Maxima and Minima 2 : Applications of Derivatives

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Detailed course in maxima and minima to gain confidence in problem solving.

2,038 students enrolled

Created by
Surya Kumari Gandikota

Last updated 6/2014

English

What Will I Learn?

- Over 49 lectures and about 6 hours of content
- The overall course goal is to lay a strong foundation of concepts for the topic maxima and minima and master the same with the help of solved examples.
- Objective 1:To understand when a function is said to attain a maximum value and a minimum value in its domain.
- Objective 2:To understand the terms local maximum value and local minimum values of a function.
- Objective 3: To understand the terms global maximum value and global minimum values of a function.
- Objective 4: To understand the terms absolute maximum value and absolute minimum values of a function in a closed interval. Also working rule to find the same.
- Objective 5: To understand the behavior of f ‘(x) at local maxima and local minima.
- Objective 6: To understand the terms stationary points, critical points and points of inflexion.
- Objective 7: To understand the concept of concavity and hence the terms concave upward and concave downward.
- Objective 8: To understand first order, second order and higher order derivative tests to find local maximum and local minimum points and their respective values.. Also working rules to find the same.
- Objective 9: To understand the techniques to solve optimization problems with the help of solved examples.
- Objective10: To be able to apply the concepts of maxima and minima in solving problems which we encounter in real life.

Requirements

- Knowledge of algebra
- Differentiation

Description

There are various applications of differentiation. In this course, we learn to apply derivatives to find the maximum and minimum values of differentiable functions in their domains. To begin with in the first section, a brief note about the need to study the topic Maxima and Minima is given.

In sections 2,3,4 the definitions and the concepts of the points of local / global /absolute maxima and minima which can be obtained by using differentiation is discussed.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.

Every concept is well explained with appropriate graphical figures.Every topic 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.

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.

Who is the target audience?

- Calculus students
- Also, anybody who wants to brush up the concepts of maxima and minima for any competitive exam within a short span of time, this course is strongly recommended.

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