Understanding Continuity and Differentiability in Mathematics for Grade XII

Derivatives play a crucial role in engineering and science, helping analyze the behavior of dynamic systems. This unit delves into the concept of continuity and differentiability, exploring the geometric interpretation of derivatives, the chain rule, differentiation of algebraic functions, and the second-order derivative. By grasping these principles, you will adeptly determine maximum and minimum values of functions and master the derivative of various functions and standard theorems.

• Mathematics
• Grade XII
• Continuity
• Differentiability
• Derivatives

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1. an International CBSE Finger Print School Coimbatore SUBJECT NAME - 041 MATHEMATICS GRADE-XII UNIT 5 TOPIC CONTINUITY AND DIFFERENTIABILITY CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY 13/06/23 1

2. CONTINUITY AND DIFFERENTIABILITY DERIVATIVES Derivatives are met in many engineering and science problems, especially when modelling the behaviour of moving objects. We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.). After studying this lesson, you will be able to : Define and interpret geometrically the derivative of the function y= f(x) at x = c; Find the derivatives of the function of the form cf(x) , (f(x) g(x)) and polynomial functions; State and apply the results concerning derivatives of the product and quotient of two functions; 13/06/23 2/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

3. state and apply the chain rule for the derivative of a function; find the derivative of algebraic functions (including rational functions); and find second order derivative of a function. Differentiability at a point. 13/06/23 3/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

4. Geometrical meaning: 13/06/23 4/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

5. Derivative of a function: This definition of derivative is also called the first principle of derivative. 13/06/23 5/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

6. Theorem: Derivative of some standard functions: 13/06/23 6/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

7. 13/06/23 7/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

8. Theorem: 13/06/23 8/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

9. DIFFERENTIATION OF A FUNCTION OF A FUNCTION Chain Rule for Differentiation: 13/06/23 9/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

10. Differentiation of Implicit Functions: LOGARITHMIC DIFFERENTIATION 13/06/23 10/18 10 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

11. Alternatively, we may write Differentiating with respect to x, we get 11/18 13/06/23 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

12. DIFFERENTIATION OF PARAMETRIC FUNCTIONS DIFFERENTIATION OF A FUNCTION with respect to ANOTHER FUNCTION 13/06/23 12/18 12 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

13. Second Order Derivative 13/06/23 13/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

14. MEAN VALUE THEOREMS ROLLE S THEOREM 13/06/23 14/18 14 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

15. Geometrical Interpretation of Rolles Theorem 13/06/23 15/18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

16. LAGRANGES MEAN VALUE THEOREM 13/06/23 16/18 16 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

17. Questions: 1. Examine whether the function f given by f (x) = x2is continuous at x = 0. 2. Discuss the continuity of the function f given by f (x) = x3+ x2 1. 3. For what value of is the function defined by continuous at x = 0? What about continuity at x = 1? 13/06/23 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY 17/18

18. References https://www.slideshare.net/SeyidKadher1/continuity-and-differentiability https://www.aees.gov.in/htmldocs/downloads/XII_Class_Content/20200831 /20200831-XII-Maths-Differentiability-2%20of%204-Ppt.pptx https://www.cyut.edu.tw/~fmchang/docs/chapter2.ppt https://d2cyt36b7wnvt9.cloudfront.net/exams/wp- content/uploads/2019/12/06154044/Maths_Ch-052.pdf https://www.vedantu.com/ncert-solutions/ncert-solutions-class-12-maths- chapter-5 13/06/23 18/18 18 CONTINUITY AND DIFFERENTIABILITY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY