Tamil Nadu Board of Secondary EducationHSC Arts Class 11

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1? f(x)=|x-1| - Mathematics

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Sum

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

`f(x) = |x - 1|`

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Solution

`f(x) = {{:(x - 1,  "if"  x > 1),(-(x - 1),  "if"  x < 1):}`

First we find left limit of `f(x)` at x = 1

When `x -> 1-` we have `f(x) = - (x - 1)`

`f"'"(1^-) =  lim_(x -> 1^-) (f(x) - f(1))/(x - 1)`

`f"'"(1^-) =  lim_(x -> 1) (-(x - 1) - (0))/(x - 1)`

= `lim_(x -> 1) (-(x - 1))/(x - 1)` = – 1  .......(1)

`f"'"(1^+) =  lim_(x -> 1^+) (f(x) - f(1))/(x - 1)`

= `lim_(x -> 1^+) ((x - 1) - 0)/(x - 1)`

`f"'"(1^+) =  lim_(x -> 1^+) (x - 1)/(x - 1)` = 1  ......(2)

From equation (1) and (2) we have

`lim_(x -> 1^-) f(x)  ≠  lim_(x -> 1^+) f(x)`

∴ `f"'"(x)` does not exist at x = 1

Concept: Differentiability and Continuity
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Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation - Exercise 10.1 [Page 147]

APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 10 Differential Calculus - Differentiability and Methods of Differentiation
Exercise 10.1 | Q 2. (i) | Page 147

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