Ex 5.1, 21 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Ex 5.1, 21 Discuss the continuity of the following functions: (a) 𝑓 (𝑥) = sin⁡𝑥+cos⁡𝑥 𝑓 (𝑥) = sin⁡𝑥+cos⁡𝑥 Let 𝑝(𝑥)=sin⁡𝑥 & 𝑞(𝑥)=cos⁡𝑥" " We know that sin⁡𝑥 & cos⁡𝑥 both continuous function ⇒ 𝑝(𝑥) & 𝑞(𝑥) is continuous at all real number By Algebra of continuous function If 𝑝(𝑥)" & " 𝑞(𝑥) are continuous for all real numbers then 𝑓(𝑥)= 𝑝(𝑥)+𝑞(𝑥) is continuous for all real numbers ∴ 𝒇(𝒙) = 𝐬𝐢𝐧⁡𝒙+𝒄𝒐𝒔⁡𝒙 continuous for all real numbers Ex 5.1, 21 Discuss the continuity of the following functions: (b) 𝑓 (𝑥) = sin⁡𝑥 – cos⁡𝑥 𝑓 (𝑥) = sin⁡𝑥 – cos⁡𝑥 Let 𝑝(𝑥)=sin⁡𝑥 & 𝑞(𝑥)=cos⁡𝑥" " We know that sin⁡𝑥 & cos⁡𝑥 are both continuous function ⇒ 𝑝(𝑥) & 𝑞(𝑥) is continuous at all real number By Algebra of continuous function If 𝑝(𝑥)" & " 𝑞(𝑥) are continuous for all real numbers then 𝑓(𝑥)= 𝑝(𝑥)−𝑞(𝑥) is continuous for all real numbers ∴ 𝒇(𝒙) = 𝒔𝒊𝒏⁡𝒙−𝒄𝒐𝒔⁡𝒙 is continuous for all real numbers Ex 5.1, 21 Discuss the continuity of the following functions: (c) 𝑓 (𝑥) = sin⁡𝑥 . cos⁡𝑥 𝑓 (𝑥) = sin⁡𝑥 . cos⁡𝑥 Let 𝑝(𝑥)=sin⁡𝑥 & 𝑞(𝑥)=cos⁡𝑥" " We know that sin⁡𝑥 & cos⁡𝑥 are both continuous functions ⇒ 𝑝(𝑥) & 𝑞(𝑥) is continuous at all real numbers By Algebra of continuous function If 𝑝(𝑥)" & " 𝑞(𝑥) are continuous for all real numbers then 𝑓(𝑥)= 𝑝(𝑥) . 𝑞(𝑥) is continuous for all real numbers ∴ 𝒇(𝒙) = 𝒔𝒊𝒏⁡𝒙.𝒄𝒐𝒔⁡𝒙 continuous for all real numbers