Lecture 12
•
0 likes•56 views
This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
Report
Share
Report
Share
Download to read offline
Recommended
Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
3.1 extrema on an interval
The document defines extrema as the extreme or maximum and minimum values of a function on an interval. An absolute extremum occurs at an endpoint of a closed interval, while a relative extremum occurs in the interior of an open interval. A function has extrema at points where the derivative is equal to 0 (critical points) or where the function is not differentiable. The Extreme Value Theorem states that a continuous function on a closed interval will have both an absolute maximum and minimum. To find the extrema of such a function, evaluate it at all critical points and endpoints. A function can have a maximum or minimum at multiple points within an interval.
Critical points
The document discusses local maxima and minima of functions. It defines a local maximum/minimum as a point where the function values in an interval around that point are less than/greater than the function value at that point. Critical points, where the derivative is zero, are important for determining local extrema. Two tests are described - the first derivative test examines the sign of the derivative on each side of the critical point, and the second derivative test uses the concavity at the critical point to determine if it is a maximum or minimum. Examples are provided to illustrate applying these tests.
Extreme values of a function & applications of derivative
This document discusses key concepts related to finding extrema of functions, including:
- Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a subinterval, respectively.
- Critical points, where the derivative is zero or undefined, and endpoints must be checked to find extrema.
- The Extreme Value Theorem states that continuous functions on closed intervals have both a maximum and minimum value.
- The first derivative determines whether a function is increasing or decreasing, and where it is zero may indicate relative extrema. The second derivative indicates concavity and points of inflection.
5.2 first and second derivative test
This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
Data Representation - Floating Point
This document discusses floating point numbers and how they are represented in computers using binary rather than decimal. It notes that floating point numbers are approximations and certain calculations may result in numbers like 0.999999999 instead of the expected 1. This is due to the limitations of binary representation. It then explains how floating point numbers use a significand and exponent to represent a wide range of values with limited precision. Finally, it provides some examples of converting between decimal and IEEE 754 single-precision floating point format.
scribe cn
This document summarizes a lesson on critical numbers:
1) Critical numbers indicate local maximums and minimums, and are found by setting the derivative f'(x) equal to 0.
2) A critical number is a number C where either f'(C) = 0 or f'(C) does not exist.
3) Not all critical points are local extrema - some are points of inflection where the second derivative changes sign.
Recommended
Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
3.1 extrema on an interval
The document defines extrema as the extreme or maximum and minimum values of a function on an interval. An absolute extremum occurs at an endpoint of a closed interval, while a relative extremum occurs in the interior of an open interval. A function has extrema at points where the derivative is equal to 0 (critical points) or where the function is not differentiable. The Extreme Value Theorem states that a continuous function on a closed interval will have both an absolute maximum and minimum. To find the extrema of such a function, evaluate it at all critical points and endpoints. A function can have a maximum or minimum at multiple points within an interval.
Critical points
The document discusses local maxima and minima of functions. It defines a local maximum/minimum as a point where the function values in an interval around that point are less than/greater than the function value at that point. Critical points, where the derivative is zero, are important for determining local extrema. Two tests are described - the first derivative test examines the sign of the derivative on each side of the critical point, and the second derivative test uses the concavity at the critical point to determine if it is a maximum or minimum. Examples are provided to illustrate applying these tests.
Extreme values of a function & applications of derivative
This document discusses key concepts related to finding extrema of functions, including:
- Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a subinterval, respectively.
- Critical points, where the derivative is zero or undefined, and endpoints must be checked to find extrema.
- The Extreme Value Theorem states that continuous functions on closed intervals have both a maximum and minimum value.
- The first derivative determines whether a function is increasing or decreasing, and where it is zero may indicate relative extrema. The second derivative indicates concavity and points of inflection.
5.2 first and second derivative test
This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
Data Representation - Floating Point
This document discusses floating point numbers and how they are represented in computers using binary rather than decimal. It notes that floating point numbers are approximations and certain calculations may result in numbers like 0.999999999 instead of the expected 1. This is due to the limitations of binary representation. It then explains how floating point numbers use a significand and exponent to represent a wide range of values with limited precision. Finally, it provides some examples of converting between decimal and IEEE 754 single-precision floating point format.
scribe cn
This document summarizes a lesson on critical numbers:
1) Critical numbers indicate local maximums and minimums, and are found by setting the derivative f'(x) equal to 0.
2) A critical number is a number C where either f'(C) = 0 or f'(C) does not exist.
3) Not all critical points are local extrema - some are points of inflection where the second derivative changes sign.
Lesson 3.3 First Derivative Information
The first derivative of a function f(x) indicates where the function is increasing, decreasing, or constant. If f'(x) is positive, f(x) is increasing, and if f'(x) is negative, f(x) is decreasing. If f'(x) is 0, f(x) is constant. To determine where a function is increasing or decreasing, identify the critical numbers and determine the sign of f'(x) in intervals between them. The first derivative test also allows identification of relative extrema - if f'(x) changes from negative to positive at a critical number c, then f(c) is a relative minimum, and if it changes from positive to negative, f(c
Application of Derivatives
This presentation discusses applications of derivatives including: extreme values of functions, the mean value theorem, monotonic functions, and concavity. It defines maximum and minimum values and explains how the mean value theorem states that between two points a and b on a continuous function, there exists a point c where the slope of the tangent line is equal to the slope of the secant line between a and b. Monotonic functions are defined as increasing or decreasing functions based on the mean value theorem. Concavity is defined based on whether the derivative of a function is increasing or decreasing, which determines if the graph is concave up or down.
Lesson 4.3 First and Second Derivative Theory
1) To determine if a function is increasing or decreasing, examine the sign of the first derivative f'(x) at points around any critical points. A change from positive to negative indicates a relative maximum, and a change from negative to positive indicates a relative minimum.
2) A curve is concave up if the second derivative f''(x) is positive, and concave down if f''(x) is negative. To determine concavity, examine the sign of f''(x) around any points where it is zero or undefined. A change in sign indicates an inflection point where the concavity changes.
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
1) The document discusses tests for determining whether a function is increasing, decreasing, or constant based on the sign of the first derivative on an interval.
2) It also describes how to use the first and second derivative tests to determine if a critical point is a relative maximum or minimum. The tests analyze how the sign of the first or second derivative changes at the critical point.
3) Finally, the document outlines the test for concavity and points of inflection based on the sign of the second derivative on an interval. A change in the sign of the second derivative indicates an inflection point.
Εφαπτομένη Ευθεία ΕΠΑΛ
This document discusses finding the tangent line to the graph of a function f. It outlines the steps as: 1) finding the domain of f, 2) finding the derivative of f, and 3) considering cases based on whether the point of tangency (x0, f(x0)) is known or unknown. If the point is known, the slope of the tangent line is f'(x0) and the equation can be found. If unknown, additional information is needed, such as if the line is parallel/perpendicular to another line or passes through a specific point. The derivative f'(x0) and this extra information can be used to find the equation of the tangent line. Care must be taken to understand
Day 2 examples
This document provides an overview of concavity, inflection points, and how to find the local extrema of functions. It explains that a graph is concave up when tangent slopes are increasing and concave down when slopes are decreasing. An inflection point is where the direction of concavity changes and occurs when the second derivative is zero or undefined, though one must check the signs of the second derivative on either side. It gives examples of finding the inflection points of f(x)=x^4+2x^3-1 and using the second derivative test to find local extrema of f(x)=x^2e^x, and provides practice problems from the textbook.
Second Derivative Information
The second derivative of a function f(x) provides information about the concavity and points of inflection of f(x). It indicates the intervals where f(x) is concave up or down. A curve is concave up where the second derivative is positive and concave down where the second derivative is negative. Points of inflection are where the concavity changes from up to down or vice versa, and occur where the second derivative is zero or undefined.
Homework graphing
Homework on identifying key properties of functions such as the domain, intercepts, asymptotes, derivative, critical numbers, intervals of increasing/decreasing behavior, extrema, second derivative, points of inflection, and sketching the graph is due the first meeting next week. Instructions provide guidance on analyzing functions to find these properties.
Lecture 12(point of inflection and concavity)
This document discusses techniques for analyzing functions based on their derivatives. It explains how to determine if a function is increasing or decreasing based on the sign of the first derivative. It also describes how to identify relative maxima and minima by examining changes in the sign of the first derivative. Additionally, it covers how to determine if a function is concave up or down using the second derivative and how to identify points of inflection where the concavity changes. The second derivative test is presented to classify critical points as maxima or minima.
Calc224FinalExamReview
This document contains notes from a calculus class covering topics including: implicit differentiation, related rates, linear approximations, maximum and minimum values, the mean value theorem, limits at infinity, and curve sketching. Example problems are provided for each topic to demonstrate key concepts and techniques.
Probability
What is the probability of winning the lottery 6/49, that is, picking the correct set of six numbers out of 49?
Get your answer here.......
Lesson 3.1
This document discusses finding the extrema (minimum and maximum values) of a function on an interval. It defines extrema on a closed interval and relative extrema on an open interval. To find extrema on a closed interval [a,b], the steps are: 1) find critical numbers where the derivative is 0 or undefined, 2) evaluate the function at critical numbers, 3) evaluate at endpoints, and 4) the least value is the minimum and greatest is the maximum. An example finds the extrema of f(x)=3x^4 - 4x^3 on [-1,2] by following these steps.
Extreme Value
1) A continuous function on a closed interval will have both a maximum and minimum value on that interval.
2) Critical points of a function occur where the derivative is equal to 0 or undefined, and can indicate local maxima or minima.
3) The second derivative determines concavity and points where it changes sign indicate inflection points.
Noise detection from the signal matlab code, Signal Diagnosis
This document describes Experiment 10 conducted by Shyamveer Singh. The aim of the experiment was to detect the original signal from a noisy signal. The input was a noisy signal and the output was the noise-free signal. The experiment involved generating a noisy signal by adding noise to an original signal, plotting the original and noisy signals, applying a filter to remove noise from the noisy signal, and plotting the filtered signal. Fourier transforms were also applied to analyze the signals in the frequency domain.
Application of derivatives
This document provides an introduction to applying derivatives, including finding maximum and minimum values of functions. It discusses increasing and decreasing functions, using the derivative to determine if a function is increasing or decreasing, and the relationship between the sign of the derivative and the tangent line slope. Extreme and critical points are defined. Two theorems are presented for determining if a point is a maximum or minimum based on the behavior of the function and its derivative around that point. An example problem finds the intervals where a sample function is increasing and decreasing. The objectives are to understand these derivative applications and concepts.
2301MaxMin.ppt
The document defines absolute and local maximum and minimum values of functions. An absolute (or global) maximum is the highest value a function can take over its entire domain, while an absolute minimum is the lowest value. A local maximum/minimum is the highest/lowest value a function takes in the neighborhood of a particular point. The Extreme Value Theorem states that continuous functions on closed intervals attain both absolute maximum and minimum values. Fermat's Theorem provides that local extrema occur where the derivative is 0 or undefined. To find absolute extrema, one evaluates the function at critical points and endpoints.
3.1
1. The document defines extrema as the minimum and maximum values of a function over an interval. On a closed interval, a continuous function is guaranteed by the Extreme Value Theorem to have both a minimum and maximum value.
2. To find the extrema of a function over a closed interval, one finds the critical numbers where the derivative is zero or undefined, evaluates the function at the endpoints and critical numbers, and the minimum and maximum values are the lowest and highest outputs.
3. The document provides examples of finding critical numbers and using them to determine the minimum and maximum values of functions over given intervals. It emphasizes the steps of finding critical numbers, evaluating the function, and identifying the extrema.
Day 1a examples
The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
Calculus Sections 4.1 and 4.3
This document discusses finding extreme values and critical points of functions by examining their first and second derivatives. It provides examples of using the first derivative test to find local maximum and minimum values when the derivative is equal to zero or undefined. The second derivative test is introduced to determine concavity and relative maximum or minimum values based on whether the second derivative is positive, negative, or zero. Graphs, sign lines, and examples are provided to demonstrate these concepts.
Ap calculus extrema v2
The document discusses extreme values (maximums and minimums) of functions. It defines local and absolute extrema and notes that not all functions have extrema. Critical points, where the derivative is zero or undefined, are discussed. The relationship between critical points and extrema is explained by Fermat's Theorem. Methods for finding extrema on closed intervals using critical points and endpoints are presented, along with an example. Rolle's Theorem relating critical points to functions with equal values at endpoints is also introduced.
extremevaluesofafunctionapplicationsofderivative-130905112534-.pdf
This document discusses key concepts related to finding extrema of functions:
- Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a sub-interval, respectively. Critical points where the derivative is zero or undefined must be checked.
- The Extreme Value Theorem states that if a function is continuous on a closed interval, it will have both a maximum and minimum value on that interval.
- To find extrema, one finds all critical points, checks the endpoints, and compares the function values to determine the maximum and minimum. The derivative indicates whether a function is increasing or decreasing and points where it is zero or undefined.
Maximums and minimum
Applications of maximums and minimums in Calculus. This includes infection points, critical points, increasing and decreasing, etc.
More Related Content
What's hot
Lesson 3.3 First Derivative Information
The first derivative of a function f(x) indicates where the function is increasing, decreasing, or constant. If f'(x) is positive, f(x) is increasing, and if f'(x) is negative, f(x) is decreasing. If f'(x) is 0, f(x) is constant. To determine where a function is increasing or decreasing, identify the critical numbers and determine the sign of f'(x) in intervals between them. The first derivative test also allows identification of relative extrema - if f'(x) changes from negative to positive at a critical number c, then f(c) is a relative minimum, and if it changes from positive to negative, f(c
Application of Derivatives
This presentation discusses applications of derivatives including: extreme values of functions, the mean value theorem, monotonic functions, and concavity. It defines maximum and minimum values and explains how the mean value theorem states that between two points a and b on a continuous function, there exists a point c where the slope of the tangent line is equal to the slope of the secant line between a and b. Monotonic functions are defined as increasing or decreasing functions based on the mean value theorem. Concavity is defined based on whether the derivative of a function is increasing or decreasing, which determines if the graph is concave up or down.
Lesson 4.3 First and Second Derivative Theory
1) To determine if a function is increasing or decreasing, examine the sign of the first derivative f'(x) at points around any critical points. A change from positive to negative indicates a relative maximum, and a change from negative to positive indicates a relative minimum.
2) A curve is concave up if the second derivative f''(x) is positive, and concave down if f''(x) is negative. To determine concavity, examine the sign of f''(x) around any points where it is zero or undefined. A change in sign indicates an inflection point where the concavity changes.
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
1) The document discusses tests for determining whether a function is increasing, decreasing, or constant based on the sign of the first derivative on an interval.
2) It also describes how to use the first and second derivative tests to determine if a critical point is a relative maximum or minimum. The tests analyze how the sign of the first or second derivative changes at the critical point.
3) Finally, the document outlines the test for concavity and points of inflection based on the sign of the second derivative on an interval. A change in the sign of the second derivative indicates an inflection point.
Εφαπτομένη Ευθεία ΕΠΑΛ
This document discusses finding the tangent line to the graph of a function f. It outlines the steps as: 1) finding the domain of f, 2) finding the derivative of f, and 3) considering cases based on whether the point of tangency (x0, f(x0)) is known or unknown. If the point is known, the slope of the tangent line is f'(x0) and the equation can be found. If unknown, additional information is needed, such as if the line is parallel/perpendicular to another line or passes through a specific point. The derivative f'(x0) and this extra information can be used to find the equation of the tangent line. Care must be taken to understand
Day 2 examples
This document provides an overview of concavity, inflection points, and how to find the local extrema of functions. It explains that a graph is concave up when tangent slopes are increasing and concave down when slopes are decreasing. An inflection point is where the direction of concavity changes and occurs when the second derivative is zero or undefined, though one must check the signs of the second derivative on either side. It gives examples of finding the inflection points of f(x)=x^4+2x^3-1 and using the second derivative test to find local extrema of f(x)=x^2e^x, and provides practice problems from the textbook.
Second Derivative Information
The second derivative of a function f(x) provides information about the concavity and points of inflection of f(x). It indicates the intervals where f(x) is concave up or down. A curve is concave up where the second derivative is positive and concave down where the second derivative is negative. Points of inflection are where the concavity changes from up to down or vice versa, and occur where the second derivative is zero or undefined.
Homework graphing
Homework on identifying key properties of functions such as the domain, intercepts, asymptotes, derivative, critical numbers, intervals of increasing/decreasing behavior, extrema, second derivative, points of inflection, and sketching the graph is due the first meeting next week. Instructions provide guidance on analyzing functions to find these properties.
Lecture 12(point of inflection and concavity)
This document discusses techniques for analyzing functions based on their derivatives. It explains how to determine if a function is increasing or decreasing based on the sign of the first derivative. It also describes how to identify relative maxima and minima by examining changes in the sign of the first derivative. Additionally, it covers how to determine if a function is concave up or down using the second derivative and how to identify points of inflection where the concavity changes. The second derivative test is presented to classify critical points as maxima or minima.
Calc224FinalExamReview
This document contains notes from a calculus class covering topics including: implicit differentiation, related rates, linear approximations, maximum and minimum values, the mean value theorem, limits at infinity, and curve sketching. Example problems are provided for each topic to demonstrate key concepts and techniques.
Probability
What is the probability of winning the lottery 6/49, that is, picking the correct set of six numbers out of 49?
Get your answer here.......
Lesson 3.1
This document discusses finding the extrema (minimum and maximum values) of a function on an interval. It defines extrema on a closed interval and relative extrema on an open interval. To find extrema on a closed interval [a,b], the steps are: 1) find critical numbers where the derivative is 0 or undefined, 2) evaluate the function at critical numbers, 3) evaluate at endpoints, and 4) the least value is the minimum and greatest is the maximum. An example finds the extrema of f(x)=3x^4 - 4x^3 on [-1,2] by following these steps.
Extreme Value
1) A continuous function on a closed interval will have both a maximum and minimum value on that interval.
2) Critical points of a function occur where the derivative is equal to 0 or undefined, and can indicate local maxima or minima.
3) The second derivative determines concavity and points where it changes sign indicate inflection points.
Noise detection from the signal matlab code, Signal Diagnosis
This document describes Experiment 10 conducted by Shyamveer Singh. The aim of the experiment was to detect the original signal from a noisy signal. The input was a noisy signal and the output was the noise-free signal. The experiment involved generating a noisy signal by adding noise to an original signal, plotting the original and noisy signals, applying a filter to remove noise from the noisy signal, and plotting the filtered signal. Fourier transforms were also applied to analyze the signals in the frequency domain.
Application of derivatives
This document provides an introduction to applying derivatives, including finding maximum and minimum values of functions. It discusses increasing and decreasing functions, using the derivative to determine if a function is increasing or decreasing, and the relationship between the sign of the derivative and the tangent line slope. Extreme and critical points are defined. Two theorems are presented for determining if a point is a maximum or minimum based on the behavior of the function and its derivative around that point. An example problem finds the intervals where a sample function is increasing and decreasing. The objectives are to understand these derivative applications and concepts.
What's hot (15)
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Noise detection from the signal matlab code, Signal Diagnosis
Noise detection from the signal matlab code, Signal Diagnosis
Similar to Lecture 12
2301MaxMin.ppt
The document defines absolute and local maximum and minimum values of functions. An absolute (or global) maximum is the highest value a function can take over its entire domain, while an absolute minimum is the lowest value. A local maximum/minimum is the highest/lowest value a function takes in the neighborhood of a particular point. The Extreme Value Theorem states that continuous functions on closed intervals attain both absolute maximum and minimum values. Fermat's Theorem provides that local extrema occur where the derivative is 0 or undefined. To find absolute extrema, one evaluates the function at critical points and endpoints.
3.1
1. The document defines extrema as the minimum and maximum values of a function over an interval. On a closed interval, a continuous function is guaranteed by the Extreme Value Theorem to have both a minimum and maximum value.
2. To find the extrema of a function over a closed interval, one finds the critical numbers where the derivative is zero or undefined, evaluates the function at the endpoints and critical numbers, and the minimum and maximum values are the lowest and highest outputs.
3. The document provides examples of finding critical numbers and using them to determine the minimum and maximum values of functions over given intervals. It emphasizes the steps of finding critical numbers, evaluating the function, and identifying the extrema.
Day 1a examples
The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
Calculus Sections 4.1 and 4.3
This document discusses finding extreme values and critical points of functions by examining their first and second derivatives. It provides examples of using the first derivative test to find local maximum and minimum values when the derivative is equal to zero or undefined. The second derivative test is introduced to determine concavity and relative maximum or minimum values based on whether the second derivative is positive, negative, or zero. Graphs, sign lines, and examples are provided to demonstrate these concepts.
Ap calculus extrema v2
The document discusses extreme values (maximums and minimums) of functions. It defines local and absolute extrema and notes that not all functions have extrema. Critical points, where the derivative is zero or undefined, are discussed. The relationship between critical points and extrema is explained by Fermat's Theorem. Methods for finding extrema on closed intervals using critical points and endpoints are presented, along with an example. Rolle's Theorem relating critical points to functions with equal values at endpoints is also introduced.
extremevaluesofafunctionapplicationsofderivative-130905112534-.pdf
This document discusses key concepts related to finding extrema of functions:
- Absolute and relative extrema refer to the maximum and minimum values of a function over its entire domain or on a sub-interval, respectively. Critical points where the derivative is zero or undefined must be checked.
- The Extreme Value Theorem states that if a function is continuous on a closed interval, it will have both a maximum and minimum value on that interval.
- To find extrema, one finds all critical points, checks the endpoints, and compares the function values to determine the maximum and minimum. The derivative indicates whether a function is increasing or decreasing and points where it is zero or undefined.
Maximums and minimum
Applications of maximums and minimums in Calculus. This includes infection points, critical points, increasing and decreasing, etc.
Application of derivatives
The document discusses several key concepts regarding derivatives:
(1) It explains how to use the derivative to determine if a function is increasing, decreasing, or neither on an interval using the signs of the derivative.
(2) It provides theorems and rules for finding local extrema (maxima and minima) of functions using the first and second derivative tests.
(3) It also discusses absolute extrema, monotonic functions, and the Rolle's Theorem and Mean Value Theorem which relate the derivative of a function to values of the function.
Chapter 4 review
This document covers key concepts in calculus including:
1) Approximating changes in a function using derivatives, finding local linearizations.
2) Identifying local extrema, maxima and minima, of a function using first derivatives.
3) Defining critical points as points where the derivative is zero or undefined.
4) Relating properties of the first and second derivatives to characteristics of the original function like concavity, inflection points, and local extrema.
Opt simple single_000
The document discusses local and absolute maxima and minima of functions. It defines open, closed, and half-open intervals and states the max-min theorem for continuous functions on closed intervals. It defines local and absolute extreme values and gives the first derivative test for local extrema. Critical points are where the derivative is zero or undefined, and these are the only possible locations for extrema. The document provides examples of finding relative maxima and minima and discusses increasing/decreasing functions. It states the first and second derivative tests for local extrema and gives examples.
Lesson 18: Maximum and Minimum Values (slides)
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 18: Maximum and Minimum Values (slides)
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Calc 3.1
This document discusses finding the extrema (maximum and minimum values) of functions over intervals. It states that a function may not have an extremum over an interval and continuity can affect whether an extremum exists. At points of relative extrema, the derivative is either zero or undefined. Examples are provided of finding the critical numbers where the derivative is zero or undefined, and then evaluating the function at those points and the endpoints to determine the extrema.
Lesson 18: Maximum and Minimum Vaues
We define what it means for a function to have a maximum or minimum value, and explain the Extreme Value Theorem, which indicates these maxima and minima must be there under certain conditions.
Fermat's Theorem says that at differentiable extreme points, the derivative should be zero, and thus we arrive at a technique for finding extrema: look among the endpoints of the domain of definition and the critical points of the function.
There's also a little digression on Fermat's Last theorem, which is not related to calculus but is a big deal in recent mathematical history.
Lesson 18: Maximum and Minimum Vaues
This document discusses finding the maximum and minimum values of functions. It introduces the Extreme Value Theorem, which states that if a function is continuous on a closed interval, then it attains both a maximum and minimum value on that interval. It also discusses Fermat's Theorem, which relates local extrema of a differentiable function to its derivative. Examples are provided to illustrate these concepts.
Derivatives in graphing-dfs
This document provides an overview of key concepts in calculus related to derivatives, including: analyzing functions to determine if they are increasing or decreasing; finding relative extrema, critical points, and inflection points; using the first and second derivative tests to determine concavity; and graphing polynomials. Examples are provided to illustrate how to apply these concepts to specific functions in order to analyze intervals of increase/decrease, locate critical points, identify relative maxima and minima, and determine intervals of concavity. Videos and Khan Academy links are also included for supplemental instruction on related topics.
Lesson 18: Maximum and Minimum Values (Section 021 handout)
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson19 Maximum And Minimum Values 034 Slides
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
maxima & Minima thoeyr&solved.Module-4pdf
In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
Lesson 19: Maximum and Minimum Values
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
Similar to Lecture 12 (20)
extremevaluesofafunctionapplicationsofderivative-130905112534-.pdf
extremevaluesofafunctionapplicationsofderivative-130905112534-.pdf
Lesson 18: Maximum and Minimum Values (Section 021 handout)
Lesson 18: Maximum and Minimum Values (Section 021 handout)
More from FahadYaqoob5
Push over analysis
A push-over analysis deforms a structure in equal drift increments, calculating the required load increment in each step without specifying load magnitude. The load distribution and direction are fixed while only the magnitude varies. PERFORM-3D allows push-over loads based on nodal patterns, mass distributions with specified height-wise displacement variations, and mass-weighted mode shapes.
Lecture 13(asymptotes) converted
This document discusses rational functions and how to find their asymptotes. A rational function is a function defined as the ratio of two polynomials. The domain of a rational function excludes any values that would make the denominator equal to zero. Vertical asymptotes occur at these excluded values. Horizontal and oblique asymptotes are determined by comparing the degrees of the polynomials in the numerator and denominator.
Lecture 11(relative extrema)
This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
Lecture 11(relative extrema)
This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
Lecture 15(graphing of cartesion curves)
1. The curve passes through the origin as there is no constant term in the equation. To find the tangents at the origin, the lowest degree terms are equated to zero, giving the x-axis as a tangent.
2. The curve meets the coordinate axes only at the origin. There are no other points of intersection.
3. The curve has no asymptotes parallel to the x-axis, but the line y=2a is an asymptote parallel to the y-axis. The curve exists only in the region 0<x<2a.
Lecture 12(point of inflection and concavity)
This document discusses techniques for analyzing functions based on their derivatives. It explains how to determine if a function is increasing or decreasing based on the sign of the first derivative. It also describes how to identify relative maxima and minima by examining changes in the sign of the first derivative. Additionally, it covers how to determine if a function is concave up or down using the second derivative and how to identify points of inflection where the concavity changes. The second derivative test is presented for classifying critical points as maxima or minima.
Lecture 10(asymptotes)
This document discusses rational functions and how to find their asymptotes. It defines a rational function as a function of the form f(x)=p(x)/q(x) where p and q are polynomials. It then explains that vertical asymptotes occur where the denominator equals 0, and how to determine if there is a horizontal or oblique asymptote based on comparing the degrees of the numerator and denominator polynomials. Specifically, if the degree of the numerator is less than the denominator, the horizontal asymptote is the x-axis; if they are equal, the horizontal asymptote is y=leading coefficient of the numerator/leading coefficient of the denominator; and if the numerator degree is greater, there is an oblique asymptote
Lecture 5(polar coordinates)
This document provides an overview of lessons on polar coordinates from a Further Pure Mathematics II course. The lessons cover key concepts like plotting curves in polar form, converting between Cartesian and polar coordinates, determining maximum and minimum values of polar curves, and calculating areas bounded by polar curves. Example problems and practice questions are presented for each topic to help students learn the relevant formulas and skills.
Lecture 10(asymptotes)
This document discusses rational functions and how to find their asymptotes. It defines a rational function as a function of the form f(x)=p(x)/q(x) where p and q are polynomials. It then explains that vertical asymptotes occur where the denominator equals 0, and how to determine if there is a horizontal or oblique asymptote based on comparing the degrees of the numerator and denominator polynomials. Specifically, if the degree of the numerator is less than the denominator, the horizontal asymptote is the x-axis; if they are equal, the horizontal asymptote is y=leading coefficient of the numerator/leading coefficient of the denominator; and if the numerator degree is greater, there is an oblique asymptote
Lecture 11(relative extrema)
This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
Lecture 5(polar coordinates)
This document provides an overview of 8 lessons on polar coordinates. The lessons cover key concepts like plotting curves in polar form, converting between Cartesian and polar coordinates, determining maximum/minimum values of polar curves, and calculating areas bounded by polar curves. Example problems and practice questions are presented for each topic to help students learn the 'classic' polar curve shapes and how to work with polar coordinates.
Lecture 15
1. The curve passes through the origin as there is no constant term in the equation. To find the tangents at the origin, the lowest degree terms are equated to zero, giving the x-axis as a tangent.
2. The curve meets the coordinate axes only at the origin. There are no other points of intersection.
3. The curve is symmetric about the x-axis as the powers of y in the equation are even. There are no other symmetries.
4. The y-axis is the only asymptote, obtained by equating the coefficient of the highest power of y to zero. There is no asymptote parallel to the x-axis.
5. The curve exists in
Lecture 10
Parametric equations define a curve where x and y are defined in terms of a third variable, called the parameter. The document provides examples of parametric equations and how to eliminate the parameter to obtain a single equation relating x and y, whose graph is the same as that defined by the original parametric equations. Examples shown include linear, circular and parabolic curves. Exercises are provided to eliminate parameters from additional parametric equations and sketch the resulting curves.
Lecture 11
This document discusses rational functions and how to find their asymptotes. It defines a rational function as a function of the form f(x)=p(x)/q(x) where p and q are polynomials. It then explains that vertical asymptotes occur where the denominator equals 0, and how to determine if there is a horizontal or oblique asymptote based on comparing the degrees of the numerator and denominator polynomials. Specifically, if the degree of the numerator is less than the denominator, the horizontal asymptote is the x-axis; if they are equal, the horizontal asymptote is y=leading coefficient of the numerator/leading coefficient of the denominator; and if the numerator degree is greater, there is an oblique asymptote
Lecture 12
This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.
Lecture 11
This document discusses rational functions and how to find their asymptotes. It defines a rational function as a function of the form f(x)=p(x)/q(x) where p and q are polynomials. It explains that vertical asymptotes occur where the denominator equals 0, and horizontal or oblique asymptotes depend on whether the degree of the numerator is less than, equal to, or greater than the degree of the denominator. The document provides examples and step-by-step instructions for determining all types of asymptotes of rational functions.
Lecture 10
Parametric equations define a curve where x and y are defined in terms of a third variable, called the parameter. The document provides examples of parametric equations and how to eliminate the parameter to obtain a single equation relating x and y, whose graph is the same as that defined by the original parametric equations. Examples shown include linear, circular and parabolic curves. Exercises are provided to eliminate parameters from additional parametric equations and sketch the resulting curves.
Lecture 8
1) The document discusses various applications of derivatives including finding the slope of tangents and normals to curves, finding points where the slope of a tangent is a given value, finding points where tangents are parallel to a given line, finding points of intersection between curves, and determining whether curves intersect orthogonally or tangentially.
2) Formulas are provided for the slope of tangents and normals. Equations are given for the tangent and normal lines to a curve at a point.
3) Examples apply these concepts and derive solutions to problems involving tangents, normals, and intersections between curves.
Lecture 5
This document provides an overview of lessons on polar coordinates from a Further Pure Mathematics II course. The lessons cover key concepts like plotting curves in polar form, converting between Cartesian and polar coordinates, determining maximum and minimum values of polar curves, and calculating areas bounded by polar curves. Example problems and practice questions are presented for each topic to help students learn the relevant formulas and skills.
More from FahadYaqoob5 (20)
Recently uploaded
办理新西兰奥克兰大学毕业证学位证书范本原版一模一样
原版一模一样【微信：741003700 】【新西兰奥克兰大学毕业证学位证书】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
Gen Z and the marketplaces - let's translate their needs
The product workshop focused on exploring the requirements of Generation Z in relation to marketplace dynamics. We delved into their specific needs, examined the specifics in their shopping preferences, and analyzed their preferred methods for accessing information and making purchases within a marketplace. Through the study of real-life cases , we tried to gain valuable insights into enhancing the marketplace experience for Generation Z.
The workshop was held on the DMA Conference in Vienna June 2024.
Should Repositories Participate in the Fediverse?
Presentation for OR2024 making the case that repositories could play a part in the "fediverse" of distributed social applications
办理毕业证(UPenn毕业证)宾夕法尼亚大学毕业证成绩单快速办理
官方原版办理宾夕法尼亚大学毕业证【微信176555708】学历认证怎么做：原版仿制宾夕法尼亚大学电子版成绩单毕业证认证【宾夕法尼亚大学毕业证成绩单】、宾夕法尼亚大学文凭证书成绩单复刻offer录取通知书、购买UPenn圣力嘉学院本科毕业证、【宾夕法尼亚大学毕业证办理UPenn毕业证书哪里买】、宾夕法尼亚大学 Offer在线办理UPenn Offer宾夕法尼亚大学Bachloer Degree。（留信学历认证永久存档查询）采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
◆◆◆◆◆ — — — — — — — — 【留学教育】留学归国服务中心 — — — — — -◆◆◆◆◆
【主营项目】
一.毕业证【微信：176555708】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【微信：176555708】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分→ 【关于价格问题（保证一手价格）
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
学历顾问：微信：176555708
Ready to Unlock the Power of Blockchain!
Imagine a world where data flows freely, yet remains secure. A world where trust is built into the fabric of every transaction. This is the promise of blockchain, a revolutionary technology poised to reshape our digital landscape.
Toptal Tech is at the forefront of this innovation, connecting you with the brightest minds in blockchain development. Together, we can unlock the potential of this transformative technology, building a future of transparency, security, and endless possibilities.
HijackLoader Evolution: Interactive Process Hollowing
CrowdStrike researchers have identified a HijackLoader (aka IDAT Loader) sample that employs sophisticated evasion techniques to enhance the complexity of the threat. HijackLoader, an increasingly popular tool among adversaries for deploying additional payloads and tooling, continues to evolve as its developers experiment and enhance its capabilities.
In their analysis of a recent HijackLoader sample, CrowdStrike researchers discovered new techniques designed to increase the defense evasion capabilities of the loader. The malware developer used a standard process hollowing technique coupled with an additional trigger that was activated by the parent process writing to a pipe. This new approach, called "Interactive Process Hollowing", has the potential to make defense evasion stealthier.
办理毕业证(NYU毕业证)纽约大学毕业证成绩单官方原版办理
退学买【纽约大学毕业证认证】【办证微信176555708】【纽约大学文凭毕业证制作】【NYU学历学位证书哪里买】办理纽约大学学位证书扫描件、办理纽约大学雅思证书！
国际留学归国服务中心【如何办纽约大学毕业证认证】【NYU学位证书扫描件哪里买】实体公司，注册经营，行业标杆，精益求精！（留信学历认证永久存档查询）采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
◆◆◆◆◆ — — — — — — — — 【留学教育】留学归国服务中心 — — — — — -◆◆◆◆◆
【主营项目】
一.毕业证【微信：176555708】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【微信：176555708】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分→ 【关于价格问题（保证一手价格）
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
学历顾问：微信：176555708
Discover the benefits of outsourcing SEO to India
"Discover the benefits of outsourcing SEO to India! From cost-effective services and expert professionals to round-the-clock work advantages, learn how your business can achieve digital success with Indian SEO solutions.
一比一原版(USYD毕业证)悉尼大学毕业证如何办理
USYD硕士毕业证成绩单【微信95270640】《如何办理悉尼大学毕业证认证》【办证Q微信95270640】《悉尼大学文凭毕业证制作》《USYD学历学位证书哪里买》办理悉尼大学学位证书扫描件、办理悉尼大学雅思证书！
国际留学归国服务中心《如何办悉尼大学毕业证认证》《USYD学位证书扫描件哪里买》实体公司，注册经营，行业标杆，精益求精！
如果您是以下情况，我们都能竭诚为您解决实际问题：【公司采用定金+余款的付款流程，以最大化保障您的利益，让您放心无忧】
1、在校期间，因各种原因未能顺利毕业，拿不到官方毕业证+微信95270640
2、面对父母的压力，希望尽快拿到悉尼大学悉尼大学毕业证offer；
3、不清楚流程以及材料该如何准备悉尼大学悉尼大学毕业证offer；
4、回国时间很长，忘记办理；
5、回国马上就要找工作，办给用人单位看；
6、企事业单位必须要求办理的；
面向美国乔治城大学毕业留学生提供以下服务:
【★悉尼大学悉尼大学毕业证offer毕业证、成绩单等全套材料，从防伪到印刷，从水印到钢印烫金，与学校100%相同】
【★真实使馆认证（留学人员回国证明），使馆存档可通过大使馆查询确认】
【★真实教育部认证，教育部存档，教育部留服网站可查】
【★真实留信认证，留信网入库存档，可查悉尼大学悉尼大学毕业证offer】
我们从事工作十余年的有着丰富经验的业务顾问，熟悉海外各国大学的学制及教育体系，并且以挂科生解决毕业材料不全问题为基础，为客户量身定制1对1方案，未能毕业的回国留学生成功搭建回国顺利发展所需的桥梁。我们一直努力以高品质的教育为起点，以诚信、专业、高效、创新作为一切的行动宗旨，始终把“诚信为主、质量为本、客户第一”作为我们全部工作的出发点和归宿点。同时为海内外留学生提供大学毕业证购买、补办成绩单及各类分数修改等服务；归国认证方面，提供《留信网入库》申请、《国外学历学位认证》申请以及真实学籍办理等服务，帮助众多莘莘学子实现了一个又一个梦想。
专业服务，请勿犹豫联系我
如果您真实毕业回国，对于学历认证无从下手，请联系我，我们免费帮您递交
诚招代理：本公司诚聘当地代理人员，如果你有业余时间，或者你有同学朋友需要，有兴趣就请联系我
你赢我赢，共创双赢
你做代理，可以帮助悉尼大学同学朋友
你做代理，可以拯救悉尼大学失足青年
你做代理，可以挽救悉尼大学一个个人才
你做代理，你将是别人人生悉尼大学的转折点
你做代理，可以改变自己，改变他人，给他人和自己一个机会过暑假的小学生快乐的日子总是过得飞快山娃尚未完全认清那几位小朋友时他们却一个接一个地回家了山娃这时才恍然发现二个月的暑假已转到了尽头他的城市生活也将划上一个不很圆满的句号了值得庆幸的是山娃早记下了他们的学校和联系方式说也奇怪在山娃离城的头一天父亲居然请假陪山娃耍了一天那一天父亲陪着山娃辗转长隆水上乐园疯了一整天水上漂流高空冲浪看大马戏大凡里面有的父亲都带着他去疯一把山娃算了算这一次足足花了老爸元很
快速办理(Vic毕业证书)惠灵顿维多利亚大学毕业证完成信一模一样
学校原件一模一样【微信：741003700 】《(Vic毕业证书)惠灵顿维多利亚大学毕业证》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
快速办理(新加坡SMU毕业证书)新加坡管理大学毕业证文凭证书一模一样
学校原件一模一样【微信：741003700 】《(新加坡SMU毕业证书)新加坡管理大学毕业证文凭证书》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
存档可查的(USC毕业证)南加利福尼亚大学毕业证成绩单制做办理
精仿办理南加利福尼亚大学毕业证成绩单（【微信176555708】）毕业证学历认证OFFER专卖国外文凭学历学位证书办理澳洲文凭|澳洲毕业证,澳洲学历认证,澳洲成绩单【微信176555708】 澳洲offer,教育部学历认证及使馆认证永久可查 ,国外毕业证|国外学历认证,国外学历文凭证书 USC毕业证，USC毕业证，USC毕业证，USC毕业证，USC毕业证，USC毕业证【微信176555708】，USC毕业证，专业为留学生办理毕业证、成绩单、使馆留学回国人员证明、教育部学历学位认证、录取通知书、Offer、（留信学历认证永久存档查询）采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
◆◆◆◆◆ — — — — — — — — 【留学教育】留学归国服务中心 — — — — — -◆◆◆◆◆
【主营项目】
一.毕业证【微信：176555708】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【微信：176555708】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分→ 【关于价格问题（保证一手价格）
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
学历顾问：微信：176555708
留学挂科(UofM毕业证)明尼苏达大学毕业证成绩单复刻办理
精仿办理明尼苏达大学学历证书＜176555708微信＞【毕业证明信-推荐信做学费单＞【微信176555708】【制作UofM毕业证文凭认证明尼苏达大学毕业证成绩单购买】【UofM毕业证书】{明尼苏达大学文凭购买}】成绩单，录取通知书，Offer，在读证明，雅思托福成绩单，真实大使馆教育部认证，回国人员证明，＞【制作UofM毕业证文凭认证明尼苏达大学毕业证成绩单购买】【UofM毕业证书】{明尼苏达大学文凭购买}留信网认证。
明尼苏达大学学历证书＜微信176555708（一对一服务包括毕业院长签字,专业课程,学位类型,专业或教育领域,以及毕业日期.不要忽视这些细节.这两份文件同样重要!毕业证成绩单文凭留信网学历认证！）（留信学历认证永久存档查询）采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
◆◆◆◆◆ — — — — — — — — 【留学教育】留学归国服务中心 — — — — — -◆◆◆◆◆
【主营项目】
一.毕业证【微信：176555708】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【微信：176555708】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分→ 【关于价格问题（保证一手价格）
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
学历顾问：微信：176555708
成绩单ps(UST毕业证)圣托马斯大学毕业证成绩单快速办理
一比一原版制作UST毕业证【微信176555708】圣托马斯大学毕业证书原版↑制作圣托马斯大学学历认证文凭办理圣托马斯大学留信网认证,留学回国办理毕业证成绩单文凭学历认证【微信176555708】专业为海外学子办理毕业证成绩单、文凭制作，学历仿制，回国人员证明、做文凭，研究生、本科、硕士学历认证、留信认证、结业证、学位证书样本、美国教育部认证百分百真实存档可查】（留信学历认证永久存档查询）采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
◆◆◆◆◆ — — — — — — — — 【留学教育】留学归国服务中心 — — — — — -◆◆◆◆◆
【主营项目】
一.毕业证【微信：176555708】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【微信：176555708】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分→ 【关于价格问题（保证一手价格）
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
学历顾问：微信：176555708
怎么办理(umiami毕业证书)美国迈阿密大学毕业证文凭证书实拍图原版一模一样
原版定制【微信:bwp0011】《(umiami毕业证书)美国迈阿密大学毕业证文凭证书》【微信:bwp0011】成绩单 、雅思、外壳、留信学历认证永久存档查询，采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
【业务选择办理准则】
一、工作未确定，回国需先给父母、亲戚朋友看下文凭的情况，办理一份就读学校的毕业证【微信bwp0011】文凭即可
二、回国进私企、外企、自己做生意的情况，这些单位是不查询毕业证真伪的，而且国内没有渠道去查询国外文凭的真假，也不需要提供真实教育部认证。鉴于此，办理一份毕业证【微信bwp0011】即可
三、进国企，银行，事业单位，考公务员等等，这些单位是必需要提供真实教育部认证的，办理教育部认证所需资料众多且烦琐，所有材料您都必须提供原件，我们凭借丰富的经验，快捷的绿色通道帮您快速整合材料，让您少走弯路。
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
【关于价格问题（保证一手价格）】
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
不能毕业如何获得(USYD毕业证)悉尼大学毕业证成绩单一比一原版制作
挂科购买☀【悉尼大学毕业证购买】【微信176555708】【USYD毕业证模板办理】加拿大文凭、本科、硕士、研究生学历都可以做,二、业务范围：
★、全套服务：毕业证、成绩单、化学专业毕业证书伪造【悉尼大学大学毕业证】微信176555708【USYD学位证书购买】◆◆◆◆◆ — — — 归国服务中心 — — -◆◆◆◆◆
【主营项目】
一.毕业证、成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
国外毕业证、学位证、成绩单办理流程：
1、客户提供办理信息：姓名、生日、专业、学位、毕业时间等（如信息不确定可以咨询顾问：【微信176555708】我们有专业老师帮你查询）；
2、开始安排制作毕业证、成绩单电子图；
3、毕业证、成绩单电子版做好以后发送给您确认；
4、毕业证、成绩单电子版您确认信息无误之后安排制作成品；
5、成品做好拍照或者视频给您确认；
6、快递给客户（国内顺丰，国外DHL、UPS等快读邮寄）。
专业服务，请勿犹豫联系我！本公司是留学创业和海归创业者们的桥梁。一次办理，终生受用，一步到位，高效服务。详情请在线咨询办理,欢迎有诚意办理的客户咨询!洽谈。
◆招聘代理：本公司诚聘英国、加拿大、澳洲、新西兰、加拿大、法国、德国、新加坡各地代理人员，如果你有业余时间，有兴趣就请联系我们咨询【微信176555708】
没文凭怎么找工作。让您回国发展信心十足！
★、真实教育部学历学位认证；（一对一专业服务，可全程监控跟踪进度）
★、真实使馆认证，可以通过大使馆查询确认；（即教育部留服认证，不成功不收费）
★、毕业证、成绩单等材料，从防伪到印刷、水印到钢印烫金，高精仿度都是跟学校原版100%相同的；（敬请放心使用）
★、可以提供钢印、水印、烫金、激光防伪、凹凸版、最新版的毕业证、百分之百让您绝对满意、
★、印刷，DHL快递毕业证、成绩单7个工作日，真实大使馆教育部认证1个月。为了达到高水准高效率。
Recently uploaded (19)
Gen Z and the marketplaces - let's translate their needs
Gen Z and the marketplaces - let's translate their needs
HijackLoader Evolution: Interactive Process Hollowing
HijackLoader Evolution: Interactive Process Hollowing
Lecture 12
- 2. Def: Absolute Extreme Values Let f be a function with domain D. Then f(c) is the… a) Absolute (or global) maximum value on D iff f(x) ≤ f(c) for all x in D. b) Absolute (or global) minimum value on D iff f(x) ≥ f(c) for all x in D. * Note: The max or min VALUE refers to the y value.
- 3. Where can absolute extrema occur? * Extrema occur at endpoints or interior points in an interval. * A function does not have to have a max or min on an interval. * Continuity affects the existence of extrema. Min Max Min No Max No Max No Min
- 4. Thm 1: Extreme Value Theorem If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval. * What is the “worst case scenario,” and what happens then?
- 5. Def: Local Extreme Values Let c be an interior point of the domain of f. Then f(c) is a… a) Local (or relative) maximum value at c iff f(x) ≤ f(c) for all x in some open interval containing c. b) Local (or relative) minimum value at c iff f(x) ≥ f(c) for all x in some open interval containing c. * A local extreme value is a max or min in a “neighborhood” of points surrounding c
- 6. Identify Absolute and Local Extrema Abs & Loc Min Abs & Loc Max Loc Max Loc Min Loc Min * Local extremes are where f ’(x)=0 or f ’(x) DNE
- 7. Thm 2: Local Extreme Values If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f ’ exists at c, then f ’(c)=0.
- 8. Def: Critical Point A critical point is a point in the interior of the domain of f at which either… 1) f ’(x)= 0, or 2) f ’(x) does not exist
- 9. Finding Absolute Extrema on [a,b]… To find the absolute extrema of a continuous function f on [a,b]… 1. Evaluate f at the endpoints a and b. 2. Find the critical numbers of f on [a,b]. 3. Evaluate f at each critical number. 4. Compare values. The greatest is the absolute max and the least is the absolute min.
- 10. Check for Understanding… (True or False??) 1. Absolute extrema can occur at either interior points or endpoints. 2. A function may fail to have a max or min value. 3. A continuous function on a closed interval must have an absolute max or min. 4. An absolute extremum is also a local extremum. 5. Not every critical point or end point signals the presence of an extreme value.