structures II. It addresses questions about Hasse diagrams and partially ordered sets (POSets). For a given Hasse diagram D, it finds the upper and lower bounds, maximal and minimal elements, maximum and minimum, upper and lower covers, and supremum and infimum for subsets E and F of D. It also shows how to construct the associated digraph of a Hasse diagram using an algorithm. Finally, it analyzes whether a given POSet is a lattice, whether it is distributive, finds complements of elements f and h, and determines if figure B is a sublattice.
Lesson 19: The Mean Value Theorem (slides)
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
AP Calculus AB Theorems and Definitions
The document defines several calculus concepts and theorems in multiple choice format:
1) It defines the limit of a function, continuity of a function, and the Extreme Value Theorem.
2) It also defines derivatives, antiderivatives, the Mean Value Theorem, the Intermediate Value Theorem, and asymptotes.
3) The document tests understanding of the definitions of the derivative, the Fundamental Theorem of Calculus, the Mean Value Theorem for Integrals, and the Inverse Function Theorem.
CAPS_Discipline_Training
This document provides definitions and formulas for calculating curl, divergence, and line integrals. It defines curl as the measure of rotation in a vector field and divergence as the measure of how a vector field spreads out from a point. Formulas are given for calculating curl, divergence, and line integrals over curves with respect to arc length, x, y, and vector fields. It also discusses properties of conservative and irrotational vector fields as they relate to these calculations.
BAS-CAL-LESSON-1-LIMITS-OF-A-FUNCTION.pdf
1. The document discusses limits of functions and how to evaluate them. It provides examples of evaluating limits from tables of values and graphs.
2. Key points are that a limit exists if the left and right side limits are equal, even if the function is undefined at that point, and that the limit is not necessarily equal to the function value.
3. Evaluating limits involves considering the behavior of a function as the variable approaches a value from the left and right, without actually being equal to that value.
real number presentations
This document provides an overview of key concepts in calculus, including:
1) Limits are defined as the values a function approaches for a given input as the input approaches a certain value.
2) Left and right hand limits describe the limit of a function as the input approaches a point from the left or right side.
3) A limit exists if the left and right hand limits are equal; a limit does not exist if they are not equal.
4) Continuity and discontinuity refer to whether a function's limit equals its value as the input approaches a point.
5) L'Hospital's rule provides a method for evaluating indeterminate limits involving infinity/infinity or 0/0 forms.
More Related Content
What's hot
Mean Value Theorem | Mathematics
The document provides an overview of the Mean Value Theorem and Rolle's Theorem. It discusses that the Mean Value Theorem states that for any function continuous on a closed interval, there exists a point where the slope of the tangent line equals the slope of the secant line through the endpoints. Rolle's Theorem is a special case where if a function is continuous on a closed interval and differentiable on the open interval, if the function is equal at the endpoints, the derivative at some interior point is zero. Graphical interpretations are also provided to illustrate these theorems.
Mean Value Theorems
- Rolle's theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in (a,b) where the derivative f'(c) = 0.
- The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists a value c in (a,b) such that the average rate of change of f over the interval [a,b] equals the instantaneous rate of change
MEAN VALUE THEOREM
The document discusses the Mean Value Theorem, which states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in (a,b) such that:
f(b) - f(a) = f'(c)(b - a)
In other words, there is at least one point where the slope of the tangent line equals the slope of the secant line between points a and b. The document provides examples and illustrations to demonstrate how to apply the Mean Value Theorem.
DBMS 11 | Design Theory [Normalization 1]
1. Anomalies
2. Normalization
3. Functional dependency
4. FD closure, Attribute closure
5. All candidate key finding
6. Extraneous attribute detection
7. Minimal cover
Abstract algebra ppt
The document defines several key concepts in abstract algebra including groups, abelian groups, subgroups, isomorphic groups, normal subgroups, quotient groups, and homomorphisms. A group is a set with a binary operation satisfying closure, associativity, identity, and inverse properties. An abelian group additionally requires the binary operation be commutative. A subgroup is a subset of a group that also forms a group. Isomorphic groups have the same structure despite notational differences. Normal subgroups allow the formation of quotient groups by considering left and right cosets. A homomorphism between groups preserves the group operation.
4.3 Logarithmic Functions
Solve an exponential equation using base 10 logarithms.
Find the missing part of a logarithm equation.
Use properties of logarithms to solve problems.
Mean value theorem
Md. Mizanur Rahaman presented on the Mean Value Theorem of differential calculus. Some key points:
1) The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that the slope of the tangent line at (c,f(c)) equals the slope of the secant line between (a,f(a)) and (b,f(b)).
2) A restricted form of the theorem was proved by Rolle in 1691 for polynomials, while the modern form was stated and proved by Cauchy in 18
Roll's theorem
This document discusses Rolle's Theorem from calculus. Rolle's Theorem states that if a function f is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), and if f(a) = f(b), then there exists at least one value c in the interval (a,b) where the derivative of f is equal to 0. The document provides an example of applying Rolle's Theorem to show that the derivative of the function f(x) = x^2 - 3x + 2 is equal to 0 at some point between the two x-intercepts of the function.
Lar calc10 ch01_sec4
This document discusses continuity and one-sided limits of functions. It defines continuity at a point and on an open interval. Functions can have removable or nonremovable discontinuities. One-sided limits are used to investigate continuity on closed intervals. The intermediate value theorem guarantees that a continuous function takes on all intermediate values between its values at the endpoints of a closed interval. Examples demonstrate determining continuity and applying properties of continuity and the intermediate value theorem.
Rolles theorem
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
Applied Calculus Chapter 1 polar coordinates and vector
The document discusses polar coordinates and vectors. It introduces parametric equations to describe the motion of a particle in the xy-plane over time. The variable t is called the parameter. Examples are provided to demonstrate forming Cartesian equations by eliminating t from parametric equations and graphing parametric equations by plugging in values of t. The document also discusses standard representation and finding direction numbers of vectors in R3.
4.1 Inverse Functions
The document discusses inverse functions. It defines inverse functions as pairs of one-to-one functions that undo each other. To find the inverse of a function, interchange the x and y variables and solve for y. The graph of an inverse function is a reflection of the original function across the line y = x. Examples are provided to illustrate how to determine if a function is one-to-one, find the inverse function, and relate the graph of a function to its inverse.
3.4 Polynomial Functions and Their Graphs
Identify the end behavior of a function
Use the Intermediate Value Theorem and the Boundedness Theorem to locate zeros of a function.
Hipster maths
This is an introduction to Analytic Combinatorics. I gave this talk as part of the PSS the 9 October 2014 at the University of Bath... needless to say I threw in a couple of hipster jokes.
Rolle's Theorem
Rolle's Theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), where f(a) = f(b), then there exists at least one number c in the open interval (a,b) where the derivative f'(c) is equal to 0. The document provides an example of applying Rolle's Theorem to the polynomial function f(x) = x^2 - 2.5x + 1.5 on the interval [1,2], showing that the derivative is 0 at x = 1.5.
Ejercicios propuestos 2 Estructuras Discretas II Verónica Torres
This document contains solutions to exercises on discrete structures II. It addresses questions about Hasse diagrams and partially ordered sets (POSets). For a given Hasse diagram D, it finds the upper and lower bounds, maximal and minimal elements, maximum and minimum, upper and lower covers, and supremum and infimum for subsets E and F of D. It also shows how to construct the associated digraph of a Hasse diagram using an algorithm. Finally, it analyzes whether a given POSet is a lattice, whether it is distributive, finds complements of elements f and h, and determines if figure B is a sublattice.
Lesson 19: The Mean Value Theorem (slides)
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
AP Calculus AB Theorems and Definitions
The document defines several calculus concepts and theorems in multiple choice format:
1) It defines the limit of a function, continuity of a function, and the Extreme Value Theorem.
2) It also defines derivatives, antiderivatives, the Mean Value Theorem, the Intermediate Value Theorem, and asymptotes.
3) The document tests understanding of the definitions of the derivative, the Fundamental Theorem of Calculus, the Mean Value Theorem for Integrals, and the Inverse Function Theorem.
CAPS_Discipline_Training
This document provides definitions and formulas for calculating curl, divergence, and line integrals. It defines curl as the measure of rotation in a vector field and divergence as the measure of how a vector field spreads out from a point. Formulas are given for calculating curl, divergence, and line integrals over curves with respect to arc length, x, y, and vector fields. It also discusses properties of conservative and irrotational vector fields as they relate to these calculations.
What's hot (20)
Applied Calculus Chapter 1 polar coordinates and vector
Applied Calculus Chapter 1 polar coordinates and vector
Ejercicios propuestos 2 Estructuras Discretas II Verónica Torres
Ejercicios propuestos 2 Estructuras Discretas II Verónica Torres
Similar to Pertemuan 4
BAS-CAL-LESSON-1-LIMITS-OF-A-FUNCTION.pdf
1. The document discusses limits of functions and how to evaluate them. It provides examples of evaluating limits from tables of values and graphs.
2. Key points are that a limit exists if the left and right side limits are equal, even if the function is undefined at that point, and that the limit is not necessarily equal to the function value.
3. Evaluating limits involves considering the behavior of a function as the variable approaches a value from the left and right, without actually being equal to that value.
real number presentations
This document provides an overview of key concepts in calculus, including:
1) Limits are defined as the values a function approaches for a given input as the input approaches a certain value.
2) Left and right hand limits describe the limit of a function as the input approaches a point from the left or right side.
3) A limit exists if the left and right hand limits are equal; a limit does not exist if they are not equal.
4) Continuity and discontinuity refer to whether a function's limit equals its value as the input approaches a point.
5) L'Hospital's rule provides a method for evaluating indeterminate limits involving infinity/infinity or 0/0 forms.
CALCULUS chapter number one presentation
This document provides an overview of key concepts in calculus, including limits, derivatives, and continuity. It discusses how limits describe the behavior of a function as its input approaches a certain value. Specific topics covered include tangent lines and limits, areas and limits, decimals and limits, one-sided limits, infinite limits, vertical asymptotes, computing limits using limit laws, and continuity of functions. The document also introduces derivatives as a new function whose value at each x is the derivative of the original function at that point.
Limits and continuity powerpoint
The document discusses limits and continuity, explaining what limits are, how to evaluate different types of limits using techniques like direct substitution, dividing out, and rationalizing, and how limits relate to concepts like derivatives, continuity, discontinuities, and the intermediate value theorem. Special trig limits, properties of limits, and how limits can be used to find derivatives are also covered.
Lecture co3 math21-1
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
Limits of a Function
The document discusses limits of functions and how to evaluate them. It explains that a limit describes the behavior of a function as its variable approaches a constant value. To evaluate a limit, tables of values are constructed showing the outputs of the function for inputs very close to the constant. The limit is the unique value the outputs approach. Examples demonstrate evaluating limits by constructing tables and interpreting the results.
Lecture 4 the limit of a function
This document discusses limits of functions. It defines limits numerically and graphically, and explains that the limit of a function as x approaches a particular value a describes the behavior of the function near a, regardless of its exact value at a. One-sided limits are introduced, where the left-sided limit considers values of x approaching a from below and the right-sided limit considers values above. The overall limit only exists if the one-sided limits exist and are equal. Examples calculating one-sided and overall limits from graphs and tables of values are provided.
Project in Calcu
This document provides an overview of key calculus concepts including:
- Functions and function notation which are fundamental to calculus
- Limits which allow defining new points from sequences and are essential to calculus concepts like derivatives and integrals
- Derivatives which measure how one quantity changes in response to changes in another related quantity
- Types of infinity and limits involving infinite quantities or areas
The document defines functions, limits, derivatives, and infinity, and provides examples to illustrate these core calculus topics. It lays the groundwork for further calculus concepts to be covered like integrals, derivatives of more complex functions, and applications of limits, derivatives, and infinity.
Section 2.4
This document discusses continuity and discontinuity in functions. It defines a function as continuous at a point c if the left and right hand limits are the same as x approaches c. A function is discontinuous if the left and right limits do not match up. Examples are given of different types of discontinuities like removable discontinuities.
Calc 1.2a
1. This document discusses finding limits graphically and numerically using tables of values and graphs. It explains how to determine if a limit exists or does not exist based on the behavior of the function as it approaches the given value.
2. Limits can exist even if the function is not defined at a point. The limit depends on whether the function values approach a single number from both sides, not on the actual function value at that point.
3. There are special types of functions like piecewise functions and greatest integer functions that require specific approaches to graph. Limits can fail to exist if the function behavior differs on each side, becomes unbounded, or oscillates without approaching a single value.
BC1-Limit-of-Function.pptx
The document provides an introduction to limits of functions in calculus. It defines the limit of a function as the value that the function approaches as the variable approaches a specific number. It discusses evaluating limits using tables of values and graphs by looking at how close the function values get to a limit as the variable approaches a given number from the left and right. The document emphasizes that the limit of a function as it approaches a number may not be equal to the actual value of the function at that number, and provides examples where the limit does not exist because the left and right limits do not match.
Limit and continuity
This document discusses limits and continuity for functions of two variables. It defines a limit of a function f(x,y) as (x,y) approaches (a,b) as making the values of f(x,y) as close to L as desired by taking the point (x,y) sufficiently close to (a,b). It states the rules of limits can be added, subtracted, multiplied, composed and divided if the denominator's limit does not equal zero. A function is continuous at (a,b) if the limit exists as (x,y) approaches (a,b) and equals the function's value at (a,b). Examples are provided to demonstrate limits that do and do
CHAP6 Limits and Continuity.pdf
The document summarizes key concepts about limits from calculus including:
1) Limits are foundational to calculus and describe the behavior of a function as its input gets closer to a target value.
2) One-sided limits describe the behavior of a function as its input approaches the target value from the left or right. For a limit to exist, the one-sided limits must be equal.
3) A limit may not exist if the one-sided limits are not equal, if the function becomes arbitrarily large (approaches positive or negative infinity) as the input approaches the target value, or if the function oscillates without settling on a value.
Function
This document contains a summary of key concepts in mathematics functions including:
- Definitions of different types of functions such as polynomial, rational, trigonometric, exponential, and logarithmic functions.
- Concepts related to functions like domain, range, and notation for functions.
- Examples demonstrating how to find the domain and range of functions.
- Key topics in trigonometry including trigonometric functions, trigonometric tables, addition formulas, laws of sines and cosines.
- Concepts of limits, average and instantaneous rates of change, techniques for finding limits, and limits involving infinity.
R lecture co4_math 21-1
This document discusses concepts related to calculus including functions, limits, continuity, and derivatives. The objective is for students to be able to evaluate limits and determine derivatives of algebraic functions. It defines functions and function notation. It discusses limits, continuity, and the definition of the derivative. It provides examples of evaluating limits using theorems and the squeeze principle. It also defines types of discontinuities and conditions for continuity.
Lemh105
This document provides an introduction to the concepts of continuity and differentiability in calculus. It begins by giving two informal examples of functions that are and aren't continuous at a point to build intuition. It then provides a formal definition of continuity as the limit of a function at a point equaling the function value at that point. Several examples are worked through to demonstrate checking continuity at points and for entire functions. The document introduces the concept of limits approaching infinity to discuss the continuity of functions like 1/x. Overall, it lays the groundwork for understanding continuity and differentiability through examples and definitions.
Limits BY ATC
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.
VISIT US @
www.anuragtyagiclasses.com
Limits BY ATC
Limits are used to describe the value a function or sequence approaches as the input approaches some value. They are essential to calculus and define concepts like continuity, derivatives, and integrals. A function is continuous at a point if the limit of the function as x approaches a is equal to the function's value at that point. If a function is continuous at every point within an interval, it is continuous on that entire interval. Key properties are that sums, differences, products, and quotients of continuous functions are also continuous.
__limite functions.sect22-24
This document discusses infinite limits, limits at infinity, and limit rules. It begins by explaining that the limits of 1/x as x approaches 0 from the left and right do not exist as real numbers, but it is useful to describe the behavior as approaching positive and negative infinity. It then discusses properties of infinite limits, including one-sided limits and examples. The document proceeds to define vertical asymptotes and provide examples of determining asymptotes. It concludes by covering limit rules that can be used to evaluate limits more easily, such as sum, difference, product, and quotient rules, as well as the sandwich theorem.
Limit 140929031133-phpapp01
The document discusses the concept of limits in calculus. It defines a limit and provides examples to build intuition. Some key points made in the document include:
1) A limit describes the behavior of a function as the input values get closer to a specific value without actually reaching it.
2) Basic theorems for limits include properties like the limit of a sum being the sum of the limits.
3) Special cases like limits with zero in the denominator require factoring to find the limit.
4) One-sided or directional limits describe the behavior of a function as the input approaches a value from the left or right.
Similar to Pertemuan 4 (20)
More from Aswar Amiruddin
Irigasi dan Bangunan Air 13.pdf
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.
Irigasi dan Bangunan Air 12.pdf
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.
Irigasi dan Bangunan Air 11.pdf
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.
Irigasi dan Bangunan Air 5.pdf
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.
MODUL-14.pdf
Dokumen tersebut membahas tentang perencanaan bangunan talang untuk jaringan irigasi, termasuk parameter perencanaan seperti kehilangan energi, dimensi, kemiringan, dan penulangan beton."
MATERI PERTEMUAN 13.pdf
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
Tugas-Observasi-Kelompok-Selumit-Pantai-Salin.docx
Laporan ini menganalisis perkembangan wilayah pesisir Selumit Pantai melalui pengamatan pola pemukiman, infrastruktur, dan tingkat perkembangannya. Pola pemukiman berkelompok karena mata pencaharian penduduk sebagai nelayan. Infrastruktur belum memadai dengan beberapa fasilitas dasar. Tingkat perkembangan wilayah masih rendah meski memiliki potensi untuk ditingkatkan.
Ppt rekayasa pantai 1 aswar
Mata kuliah Rekayasa Pantai membahas tentang definisi, istilah, dan bidang kajian rekayasa pantai serta outcome dari mata kuliah tersebut."
Pertemuan 1
Mata kuliah ini membahas konsep integral tak tentu, tertentu, dan operasi matriks untuk memecahkan permasalahan teknik. Mahasiswa akan mempelajari teknik-teknik integral seperti integral dasar, integral pecah rasional, dan integral parsial beserta penerapannya dalam masalah teknik sipil menggunakan integral. Penilaian terdiri atas tugas, UTS, dan UAS. Kuliah dilaksanakan secara tatap muka atau daring tergantung kebijakan univers
Psda 5
Pengembangan Sumber Daya Air
Ketersediaan Air
Jurusan Teknik Sipil
Fakultas Teknik
Universitas Borneo Tarakan
Rekayasa hidrologi pertemuan 2
This document discusses rainfall (presipitasi) in Indonesia. It defines presipitasi as water falling from the atmosphere to Earth's surface, including rain, snow, fog, dew and hail. In tropical areas like Indonesia, the main contributor is rain. Rain occurs when moist air rising in the atmosphere cools and condenses. Factors like wind, temperature, atmospheric and local pressures influence rainfall amounts. The types of rain discussed are convective (from air masses lifted by heating), cyclonic (from uneven solar heating creating pressure systems), and orographic (from moist air lifted over mountains). Methods for measuring and analyzing rainfall data using rainfall stations are also outlined.
Rekayasa hidrologi pertemuan 1
Rek Hidrologi 1
Jurusan Teknik Sipil
Fakultas Teknik
Universitas Borneo Tarakan
Rekayasa hidrologi pertemuan 3
The document discusses methods for analyzing rainfall data and estimating areal rainfall, including:
1. Methods for correcting missing data like the normal ratio method and reciprocal method.
2. Consistency testing of rainfall data using double mass curve analysis.
3. Methods for estimating areal rainfall including arithmetic mean, Thiessen polygon, and isohyetal methods. Worked examples are provided for the Thiessen polygon and isohyetal methods.
Rekayasa hidrologi pertemuan 4
The document discusses evapotranspiration and methods for calculating it. It defines evaporation, transpiration, and evapotranspiration. It also discusses factors that influence evaporation and several methods for measuring and calculating evapotranspiration, including the Penman, Blaney-Criddle, Thornthwaite, and Penman-Monteith methods. An example calculation is provided using the Penman modified method.
Rekayasa hidrologi pertemuan 5
This document discusses frequency analysis of hydrological data. The goals of frequency analysis are to understand the relationship between extreme events and their frequency using probability distributions. Four common distributions used in hydrology are discussed: normal, log normal, log Pearson III, and Gumbel. Key statistical parameters like mean, standard deviation, and coefficients are defined. Chi-square and Smirnov-Kolmogorov tests are introduced to select the best fitting distribution. Tables of critical values for these tests are also included.
More from Aswar Amiruddin (20)
Tugas-Observasi-Kelompok-Selumit-Pantai-Salin.docx
Tugas-Observasi-Kelompok-Selumit-Pantai-Salin.docx
Recently uploaded
International Conference on NLP, Artificial Intelligence, Machine Learning an...
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Computational Engineering IITH Presentation
This Presentation will give you a brief idea about what Computational Engineering at IIT Hyderabad has to offer.
Unit-III-ELECTROCHEMICAL STORAGE DEVICES.ppt
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
Casting-Defect-inSlab continuous casting.pdf
Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...
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
BRAIN TUMOR DETECTION for seminar ppt.pdf
BRAIN TUMOR DETECTION
AND CLASSIFICATION USING
ARTIFICIAL INTELLIGENCE
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressions
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
22CYT12-Unit-V-E Waste and its Management.ppt
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Literature Review Basics and Understanding Reference Management.pptx
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Understanding Inductive Bias in Machine Learning
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Hematology Analyzer Machine - Complete Blood Count
The CBC machine is a common diagnostic tool used by doctors to measure a patient's red blood cell count, white blood cell count and platelet count. The machine uses a small sample of the patient's blood, which is then placed into special tubes and analyzed. The results of the analysis are then displayed on a screen for the doctor to review. The CBC machine is an important tool for diagnosing various conditions, such as anemia, infection and leukemia. It can also help to monitor a patient's response to treatment.
Recently uploaded (20)
International Conference on NLP, Artificial Intelligence, Machine Learning an...
International Conference on NLP, Artificial Intelligence, Machine Learning an...
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressions
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressions
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf
BPV-GUI-01-Guide-for-ASME-Review-Teams-(General)-10-10-2023.pdf
Literature Review Basics and Understanding Reference Management.pptx
Literature Review Basics and Understanding Reference Management.pptx
Introduction to AI Safety (public presentation).pptx
Introduction to AI Safety (public presentation).pptx
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...
Hematology Analyzer Machine - Complete Blood Count
Hematology Analyzer Machine - Complete Blood Count
Pertemuan 4
- 1. MATEMATIKA DASAR Aswar Amiruddin, ST., MT.
- 3. Sub CP-MK Memahami konsep limit di suatu titik Indikator Menetukan nilai limit fungsi di satu titik dan sifatnya
- 4. Konsep Limit Misalkan f(x) suatu fungsi, c dan L bilangan real sedemikian hingga: • Bila x dekat c tetapi tidak sama dgn c (x c), maka f(x) dekat ke L • Bila x mendekati c tetapi x c, maka f(x) mendekati L • Misalkan f(x) dapat kita buat sedekat mungkin ke L dg membuat x cukup dekat c tetapi tidak sama dengan c • Maka dapat dikatakan bahwa limit f(x) bila x mendekati c adalah L, 𝐥𝐢𝐦 𝒙→𝒄 𝒇 (𝒙) = 𝑳
- 5. Konsep Limit Perhatikan fungsi berikut : lim 𝑥→𝑐 𝑓(𝑥) = 𝒙𝟐−𝟏 𝒙−𝟏 , terdefinisi untuk setiap x ϵ R kecuali untuk x = 1, fungsi tidak terdefinisi jika x=1, akan tetapi dengan mengambil nilai mendekati 1 maka fungsi di atas terdefinisi.
- 6. Konsep Limit Dari tabel di atas dan grafik disamping terlihat bahwa f(x) mendekati 2 jika x mendekati 1 Secara matematis dapat dituliskan Sebagai berikut 𝐥𝐢𝐦 𝒙→𝟏 𝒙𝟐 − 𝟏 𝒙 − 𝟏 = 𝟐
- 8. Pengertian limit secara intuitif Kita tuliskan lim 𝑥→𝑐 𝑓(𝑥) = 𝐿 apabila jika x dekat ke c, maka nilai f(x) dekat ke L lim 𝑥→𝑐 𝑓(𝑥) disebut “limit f di c”. Bilangan L merupakan nilai limit tersebut. Contoh lim 𝑥→𝑐 𝑓(𝑥) = 𝒙𝟐 − 𝟏 𝒙 − 𝟏
- 10. Arti ”mendekati” pada konsep limit, bisa dari kiri titik c atau dari kanan titik c. Dengan kata lain, nilai limit f (x) untuk x mendekati c ada, jika limit kiri dan limit kanan ada. Limit Kanan dan Limit Kiri Limit kanan 𝑙𝑖𝑚 𝑥→𝑐+ 𝑓(𝑥) = 𝐿 berarti jika x > c dan dekat ke c, maka f(x) dekat ke L. Limit kanan 𝑙𝑖𝑚 𝑥→𝑐− 𝑓(𝑥) = 𝐿 berarti jika x < c dan dekat ke c, maka f(x) dekat ke L. Limit Kanan dan Limit Kiri
- 11. Bilakah Limit Ada/Tidak Ada? Limit fungsi f di c ada jika dan hanya jika limit kanan dan limit kiri f di c ada dan nilainya sama dengan L (perhatikan definisi limit). Limit f di c tidak ada bila salah satu di antara beberapa kemungkinan berikut terjadi: 1. Limit kanan dan limit kiri f di c ada, tetapi nilainya tidak sama. 2. Limit kanan atau limit kiri f di c tidak ada, karena a. Nilai f di dekat c menuju tak terhingga. b. Nilai f di dekat c berosilasi.
- 12. Limit Fungsi Bila sebelumnya kita telah mencoba memaknai limit fungsi di suatu titik secara intuitif, maka skrg kita akan mendefinisikannya secara formal. Definisi: 𝑙𝑖𝑚 𝑥→𝑐 𝑓(𝑥) = 𝐿 jika dan hanya jika: “untuk setiap ε > 0 terdapat δ > 0 sehingga: jika 0 < | x – c | < δ, maka | f(x) – L | < ε.”
Editor's Notes
- Are your classroom colors different than what you see in this template? That’s OK! Click on Design -> Variants (the down arrow) -> Pick the color scheme that works for you! Feel free to change any “You will…” and “I will…” statements to ensure they align with your classroom procedures and rules!