Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

Ex 5.1, 17 Find the relationship between a and b so that the function f defined by 𝑓(π‘₯)={β–ˆ(π‘Žπ‘₯+1, 𝑖𝑓 π‘₯≀3@&𝑏π‘₯+3, 𝑖𝑓 π‘₯>3)─ is continuous at x = 3.Given function is continuous at x = 3 f(x) is continuous at π‘₯ =3 if L.H.L = R.H.L = 𝒇(πŸ‘) if lim┬(xβ†’3^βˆ’ ) 𝑓(π‘₯)=lim┬(xβ†’3^+ ) " " 𝑓(π‘₯)= 𝑓(3) Since there are two different functions on the left & right of 3, we take LHL & RHL . LHL at x β†’ 3 lim┬(xβ†’3^βˆ’ ) f(x) = lim┬(hβ†’0) f(3 βˆ’ h) = lim┬(hβ†’0) π‘Ž(3βˆ’β„Ž)+1 = π‘Ž(3βˆ’0)+1 = 3a + 1 RHL at x β†’ 3 lim┬(xβ†’3^+ ) f(x) = lim┬(hβ†’0) f(3 + h) = lim┬(hβ†’0) 𝑏(3+β„Ž)+3 = b(3 + 0) + 3 = b + 3 And 𝑓(3)=π‘Žπ‘₯+1 𝒇(πŸ‘)=πŸ‘π’‚ +𝟏 Now, lim┬(xβ†’3^βˆ’ ) 𝑓(π‘₯) = lim┬(xβ†’3^+ ) 𝑓(π‘₯) = 𝑓(1) 3π‘Ž + 1 = 3𝑏 + 3 = 3π‘Ž + 1 Comparing values πŸ‘π’‚ + 𝟏 = πŸ‘π’ƒ + πŸ‘ 3π‘Žβˆ’3b=3βˆ’1 3π‘Ž βˆ’3𝑏=2 3(π‘Žβˆ’π‘)=2 π‘Žβˆ’π‘=2/3 𝒂=𝒃+ 𝟐/πŸ‘ Thus , for any value of b, We can find value of a

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.