Illustration 1: Let [.] denotes the greatest integer function and f(x) = [tan^{2}x], then does the limit exist or is the function differentiable or continuous at 0? (1999)
Solution: Given f(x) = [tan^{2}x]
Now, 45°< x < 45°
tan(45°)< tanx < tan45°
tan 45°< tan x < tan 45°
1< tan x <1
So, 0 <tan^{2}x < 1
[tan^{2}x] = 0
So, f(x) is zero for all values of x form x = 45° to 45°.
Hence, f is continuous at x =0 and f is also differentiable at 0 and has a value zero.
Illustration 2: A function is defined as follows:
f(x)= x^{3} , x^{2}< 1
x , x^{2}≥ 1
Discuss the differentiability of the function at x=1.
Solution:We have R.H.D. = Rf'(1)
= limh→0 (f(1h)f(1))/h
= limh→0 (1+h1)/h = 1
and L.H.D. = Lf'(1)= lim_{h→0} (f(1h)f(1))/(h)
= lim_{h→0} ((1h)31)/(h)
= lim_{h→0} (33h+h^{2}) = 3
?Rf'(1)≠ Lf'(1)⇒ f(x) is not differentiable at x=1.
Illustration 3:If y = (sin1x)^{2} + k sin1x, show that (1x^{2}) (d^{2 }y)/dx^{2}  x dy/dx = 2
Solution: Here y = (sin1x)^{2} + k sin1x.
Differentiating both sides with respect to x, we have
dy/dx = 2(sin1 x)/√(1x^{2 }) + k/√(1x^{2} )
⇒(1x^{2} ) (dy/dx)^{2} = 4y + k^{2}
Differentiating this with respect to x, we get
(1x^{2}) 2 dy/dx.(d^{2} y)/(dx^{2 })  2x (dy/dx)^{2} = 4(dy/dx)
⇒(1x^{2} ) ( d^{2} y)/dx^{2 }x dy/dx = 2
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