Maxima and Minima 2 : Applications of Derivatives

Detailed course in maxima and minima to gain confidence in problem solving.
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  • Lectures 49
  • Length 6 hours
  • Skill Level All Levels
  • Languages English
  • Includes Lifetime access
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About This Course

Published 6/2014 English

Course 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.

What are the requirements?

  • Knowledge of algebra
  • Differentiation

What am I going to get from this course?

  • 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.

What 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?

Not for you? No problem.
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Certificate of completion.

Curriculum

Section 1: About the Course
7 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.

06:54

In this video you will understand global maximum value and global minimum values 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.
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 terms stationary points, critical points and points of inflexion.

10:43

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.

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

Lecture Description:

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: Examples on maximum and minimum
06:04

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

05:24

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

02:32

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

03:26

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

02:45

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives

05:06

In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives.

03:17

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

02:58

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

02:02

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

02:54

In this video we will discuss an example to find the maximum or minimum values, if any

of a given function in its domain without using derivatives.

Section 6: Examples on First Derivative Test
07:56

In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test.

12:42

In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test.

14:16

In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test

06:41

In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test.

09:10

In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test.

06:10

In this video we will discuss examples of some functions that do not have a local maximum or a local minimum.

Section 7: Examples on Second Derivative Test
06:21

In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test.

09:23

In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test.

08:51

In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test.

07:09

In this video we will discuss an example to find local maximum value and local minimum values of a function where the second order derivative test fails and hence use first derivative test to find the same.

11:38

In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test.

14:12

In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test.

09:24

In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test.

Section 8: Examples on Absolute Maximum and Absolute Minimum in a closed interval
06:47

An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval.

07:55

An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval.

08:03

An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval.

06:42

An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval.

Section 9: Optimization problems
9 pages

A brief note on optimization problems will be given.

05:25

To show that of all the rectangles with a

given perimeter, the square has the largest area.

Example2_ Square has the smallest perimeter
06:42
08:23

Solution of the problem: "A square sheet of metal is to be made into an open box by cutting smaller squares from its corners and folding up the flaps to form the sides. What should be the side of the square to be cut off so that the volume of the box is maximum?"

10:09

Solution of the problem: To show that of all the rectangles inscribed

in a given fixed circle,the square has

the maximum area.

08:01

Solution of the problem: To show that the height of a closed cylinder of given surface and maximum volume is equal to the diameter of its base.

10:54

Solution of the problem: A wire of length 40 metres is to be cut into two pieces. One of the two pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum?

08:05

Solution to the problem: Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is maximum.

07:03

Solution to the problem: A closed right circular cylinder has a volume of 2156 cubic units.What should be the radius of its base so that its total surface area may be minimum?

09:37

Solution to the problem: Prove that the area of a right-angled triangle of a given hypotenuse is maximum when the triangle is isosceles.

Section 10: Course wrap-up
5 pages

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

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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]

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