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Ex. 5.2 ii.jpg
Ex. 5.2 iii.jpg Ex. 5.2 iv.jpg

  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.2, 10 (Introduction) Greatest Integer Function F(x) = [x] Example: [1] = 1 [1.2] = 1 [1.9999] = 1 [2] = 2 [2.01] = 2 [20001] = 2 ∴ , [1 + h] = 1 [1 − h] = 0 [1] = 1 Ex 5.2, 10 Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differentiable at 𝑥=1 𝑎𝑛𝑑 𝑥= 2. f (x) = [x] At x = 1 f (x) is differentiable at x = 1 y if L H D = R H D (𝑙𝑖𝑚)┬(h→0) (𝑓(𝑥) − 𝑓(𝑥 − ℎ))/ℎ = (𝑙𝑖𝑚)┬(h→0) (𝑓(1) − 𝑓(1 − ℎ))/ℎ = (𝑙𝑖𝑚)┬(h→0) ([1] − [(1 − ℎ)])/ℎ = (𝑙𝑖𝑚)┬(h→0) (1 − 0)/ℎ = (𝑙𝑖𝑚)┬(h→0) 1/ℎ = 1/0 = Not defined (𝑙𝑖𝑚)┬(h→0) (𝑓(𝑥 + ℎ) − 𝑓(𝑥))/ℎ = (𝑙𝑖𝑚)┬(h→0) (𝑓(1 + ℎ) − 𝑓(1))/ℎ = (𝑙𝑖𝑚)┬(h→0) ([(1 + ℎ)] − [1])/ℎ = (𝑙𝑖𝑚)┬(h→0) (1 − 1)/ℎ = (𝑙𝑖𝑚)┬(h→0) 0/ℎ = (𝑙𝑖𝑚)┬(h→0) 0 = 0 Since LHD ≠ RHD ∴ f(x) is not differentiable at x = 1 Hence proved (𝑙𝑖𝑚)┬(h→0) (𝑓(𝑥) − 𝑓(𝑥 − ℎ))/ℎ = (𝑙𝑖𝑚)┬(h→0) (𝑓(2) − 𝑓(2 − ℎ))/ℎ = (𝑙𝑖𝑚)┬(h→0) ([2] − [(2 − ℎ)])/ℎ = (𝑙𝑖𝑚)┬(h→0) (2 − 1)/ℎ = (𝑙𝑖𝑚)┬(h→0) 1/ℎ = 1/0 = Not defined (𝑙𝑖𝑚)┬(h→0) (𝑓(𝑥 + ℎ) − 𝑓(𝑥))/ℎ = (𝑙𝑖𝑚)┬(h→0) (𝑓(2 + ℎ) − 𝑓(2))/ℎ = (𝑙𝑖𝑚)┬(h→0) ([(2 + ℎ)] − [2])/ℎ = (𝑙𝑖𝑚)┬(h→0) (2 − 2)/ℎ = (𝑙𝑖𝑚)┬(h→0) 0/ℎ = (𝑙𝑖𝑚)┬(h→0) 0 = 0 Since LHD ≠ RHD ∴ f(x) is not differentiable at x = 2 Hence proved

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
  • Ahmed Raza's image

    Screenshot_2017-06-08-19-46-59.jpg class 12  , ex. 5.2 - Question no. 10


    In the RHD please can the solution of RHD 3rd step [1 h]

    How we solved it and why the solution is differentfrom the LHD solution of [1-h]

    View answer
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