Maxima and Minima concepts : Applications of Derivatives
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There are various applications of differentiation in Calculus. In this course "Maxima and Minima Concepts", we learn to apply derivatives to find the maximum and minimum values of differentiable functions in their domains. We will also define the points of local / global /absolute maxima and minima which can be obtained by using differentiation.
To begin with in the first section, a brief note about the need to study the topic Maxima and minima is given. In the second section when a function is said to attain a maximum value and a minimum value in its domain is discussed. Then the terms local maxima and minima, global maxima and minima, and absolute maxima and minima in a closed interval are introduced. Also the behavior of f ‘(x) at local maxima and local minima points is discussed.
In section three, the terms stationary points, critical points and points of inflexion are taken up. In this section we also discuss about the concept of concavity, concave upward curves and concave downward curves. Also we see how the concept of concavity is applied to identify the points of inflexion.
The next section deals with various derivative tests for local maximum and local minimum. The tests discussed are the first derivative test, the second derivative test and in general the higher order derivative test. Working rule to use these tests is also included at the end of the lectures. Also downloadable supplementary material is provided under the heading "Concepts to Remember" . This covers the key concepts covered lecture-wise.
Finally the summary of the course is given in the wrap-up lecture.
Every concept is well explained with appropriate graphical figures. The course can be completed in about 1 hour 30 minutes.
This topic is very important and useful for higher studies in Science, Technology and Economics in optimization problems. For example in Economics, we can tackle the problems like 1.Minimize cost production.i.e. expenses, effort etc.
2.Maximize profits, efficiency and outputs etc.
In Mensuration, we can find the solutions to the problems where we need to maximise or minimise the volumes or areas of geometric figures such as cylinder, cuboid etc.
However, we are today equipped with graphing calculators and computers to find the maximum and minimum values of functions.
But having said that it is still required to study this topic of “ Maxima and Minima” in Calculus to increase our understanding of functions and the mathematics involved.
This is purely a conceptual course. Part 2 of this course includes videos of examples which have been carefully selected and properly graded and solved to illustrate the concepts and techniques. Wherever possible the solutions include graphical explanations as well. At the end of the course the applications of maxima and minima under the heading 'optimization problems' have been discussed.
So after completing this course I strongly recommend to take part 2 of this course "Maxima and Minima: Concepts and Problem Solving", as well so that the applications of these concepts can be mastered through various solved examples. This course will be published shortly.
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|Section 1: About the Course|
|Lecture 1||8 pages|
In this lecture we understand the need to study the topic Maxima and Minima.
|Section 2: Maximum and Minimum values of a function.|
In this video you will understand when a function is said to attain a maximum value and a minimum value in its domain.
In this video you will understand local maximum value and local minimum values of a function. i.e Maximum and minimum values of a function in a small interval.
In this video you will understand global maximum value and global minimum value of a function.
In this video you will understand absolute maximum value and absolute minimum values of a function in a closed interval. Also working rule to find the same.
In this video you will understand the behavior of f ‘(x) at local maxima and local minima.
|Section 3: Stationary, Critical and points of Inflexion and Concavity|
In this video you will understand the the concept of concavity and hence the terms concave upward and concave downward. Understand points of inflexion with the help of concavity
In this video you will understand the terms stationary points, critical points and points of inflexion.
|Section 4: Derivative Tests for Local Maximum and Local Minimum|
In this video you will understand first order derivative test to find local maximum and local minimum points and their respective values.. Also working rule to find the same.
In this video you will understand second order derivative test to find local maxima and local minima. Also working rule to find the same
|Lecture 11||9 pages|
In this lecture you will understand the higher order derivative test to find local maxima and local minima. Also working rule to find the same.
|Section 5: Course wrap-up|
|Lecture 12||5 pages|
This gives a summary of what was taught in this course.
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