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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

[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

(iii) f(x) = -x^{2} + 2

Given f(x) = -x^{2} + 2

f(x + h) = -(x + h)^{2} + 2

= -x^{2} – h^{2} – 2xh + 2

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:

f'(1) does not exist

∴ ‘f’ is not differentiable at x = 1.

(ii) f(x) = \(\sqrt{1-x^{2}}\)

Solution:

∴ ‘f’ is not differentiable at x = 1.

(iii)

Solution:

‘f’ is not differentiable at x = 1

Question 3.

Determine whether the following functions is differentiable at the indicated values.

(i) f(x) = x |x| at x = 0

Solution:

Limits exists

Hence ‘f’ is differentiable at x = 0.

(ii) f(x) = |x^{2} – 1| at x = 1

Solution:

f(x) is not differentiable at x = 1.

(iii) f(x) = |x| + |x – 1| at x = 0, 1

Solution:

∴ f(x) is not differentiable at x = 0.

∴ f(x) is not differentiable at x = 1.

(iv) f(x) = sin |x| at x = 0

Solution:

∴ 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)

Solution:

f(x) is not differentiable at x = 2.

(ii)

Solution:

f(x) is not differentiable at x = 0.

Question 5.

The graph off is shown below. State with reasons that x values (the numbers), at which f is not differentiable.

Solution:

(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| + x^{2}, test whether f’ (-100) exists.

Solution:

f(x) = |x + 100| + x^{2}

Question 7.

Examine the differentiability of functions in R by drawing the diagrams.

(i) |sin x|

Solution:

The limit exists and continuous for all x ∈ R clearly, differentiable at R — {nπ n ∈ z) Not differentiable at x = nπ, n ∈ z.

(ii) |cos x|

Solution:

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:

Question 2.

Discuss the differentiability of the functions:

Solution:

∴ f(2) is not differentiable at x = 2. Similarly, it can be proved for x = 4.