Create your Smart Home App in the Free Webinar on Home Automation. Register Now
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-1023-196
+91-120-4616500
CART 0
Use Coupon: CART20 and get 20% off on all online Study Material
Welcome User
OR
LOGIN
Complete Your Registration (Step 2 of 2 )
Free webinar on App Development Learn to create your own Smart Home App
16th Jan @ 5:00PM for Grade 1 to 10
Sit and relax as our customer representative will contact you within 1 business day
Solved Examples on Differentiability Illustration 1: Let [.] denotes the greatest integer function and f(x) = [tan2x], then does the limit exist or is the function differentiable or continuous at 0? (1999) Solution: Given f(x) = [tan2x] Now, -45°< x < 45° tan(-45°)< tanx < tan45° -tan 45°< tan x < tan 45° -1< tan x <1 So, 0 <tan2x < 1 [tan2x] = 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)= x3 , x2< 1 x , x2≥ 1 Discuss the differentiability of the function at x=1. Solution:We have R.H.D. = Rf'(1) = limh→0 (f(1-h)-f(1))/h = limh→0 (1+h-1)/h = 1 and L.H.D. = Lf'(1)= limh→0 (f(1-h)-f(1))/(-h) = limh→0 ((1-h)3-1)/(-h) = limh→0 (3-3h+h2) = 3 ?Rf'(1)≠ Lf'(1)⇒ f(x) is not differentiable at x=1. Illustration 3:If y = (sin-1x)2 + k sin-1x, show that (1-x2) (d2 y)/dx2 - x dy/dx = 2 Solution: Here y = (sin-1x)2 + k sin-1x. Differentiating both sides with respect to x, we have dy/dx = 2(sin-1 x)/√(1-x2 ) + k/√(1-x2 ) ⇒(1-x2 ) (dy/dx)2 = 4y + k2 Differentiating this with respect to x, we get (1-x2) 2 dy/dx.(d2 y)/(dx2 ) - 2x (dy/dx)2 = 4(dy/dx) ⇒(1-x2 ) ( d2 y)/dx2 -x dy/dx = 2
Illustration 1: Let [.] denotes the greatest integer function and f(x) = [tan2x], then does the limit exist or is the function differentiable or continuous at 0? (1999)
Solution: Given f(x) = [tan2x]
Now, -45°< x < 45°
tan(-45°)< tanx < tan45°
-tan 45°< tan x < tan 45°
-1< tan x <1
So, 0 <tan2x < 1
[tan2x] = 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)= x3 , x2< 1
x , x2≥ 1
Discuss the differentiability of the function at x=1.
Solution:We have R.H.D. = Rf'(1)
= limh→0 (f(1-h)-f(1))/h
= limh→0 (1+h-1)/h = 1
and L.H.D. = Lf'(1)= limh→0 (f(1-h)-f(1))/(-h)
= limh→0 ((1-h)3-1)/(-h)
= limh→0 (3-3h+h2) = 3
?Rf'(1)≠ Lf'(1)⇒ f(x) is not differentiable at x=1.
Illustration 3:If y = (sin-1x)2 + k sin-1x, show that (1-x2) (d2 y)/dx2 - x dy/dx = 2
Solution: Here y = (sin-1x)2 + k sin-1x.
Differentiating both sides with respect to x, we have
dy/dx = 2(sin-1 x)/√(1-x2 ) + k/√(1-x2 )
⇒(1-x2 ) (dy/dx)2 = 4y + k2
Differentiating this with respect to x, we get
(1-x2) 2 dy/dx.(d2 y)/(dx2 ) - 2x (dy/dx)2 = 4(dy/dx)
Signing up with Facebook allows you to connect with friends and classmates already using askIItians. It’s an easier way as well. “Relax, we won’t flood your facebook news feed!”
Post Question
Dear , Preparing for entrance exams? Register yourself for the free demo class from askiitians.