Maxima and Minima 2 : Applications of Derivatives

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

2,364 students enrolled

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.

Compare to Other Derivatives Courses

Curriculum For This Course

49 Lectures

06:09:58
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About the Course
1 Lecture
00:00

In this lecture we understand the need to study the topic Maxima and minima.

Why study maxima and minima?

7 pages

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Maximum and Minimum values of a function.
5 Lectures
44:28

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

Understanding maximum and minimum .

08:20

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

Understanding Local Maximum and Local Minimum

11:51

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

Understanding Global Maximum and Global Minimum

06:54

- In this video you will understand absolute maximum value and absolute minimum values of a function in a closed interval.

Understanding Absolute Maximum and Absolute Minimum in a closed interval.

07:31

In this video you will understand the behavior of f ‘(x) at local maxima and local minima.

Understanding the behavior of f ‘(x) at Local Maxima and Local Minima.

09:52

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Stationary, Critical and points of Inflexion, and Concavity
2 Lectures
20:05

In this video you will understand the terms stationary points, critical points and points of inflexion.

Understanding Stationary, Critical and points of Inflexion.

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.

Understanding the concept of concavity. More about points of Inflexion.

10:43

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Derivative Tests for Local Maximum and Local Minimum
3 Lectures
11:18

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.

Understanding First Derivative Test for Local Maximum and Local Minimum.

05:52

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.

Understanding Second Order Derivative Test for Local Maximum and Local Minimum

05:26

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.

Understanding Higher Order Derivative Test for Local Maximum and Local Minimu

9 pages

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Examples on maximum and minimum
10 Lectures
36:28

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.

Example1

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.

Example 2

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.

Example 3

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.

Example 4

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

Example 5

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.

Example 6

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.

Example 7

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.

Example 8

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.

Example 9

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.

Example 10

02:54

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Examples on First Derivative Test
6 Lectures
56:55

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.

Example1

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.

Example2

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

Example3

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.

Example4

06:41

Example5

09:10

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

Example6

06:10

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Examples on Second Derivative Test
7 Lectures
01:06:58

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.

Example1

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.

Example2

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.

Example3

08:51

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.

Example4

07:09

Example5

11:38

Example6

14:12

Example7

09:24

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Examples on Absolute Maximum and Absolute Minimum in a closed interval
4 Lectures
29:27

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

Example1

06:47

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

Example2

07:55

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

Example3

08:03

Example4

06:42

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Optimization problems
10 Lectures
01:14:19

A brief note on optimization problems will be given.

About optimization

9 pages

To show that of all the rectangles with a

given perimeter, the square has the largest area.

Example1_Square has the largest area

05:25

Example2_ Square has the smallest perimeter

06:42

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?"

Example3_ Maximum volume of the open box

08:23

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

in a given fixed circle,the square has

the maximum area.

Example4_ Rectangle is a square of maximum area inscribed in a circle

10:09

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.

Example5_ Height of a closed cylinder of maximum volume is equal to the diameter

08:01

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?

Example6_Minimise the combined area of the square and the circle cut from a wire

10:54

Solution to the problem: Find two positive numbers x and y such that their sum is 35 and the product x^{2 }y^{5} is maximum.

Example7_ To maximise the product ( x^ 2 )(y^5 ) , x and y are positive numbers

08:05

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?

Example8_Minimise total surface area of right circular cylinder, given volume

07:03

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

Example9_ Area right triangle maximum when isosceles and hypotenuse given

09:37

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Course wrap-up
1 Lecture
00:00

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

Course Summary

5 pages

About the Instructor