Lesson Plan For CBSE Class 12 (Chapter 5) For Mathematics Teacher

E- LESSON PLAN   SUBJECT MATHEMATICS    CLASS 10+2

Lesson Plan For CBSE Class 12 (Chapter 2) For Mathematics Teacher 

Board – CBSE	CLASS –XII	SUBJECT- MATHEMATICS
CHAPTER 5  :-  Continuity and Differentiability

TOPIC:-

Chapter : 5 : Continuity and Differentiability

DURATION:-  

This chapter is divided into ten modules and is completed in 20 class meetings.

PRE- REQUISITE KNOWLEDGE:-

Knowledge of trigonometry class 10+1, inverse trigonometry class 10+2, Limits and continuity class 10+1

Lesson Plan For CBSE Class 12 (Chapter 2) For Mathematics Teacher 

TEACHING AIDS:- 

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

METHODOLOGY:-   Lecture method

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.

• Roll’s and mean value theorem (with proof) and their geometric interpretation.

Lesson Plan For CBSE Class 12 (Chapter 2) For Mathematics Teacher 

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.

S. No.	Topic	Explanation
1	Introduction of limits and continuity, left hand limit and right hand limit with some practical problems.	DIFFERENTIABILITY f(x) is said to be differentiable if left hand differentiation is equal to the right hand differentiation. limx→0−f(x)−f(a)x−a=limx→0+f(x)−f(a)x−a$$\underset{x\to 0^{-}}{lim}\frac{f(x)-f(a)}{x-a}=\underset{x\to 0^{+}}{lim}\frac{f(x)-f(a)}{x-a}$$ DERIVATIVE BY FIRST PRINCIPAL f′(x)=limx→0f(x+h)−f(x)h(OR)f′(x)=limx→af(x)−f(a)x−a$$f^{{}^{\prime }}(x)=\underset{x\to 0}{lim}\frac{f(x+h)-f(x)}{h}(OR)f^{{}^{\prime }}(x)=\underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}$$ PRODUCT RULE OF DIFFERENTIATION (uv)′=u′v+uv′$${\left(uv\right)}^{{}^{\prime }}=u^{{}^{\prime }}v+u{v}^{{}^{\prime }}$$ QUOTIENT RULE OF DIFFERENTIATION (uv)′=u′v−uv′v2−$${\left(\frac{u}{v}\right)}^{{}^{\prime }}=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}-$$ DIFFERENTIATION OF TRIGONOMETRIC AND OTHER FUNCTIONS Differentiation formulas for different maths functions for class 12-cbse mathematics ddxxn=nxn−1$$\frac{d}{dx}{x}^n=n{x}^{n-1}$$ ddxex=ex$$\frac{d}{dx}{e}^x=e^x$$ ddxlogx=1x$$\frac{d}{dx}logx=\frac{1}{x}$$ ddxsinx=cosx$$\frac{d}{dx}sinx=cosx$$ ddxcosx=−sinx$$\frac{d}{dx}cosx=-sinx$$ ddxtanx=sec2x$$\frac{d}{dx}tanx=se{c}^{2}x$$ ddxcotx=−cosec2x$$\frac{d}{dx}cotx=-cose{c}^{2}x$$ ddxsecx=secxtanx$$\frac{d}{dx}secx=secxtanx$$ ddxcosecx=−cosecxcotx$$\frac{d}{dx}cosecx=-cosecxcotx$$ DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS  ddxsin−1x=11−x2−−−−−√$$\frac{d}{dx}si{n}^{-1}x=\frac{1}{\sqrt{1-x^2}}$$ ddxcos−1x=−11−x2−−−−−√$$\frac{d}{dx}co{s}^{-1}x=\frac{-1}{\sqrt{1-x^2}}$$ ddxsec−1x=1xx2−1−−−−−√$$\frac{d}{dx}se{c}^{-1}x=\frac{1}{x\sqrt{x^2-1}}$$ ddxcosec−1x=−1xx2−1−−−−−√$$\frac{d}{dx}cose{c}^{-1}x=\frac{-1}{x\sqrt{x^2-1}}$$ ddxtan−1x=11+x2$$\frac{d}{dx}ta{n}^{-1}x=\frac{1}{1+x^2}$$ ddxcot−1x=−11+x2$$\frac{d}{dx}co{t}^{-1}x=\frac{-1}{1+x^2}$$ ROLL'S THEOREM If f(x) is a continuous function in [a, b] and is differentiable in (a, b) and f(a) = f(b) then there exist some c ∊ (a, b) such that f ' (c) = 0 Differentiation formulas for different maths functions for class 12-cbse mathematics MEAN VALUE THEOREM If f(x) is a continuous function in [a,b] and is differentiable in (a,b) then there exist some c ∊ (a,b)  such that    f′(c)=f(b)−f(a)b−a$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$$ Differentiation formulas for different maths functions for class 12-cbse mathematics LOGARITHMIC FUNCTIONS Anyfunctionoftypey=logbxiscalledLogarithmicfunction.$$Anyfunctionoftypey=lo{g}_{b}xiscalledLogarithmicfunction.$$ Ifbasebisreplacedby10thenitiscalledcommonlogarithm.$$Ifbasebisreplacedby10thenitiscalledcommonlogarithm.$$ Ifbasebisreplacedbyethenitiscallednaturallogarithmandisdenotedbylnx$$Ifbasebisreplacedbyethenitiscallednaturallogarithmandisdenotedbylnx$$ Inplaceoflnxwesimplywritelogx$$Inplaceoflnxwesimplywritelogx$$ logmn=logm+logn$$logmn=logm+logn$$ logmn=logm−log$$log\frac{m}{n}=logm-log$$ logmn=nlogm$$log{m}^n=nlogm$$ Differentiation formulas for different maths functions for class 12-cbse mathematics
2	Introduction of differentiability, left hand and right hand differentiability, with some practical problems.
3	Basic Formulas of  differentiation, differentiation of composite functions and chain rule.
4	Differentiation of inverse trigonometric functions.
5	Differentiation of implicit functions.
6	Concept of exponential and logarithmic functions.
7	Derivatives of exponential and logarithmic functions.
8	Derivatives of the functions expressed in parametric forms.
9	Concept of second order derivatives.
10	Roll’s theorem and Lagrange’s Mean Value Theorem(with proof) and their geometric interpretation.

Lesson Plan For CBSE Class 12 (Chapter 2) For Mathematics Teacher 

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 and the applicability of Roll’s and mean value theorem.

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.

Lesson Plan For CBSE Class 12 (Chapter 2) For Mathematics Teacher