Maxima and Minima concepts : Applications of Derivatives

An exclusive course for learning concepts of maxima and minima.
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• Lectures 12
• Length 1.5 hours
• Skill Level All Levels
• Languages English
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Published 4/2014 English

Course Description

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.

What are the requirements?

• Knowledge of algebra
• Differentiation

What am I going to get from this course?

• Over 12 lectures and about 1 hour 30 minutes of content.
• The overall course goal is to lay a strong foundation of concepts for the topic maxima and minima.
• 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.

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.

What you get with this course?

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Curriculum

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.
08:20

In this video you will understand when a function is said to attain a maximum value and a minimum value in its domain.

11:51

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.

06:54

In this video you will understand global maximum value and global minimum value of a function.

07:31

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.

09:52

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
09:22

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

10:43

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
05:52

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.

05:26

In this video you will understand second order derivative test to find local maxima and local minima. Also working rule to find the same

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
5 pages

This gives a summary of what was taught in this course.

Instructor Biography

I am Mathematics Subject Matter Expert.

I am a certified (post graduate trained) mathematics instructor (private tutor) with over 24 years of teaching experience to middle school, high school, senior secondary level and intermediate level covering various school boards including CBSE, ICSE, ISC, IGCSE, 2-year IB Diploma (International Baccalaureate) covering AL,SL and HL courses. I have so far made 200+ videos in various Math topics.

Worked for:

FIITJEE EDU SOFT Ltd. as

CHIEF CONTENT MODERATOR under the department of

SYSTEM DEVELOPMENT associated with the project

EDFORA (Education for All) in Gurgaon.

Key Responsibilities held: 1. Overall content direction ( in making educational videos).

2. Managing quality of content created by content contributors.

3. Refining checklist and checklist of content.

4. Leading the team of HODs and Domain Auditors.

5. Creating training collateral for content creators.

6. Establish QA/QC Benchmarks for Videos

7. Work closely with HODs to achieve domain wise targets and drive the process from Review to Publish

8 .Providing overall guidance/ direction to the content team for preparing content for different state boards.[ CBSE, ICSE, ISC, Andhra Pradesh Board, Madhya Pradesh Board, Maharashtra Board, Rajasthan Board, Uttar Pradesh Board]