Lesson Plan Math Class XII (Ch-5) Continuity & Differentiability

LESSON PLAN MATHEMATICS
CLASS - XII
CH - 5 CONTINUITY & DIFFERENTIABILITY

Lesson Plan Class 12th Subject Mathematics for Mathematics Teacher. Effective way of Teaching Mathematics. Top planning by the teacher for effective teaching in the class.

Rmb dav centnary public school Nawanshahr
NAME OF THE TEACHER				DINESH KUMAR
CLASS	10+2	CHAPTER	05	SUBJECT	MATHEMATICS
TOPIC	CONTINUITY & DIFFERENTIABILITY			DURATION : 20 CLASS MEETINGS

PRE- REQUISITE KNOWLEDGE:-

Knowledge of Trigonometry class 10+1,

Knowledge Inverse Trigonometric Functions class 10+2,

Knowledge of Limits and continuity class 10+1

TEACHING AIDS:

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

METHODOLOGY: Lecture method

LEARNING OBJECTIVES:-

Continuity and differentiability.

Derivative of composition functions and chain rule.

Derivatives of inverse trigonometric functions and implicit functions.

Concept of exponential and logarithmic functions.

Derivative of exponential and logarithmic functions.

Derivative of the functions expressed in parametric forms.

Second order derivatives.

EXPECTED OUTCOMES:

After studying this lesson students should know the

Concept of limits, continuity and differentiability.

Students should learn all formulas of differentiations and should know the applicability of them in different problems.

Students should be able to find the derivatives of composite functions, inverse trigonometric functions, implicit functions, exponential functions and logarithmic functions.

Students should know the second order derivatives

RESOURCES

NCERT Text Book,

NCERT Exemplar Book of mathematics,

RESOURCE MATERIAL :

Worksheets , E-content, Basics and formulas from (cbsemathematics.com)

KEY WORDS

Continuity : Continuous function, Discontinuous function, Left-hand limit (LHL), Right-hand limit (RHL), Limit of a function, Continuity at a point, Algebra of continuous functions

Differentiability : Derivative, Differentiable function, Differentiability at a point, Differentiability in an interval

Derivatives of Composite and Implicit Functions : Chain rule, Implicit differentiation, Logarithmic differentiation

Second Order Derivative : Second derivative (d²y/dx²) Notation: f′′(x),y′′f''(x), y''f′′(x),y′′

CONTENT OF THE TOPIC

Continuity of different types of functions.

Differentiability.

Chain Rule,

Implicit differentiation,

logarithmic differentiation.

Derivative of inverse trigonometric functions.

Second order derivative.

LEARNING ACTIVITIES:

INTRODUCTORY ACTIVITY

EXPERIENTIAL ACTIVITY

ART INTEGRATED ACTIVITY

PROCEDURE :

Start the session by asking the questions related to the trigonometry, inverse trigonometry and limits and continuity. Now introduce the topic continuity and differentiability step by step as follows.

COMPLETE EXPLANATION

CLICK HERE : [Complete Explanation]

Introduction of limits and continuity, left hand limit and right hand limit with some practical problems.

Introduction of differentiability, left hand and right hand differentiability, with some practical problems.

CONCEPT OF DIFFERENTIABILITY

A function is formally considered differentiable if its derivative exists at each point in its domain.

For a function to be differentiable it nust be continuous.

DEFINITION OF DIFFERENTIABILITY

f(x) is said to be differentiable at the point x = a if the derivative f ‘(a) exists at every point in its domain. It is given by

$f^{'}(x)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$

This formula is derived from Lagranges Mean Value Theorem.

Putting x = a + h, as x → a, h → 0 so this can be written as

$f'(a)=\displaystyle\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$

Now Explain the Basic Formulas of differentiation, differentiation of composite functions

Explain the chain rule of differentiation

$\frac{d}{dx}\left(ax+b\right)^{3}=3(ax+b)^{3-1}\frac{d}{dx}(ax+b)=3a(ax+b)^{2}$

Explain the Product rule of differentiation

(uv)' = u'v + uv' OR

$\frac{d(uv)}{dx}=\left(\frac{du}{dx}\right)v+u\left(\frac{dv}{dx}\right)$

Example : Differentiate y = x²sinx, w.r.t. x

Solution

$\frac{dy}{dx}=\left(\frac{d}{dx}x^{2}\right)sinx+x^{2}\left(\frac{d}{dx}sinx\right)$

$\frac{dy}{dx}=2x\:sinx+x^{2}\:cosx$

Explain the quotient rule of differentiation

$\left(\frac{u}{v}\right)^{'}=\frac{u'v-uv'}{v^{2}}$

(OR)

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{\left(\frac{d}{dx}u\right)v-u\left(\frac{d}{dx}v\right)}{v^{2}}$

Example: Differentiate $\frac{x^{2}}{sinx}$ , w. r. t. x

Solution:

$\frac{dy}{dx}=\frac{\left(\frac{d}{dx}x^{2}\right)sinx-x^{2}\left(\frac{d}{dx}sinx\right)}{(sinx)^{2}}$

$\frac{dy}{dx}=\frac{2xsinx-x^{2}cosx}{(sinx)^{2}}$

Explain the Differentiation of trigonometric functions.

Explain Differentiation of inverse trigonometric functions.

$If\:y=sin^{-1}x\:\:then\:\:\frac{dy}{dx}=\frac{1}{\sqrt{1-x^{2}}}$

$If\:y=cos^{-1}x\:\:then\:\:\frac{dy}{dx}=\frac{-1}{\sqrt{1-x^{2}}}$

$If\:y=tan^{-1}x\:\:then\:\:\frac{dy}{dx}=\frac{1}{1+x^{2}}$

$If\:y=cot^{-1}x\:\:then\:\:\frac{dy}{dx}=\frac{-1}{1+x^{2}}$

$If\:y=sec^{-1}x\:\:then\:\:\frac{dy}{dx}=\frac{1}{|x|\sqrt{x^{2}-1}}$

$If\:y=cosec^{-1}x\:\:then\:\:\frac{dy}{dx}=\frac{-1}{|x|\sqrt{x^{2}-1}}$

Explain the Differentiation of implicit functions.

Explain the Concept of exponential and logarithmic functions.

Explain the Derivatives of exponential and logarithmic functions.

$\frac{d}{dx}logx=\frac{1}{x}$

Explain the Derivatives of the functions expressed in parametric forms.

Explain the Concept of second order derivatives.

STUDENTS DELIVERABLES:

Review questions given by the teacher.

Students should prepare the presentation on the formulas of derivatives.

Solve NCERT problems with examples,

Solve assignment given by the teacher.

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 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.
Competency based assessment can be taken so as to ensure if the learning outcomes have been achieved or not. e.g.
Puzzle
Quiz
Misconception check
Peer check
Students discussion
Competency Based Assessment link: M C Q

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