Lecture 11(relative extrema)
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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.
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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.
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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.
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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.
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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
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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.
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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 11(relative extrema) (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)
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5. The curve exists in
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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.
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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.
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3) Examples apply these concepts and derive solutions to problems involving tangents, normals, and intersections between curves.
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- 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.