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Ex 5.1, 31 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at April 16, 2024 by Teachoo
Continuity of composite functions
Chapter 5 Class 12 Continuity and Differentiability
Concept wise
- Checking continuity at a given point
- Checking continuity at any point
- Checking continuity using LHL and RHL
- Algebra of continous functions
-
Continuity of composite functions
- Checking if funciton is differentiable
- Finding derivative of a function by chain rule
- Finding derivative of Implicit functions
- Finding derivative of Inverse trigonometric functions
- Finding derivative of Exponential & logarithm functions
- Logarithmic Differentiation - Type 1
- Logarithmic Differentiation - Type 2
- Derivatives in parametric form
- Finding second order derivatives - Normal form
- Finding second order derivatives- Implicit form
- Proofs
- Verify Rolles theorem
- Verify Mean Value Theorem
Transcript
Ex 5.1, 31 Show that the function defined by 𝑓(𝑥)=cos(𝑥^2 ) is a continuous function.𝑓(𝑥) = cos(𝑥^2 ) Let 𝒈(𝒙) = cos𝑥 & 𝒉(𝒙) = 𝑥^2 Now, 𝒈𝒐𝒉(𝒙) = g(ℎ(𝑥)) = 𝑔(𝑥^2 ) = cos(𝑥^2 ) = 𝒇(𝒙) Hence, 𝑓(𝑥) = 𝑔𝑜ℎ(𝑥) We know that 𝒈(𝒙) = cos𝑥 is continuous as cos x is always continuous & 𝒉(𝒙) = 𝑥^2 is continuous as it is a polynomial Hence, 𝑔(𝑥) & ℎ(𝑥) are both continuous . We know that If two function of 𝑔(𝑥) & ℎ(𝑥) both continuous, then their composition 𝒈𝒐𝒉(𝒙) is also continuous Hence, 𝒇(𝒙) is continuous .
