### Dictionary Rank of a Word | Permutations & Combinations

PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

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

E- LESSON PLAN  (SUBJECT MATHEMATICS)  CLASS 10+2
Lesson Plan, Class XII Subject 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.

 Board – CBSE CLASS –XII SUBJECT- MATHEMATICS CHAPTER 6  :-  Application of derivative

TOPIC:- Chapter : 6 : Applications of derivatives

DURATION:-

This chapter is divided into 11 modules and is completed in 25 class meetings.

PRE- REQUISITE KNOWLEDGE:-
TEACHING AIDS:-

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

METHODOLOGY:-   Lecture method

OBJECTIVES:-
• Rate of change of bodies.
• Increasing and decreasing functions.
• Tangents and Normals.
• Use of derivatives in approximation.
• Maxima and minima using first and second derivative test.
• Word problems which involves maxima and minima.

PROCEDURE :-

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.

S. No.

For Complete Explanation of Topic

## 1

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.

2

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)

3

Slope and Tangent of a line or curve

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.

Slope of a line:

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

Equation of the tangent at (x1, y1) is given by

y – y1 = m(x - x1) where m is the slope of the tangent

4

Slope and Normal of a line or curve

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 Normal

Equation of the Normal at (x1, y1) is given by

y – y1 = m(x - x1) where m is the slope of the normal

5

Method of Approximation

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

6

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 developed in the mind of the students.

Definition of Maximum 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 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

7

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.

8

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.

9

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

Algorithms used in 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.

10

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

11

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

EXPECTED OUTCOMES:-
• After studying this lesson students should know the
• Concept rate of change of quantities,
• Method of finding slopes of tangent and normal,
• Method of finding the equations of slopes and tangent,
• Method of finding the approximate value of a number.
• Students should be able to find the maximum and minimum value of the function by using first and second derivative test.
STUDENTS DELIVERABLES:-
• Review questions given by the teacher.
• Students should prepare the presentation in groups on different topics like rate of change of quantities, tangents and normals, increasing, decreasing , maxima and minima of a function.
• Solve NCERT problems with examples.
EXTENDED LEARNING:-

Students can extend their learning in Mathematics through the RESOURCE CENTRE.  Students can also find many interesting topics on mathematics at cbsemathematics.com

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  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.

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