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### Lesson Plan Math Class XII Ch-6 | Application of Derivatives

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**LESSON PLAN MATHEMATICS CH-6**

**APPLICATION OF DERIVATIVES**

*Lesson Plan, Class XII Mathematics, chapter 6, Applications of Derivatives , for Mathematics Teacher. Effective way of Teaching Mathematics. Top planning by the teacher for effective teaching in the class. E lesson planning for mathematics.*

__PRE- REQUISITE KNOWLEDGE:-__

__TEACHING AIDS:-__

**Green Board, Chalk, Duster, Charts, smart board, projector, laptop etc.**

__METHODOLOGY:-__

**Lecture cum demonstration and Learning by doing method.**

__LEARNING OBJECTIVES:__

**To explain the**

**Rate of change of bodies.**

**Increasing and decreasing of functions.**

**Tangents and Normals (Deleted)**

**Use of derivatives in approximation(Deleted)**

**Maxima and minima using first derivative and second derivative test.**

**Word problems which involves maxima and minima.**

__LEARNING OUTCOMES:__**After studying this lesson students should know the**

**Concept rate of change of quantities,****Increasing decreasing, strictly increasing and strictly decreasing of functions.****Method of finding slopes of tangent and normal****(Deleted)****Method of finding the equations of slopes and tangent****(Deleted)****Method of finding the approximate value of a number****(Deleted)****Students should be able to find the maximum and minimum value of the function by using first and second derivative test.****Method of finding the absolute maxima and absolute minima.**

**RESOURCES**

**NCERT Text Book,**

NCERT Exemplar Book of mathematics,

NCERT Exemplar Book of mathematics,

**Resource Material :**

**KEY WORDS**

**Rate of change of quantities, Strictly increasing and decreasing, Local maxima and minima, Absolute maxima and minima, monotonic increasing and decreasing of functions, Critical Points and point of inflexion.**

**CONTENT OF THE TOPIC**

**Rate of change of quantities.****Critical Points and baby curve method.****Increasing and decreasing of functions.****Stictly increasing and strictly decreasing of functions.****Maxima and minima of functions.****Local maxima and local minima of functions.****First derivative and second derivative test.****Absolute maxima and absolute minima**

**LEARNING ACTIVITIES:**

**INTRODUCTORY ACTIVITY**

__PROCEDURE & EXPLANATION__

__Click Here for : Complete Explanation__

**Start the session by asking the questions related to the trigonometry, inverse trigonometry and formulas of derivatives. Now introduce the topic application of derivatives step by step as follows.**

**Introduction:**

First of all teacher will give Introduction of the application of derivatives in different fields of mathematics like engineering, science and social science etc.

Now teacher will explain the term rate of change of bodies with different examples and application of this concept in word problems.

First of all teacher will give Introduction of the application of derivatives in different fields of mathematics like engineering, science and social science etc.

Now teacher will explain the term rate of change of bodies with different examples and application of this concept in word problems.

It is the tangent of the angle made by the line with the positive direction of x - axis measured anti-clockwise direction. Slope of any line or curve can be find by finding dy / dx

Equation of Tangent (Deleted)

Equation of the tangent at (x

Equation of the Normal at

Let f(x) be a real function defined on an interval I. Then f(x) is said to have the maximum value in I, if there exist a point c in I such that f(x) ≤ f(c) for all x ϵ I

In this case f(c) is called the maximum value

Definition of Minimum Value :

Let f(x) be a real function defined on an interval I. Then f(x) is said to have the maximum value in I, if there exist a point c in I such that f(x) ≥ f(c) for all x ϵ I

In this case f(c) is called the minimum value

Algorithms used in First Derivative Test

i) Put y = f(x)

ii) Find dy/dx and put dy/dx = 0 to find the critical points.

iii) Place all the critical points on the horizontal line and write signs ( +ve or -ve) as in baby curve method.

iv) The critical point at which dy/dx changes its sign from positive to negative is called the point of local maximum and the function have maximum value at this point.

v) The critical point at which dy/dx changes its sign from negative (- ve) to positive (+ ve) is called the point of local minimum and the function have local minimum value at this point.

vi) The critical point at which dy/dx does not change its sign is called the point of inflexion.

i) Put y = f(x)

ii) Find f '(x) and put f '(x) = 0 to find the critical points.

iii) Find f ''(x) and find the value of f''(x) at the critical points.

iv) If f ''(c) < 0 , then f(x) has local maximum value at x = c

v) If f ''(c) > 0, then f(x) has local minimum value at x = c

vi) If f '' (c) = 0 then test fail.

Increasing and Decreasing of Functions

Now teacher will explain the Concept of increasing and decreasing and their graphical representation. Also teacher will help the students in finding the interval in which the given function is increasing or decreasing with the help of Baby Curve Method.

a) f is strictly increasing in (a, b) if f '(x) > 0 for each x ϵ (a, b)

b)f is strictly decreasing in (a, b) if f '(x) < 0 for each x ϵ (a, b)

Slope and Tangent of a line or curve (Deleted)

First of all teacher will explain the term slope of a line and explain different methods of finding the slopes of lines or curves. Teacher will also explain the method of finding slope of tangent and equation of tangent by taking some examples.

It is the tangent of the angle made by the line with the positive direction of x - axis measured anti-clockwise direction. Slope of any line or curve can be find by finding dy / dx

Equation of Tangent (Deleted)

Equation of the tangent at (x

_{1}, y_{1}) is given byy – y_{1} = m(x - x_{1}) where m is the slope of the tangent

Slope and Normal of a line or curve (Deleted)

Here teacher will explain the Method of finding the slope of normal and equation of normal. Also explain the relationship between slope of tangent and normal.

Equation of the Normal at

**(x**is given by_{1}, y_{1})y – y_{1} = m(x - x_{1}) where m is the slope of the normal

Method of Approximation (Deleted)

Teacher will explain the method of approximation, approximate change in surface area and volumes and approximate error. Explain the implementation of the formula used in the method of approximation in different problems.

f(x + Δx) = f(x) + f '(x) x Δx

## MAXIMA AND MINIMA

Maximum Value and Minimum Value

First of all teacher will provide the definition of maxima and minima and then explain the definition by giving examples. Teacher will explain the concept of maxima and minima graphically and geometrically so that a clear picture will be developed in the mind of the students.

Let f(x) be a real function defined on an interval I. Then f(x) is said to have the maximum value in I, if there exist a point c in I such that f(x) ≤ f(c) for all x ϵ I

In this case f(c) is called the maximum value

Definition of Minimum Value :

Let f(x) be a real function defined on an interval I. Then f(x) is said to have the maximum value in I, if there exist a point c in I such that f(x) ≥ f(c) for all x ϵ I

In this case f(c) is called the minimum value

Critical Points

Now explain the concept of Critical points, local maxima and local minima and its geometrical meaning and different methods of finding local maxima and local minima as given below.

Explain the Method of finding maxima and minima by using first derivative test.

Algorithms used in First Derivative Test

i) Put y = f(x)

ii) Find dy/dx and put dy/dx = 0 to find the critical points.

iii) Place all the critical points on the horizontal line and write signs ( +ve or -ve) as in baby curve method.

iv) The critical point at which dy/dx changes its sign from positive to negative is called the point of local maximum and the function have maximum value at this point.

v) The critical point at which dy/dx changes its sign from negative (- ve) to positive (+ ve) is called the point of local minimum and the function have local minimum value at this point.

vi) The critical point at which dy/dx does not change its sign is called the point of inflexion.

Explain the Method of finding maxima and minima by using second derivative test.

i) Put y = f(x)

ii) Find f '(x) and put f '(x) = 0 to find the critical points.

iii) Find f ''(x) and find the value of f''(x) at the critical points.

iv) If f ''(c) < 0 , then f(x) has local maximum value at x = c

v) If f ''(c) > 0, then f(x) has local minimum value at x = c

vi) If f '' (c) = 0 then test fail.

Now explain the concept of Absolute maximum value and absolute minimum value and the method of finding these.

Explain the Implementations of Concept of maxima and minima in daily life word problems.

**REFLECTION OF ACTIVITY**

Students will be able to know the

Students will be able to know the

**1) Rate of change of quantities.**

**2) Increasing, Decreasing,
strictly increasing and strictly decreasing of functions. **

**3) Method of finding the maxima
and minima of functions.**

**4) Local maxima and local minima
of functions.**

**5) Absolute maxima and absolute
minima of functions.**

**Teacher will ask few questions related to the above topics and note the reflection of the students.**

**IMMEDIATE FEEDBACK**

**After completing the above
activities students will be able to **

**1) Identify the rate of change
of different quantities with respect to different variables.**

**2) Identify different methods of
finding the increasing, decreasing, strictly increasing and strictly decreasing
of functions.**

**3) Able to find the maximum value,
minimum value, local maximum value, local minimum value, absolute maximum value
and absolute minimum value.**

**4) Understand the difference
between the first derivative and second derivative test.**

**CREATION (e.g. MIND-MAP, COLLAGE, GRAPH, MAP etc.)**

MIND MAP / CONCEPT MAP APPLICATIONS OF DERIVATIVES

SUBJECTS INTEGRATED

English : Write a brief note on the difference between between Local maxima, local minima Absolute maxima and absolute minima.

Science: Explanation of rate of change of quantities w.r.t time, radius, displacement etc.

Drawing : Students can draw the free hand pictures of different types of functions, their way of increasing and decreasing or can also use some other material like thread, match sticks etc.

**DIFFERENTIAL LEARNING**

For Below Average StudentsFor Average StudentsFor Above Average Students:SKILLS ENHANCEDObservation skill, analytical skill, critical thinking, team work, constructive approach, interpersonal skill, engagement in learning process etc.

- Mind/ Concept maps
- Charts , Models and activity
- Simple questions

- Group Discussion
- Higher Order Thinking Skill questions

__ASSESSMENT TECHNIQUES:-__

- Assignment sheet will be given as home work at the end of the topic.
- Separate sheets which will include questions of logical thinking and Higher order thinking skills will be given to the above average students.
- Class Test , Oral Test , Worksheet and Assignments. can be made the part of assessment.
- Re-test(s) will be conducted on the basis of the performance of the students in the test.

- Puzzle
- Quiz based on MCQ
- Misconception check
- Peer check
- Students discussion
- Competency Based Assessment link: Multiple Choice Questions

__THANKS FOR YOUR VISIT____PLEASE COMMENT BELOW____🙏__

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