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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1

Question 1.
Find the derivatives of the following functions using first principle.
(i) f(x) = 6
Solution:
Given f(x) = 6
f(x + h) = 6
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 1
[h → 0 means h is very nears to zero from left to right but not zero]

(ii) f(x) = -4x + 7
Solution:
Given f(x) = -4x + 7
f(x + h) = -4(x + h) + 7
= -4x – 4h + 7
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 2
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 3

(iii) f(x) = -x2 + 2
Given f(x) = -x2 + 2
f(x + h) = -(x + h)2 + 2
= -x2 – h2 – 2xh + 2
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 4

Question 2.
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?
(i) f(x) = |x – 1|
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 5
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 6
f'(1) does not exist
∴ ‘f’ is not differentiable at x = 1.

(ii) f(x) = \(\sqrt{1-x^{2}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 7
∴ ‘f’ is not differentiable at x = 1.

(iii)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 8
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 9
‘f’ is not differentiable at x = 1

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1

Question 3.
Determine whether the following functions is differentiable at the indicated values.
(i) f(x) = x |x| at x = 0
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 10
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 11
Limits exists
Hence ‘f’ is differentiable at x = 0.

(ii) f(x) = |x2 – 1| at x = 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 12
f(x) is not differentiable at x = 1.

(iii) f(x) = |x| + |x – 1| at x = 0, 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 13
∴ f(x) is not differentiable at x = 0.
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 14
∴ f(x) is not differentiable at x = 1.

(iv) f(x) = sin |x| at x = 0
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 15
∴ f(x) is not differentiable at x = 0.

Question 4.
Show that the following functions are not differentiable at the indicated value of x.
(i)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 16
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 17
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 18
f(x) is not differentiable at x = 2.

(ii)
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 19
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 20
f(x) is not differentiable at x = 0.

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1

Question 5.
The graph off is shown below. State with reasons that x values (the numbers), at which f is not differentiable.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 21
(i) at x = – 1 and x = 8. The graph ‘ is not differentiable since ‘ has vertical tangent at x = -1 and x = 8(also At x = -1. The graph has shape edge v] and at x = 8;The graph has shape peak ^]
(ii) At x = 4: The graph f is not differentiable, since at x =4. The graph f’ is not continuous.
(iii) At x = 11; The graph f’ is not differentiable, since at x = 11. The tangent line of the graph is perpendicular.

Question 6.
If f(x) = |x + 100| + x2, test whether f’ (-100) exists.
Solution:
f(x) = |x + 100| + x2
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 22

Question 7.
Examine the differentiability of functions in R by drawing the diagrams.
(i) |sin x|
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 23
Limit exist and continuous for all x ∈ R clearly, differentiable at R — {nπ n ∈ z) Not differentiable at x = nπ , n ∈ z.

(ii) |cos x|
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 24
Limit exist and continuous for all x ∈ R clearly, differentiable at R {(2n + 1)π/2/n ∈ z} Not differentiable at x = (2n + 1) \(\frac{\pi}{2}\), n ∈ Z.

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 Additional Questions

Question 1.
Is the function f(x) = |x| differentiable at the origin. Justify your answer.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 25

Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1

Question 2.
Discuss the differentiability of the functions:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 26
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 10 Differentiability and Methods of Differentiation Ex 10.1 27
∴ f(2) is not differentiable at x = 2. Similarly, it can be proved for x = 4.