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Page 1 10. Differentiability Exercise 10.1 1. Question Show that f(x) = |x - 3| is continuous but not differentiable at x = 3. Answer f(x) = |x – 3| Therefore we can write it as, f(3) = 3 - 3 = 0 LHL = = = = RHL = = = = 0 LHL = RHL = f(3) Since, f(x) is continuous at x = 3 (LHD at x = 3) = = = = = - 1 (RHD at x = 3) = = = = = 1 (LHD at x = 3) (RHD at x = 3) Page 2 10. Differentiability Exercise 10.1 1. Question Show that f(x) = |x - 3| is continuous but not differentiable at x = 3. Answer f(x) = |x – 3| Therefore we can write it as, f(3) = 3 - 3 = 0 LHL = = = = RHL = = = = 0 LHL = RHL = f(3) Since, f(x) is continuous at x = 3 (LHD at x = 3) = = = = = - 1 (RHD at x = 3) = = = = = 1 (LHD at x = 3) (RHD at x = 3) Hence, f(x) is continuous but not differentiable at x = 3. 2. Question Show that f(x) = is not differentiable at x = 0. Answer For differentiability, LHD(at x = 0) = RHD (at x = 0) (LHD at x = 0) = = = = = = = Not defined (RHD at x = 3) = = = = = = Not defined Since, LHD and RHD does not exist at x = 0 Hence, f(x) is not differentiable at x = 0 3. Question Show that is differentiable at x = 3. Also, find f’(3). Answer For differentiability, LHD(at x = 3) = RHD (at x = 3) (LHD at x = 3) = Page 3 10. Differentiability Exercise 10.1 1. Question Show that f(x) = |x - 3| is continuous but not differentiable at x = 3. Answer f(x) = |x – 3| Therefore we can write it as, f(3) = 3 - 3 = 0 LHL = = = = RHL = = = = 0 LHL = RHL = f(3) Since, f(x) is continuous at x = 3 (LHD at x = 3) = = = = = - 1 (RHD at x = 3) = = = = = 1 (LHD at x = 3) (RHD at x = 3) Hence, f(x) is continuous but not differentiable at x = 3. 2. Question Show that f(x) = is not differentiable at x = 0. Answer For differentiability, LHD(at x = 0) = RHD (at x = 0) (LHD at x = 0) = = = = = = = Not defined (RHD at x = 3) = = = = = = Not defined Since, LHD and RHD does not exist at x = 0 Hence, f(x) is not differentiable at x = 0 3. Question Show that is differentiable at x = 3. Also, find f’(3). Answer For differentiability, LHD(at x = 3) = RHD (at x = 3) (LHD at x = 3) = = = = = = 12 (RHD at x = 3) = = = = = = = 12 Since, (LHD at x = 3) = (RHD at x = 3) Hence, f(x) is differentiable at x = 3. 4. Question Show that the function f defined as follows, Is continuous at x = 2, but not differentiable there at x = 2. Answer For continuity, LHl(at x = 2) = RHL (at x = 2) f(2) = 2(2) 2 - 2 = 8 - 2 = 6 LHL = = = = 8 - 2 = 6 RHL = Page 4 10. Differentiability Exercise 10.1 1. Question Show that f(x) = |x - 3| is continuous but not differentiable at x = 3. Answer f(x) = |x – 3| Therefore we can write it as, f(3) = 3 - 3 = 0 LHL = = = = RHL = = = = 0 LHL = RHL = f(3) Since, f(x) is continuous at x = 3 (LHD at x = 3) = = = = = - 1 (RHD at x = 3) = = = = = 1 (LHD at x = 3) (RHD at x = 3) Hence, f(x) is continuous but not differentiable at x = 3. 2. Question Show that f(x) = is not differentiable at x = 0. Answer For differentiability, LHD(at x = 0) = RHD (at x = 0) (LHD at x = 0) = = = = = = = Not defined (RHD at x = 3) = = = = = = Not defined Since, LHD and RHD does not exist at x = 0 Hence, f(x) is not differentiable at x = 0 3. Question Show that is differentiable at x = 3. Also, find f’(3). Answer For differentiability, LHD(at x = 3) = RHD (at x = 3) (LHD at x = 3) = = = = = = 12 (RHD at x = 3) = = = = = = = 12 Since, (LHD at x = 3) = (RHD at x = 3) Hence, f(x) is differentiable at x = 3. 4. Question Show that the function f defined as follows, Is continuous at x = 2, but not differentiable there at x = 2. Answer For continuity, LHl(at x = 2) = RHL (at x = 2) f(2) = 2(2) 2 - 2 = 8 - 2 = 6 LHL = = = = 8 - 2 = 6 RHL = = = = 6 Since, LHL = RHL = f(2) Hence, F(x) is continuous at x = 2 For differentiability, LHD(at x = 2) = RHD (at x = 2) (LHD at x = 2) = = = = = = = = 6 (RHD at x = 2) = = = = = 5 Since, (RHD at x = 2) (LHD at x = 2) Hence, f(2) is not differentiable at x = 2. 5. Question Discuss the continuity and differentiability of f(x) = |x| + |x - 1| in the interval ( - 1,2). Answer f(x) = f(x) = We know that a polynomial and a constant function is continuous and differentiable every where. So, f(x) is continuous and differentiable for x ( - 1,0) and x (0,1) and (1,2). Page 5 10. Differentiability Exercise 10.1 1. Question Show that f(x) = |x - 3| is continuous but not differentiable at x = 3. Answer f(x) = |x – 3| Therefore we can write it as, f(3) = 3 - 3 = 0 LHL = = = = RHL = = = = 0 LHL = RHL = f(3) Since, f(x) is continuous at x = 3 (LHD at x = 3) = = = = = - 1 (RHD at x = 3) = = = = = 1 (LHD at x = 3) (RHD at x = 3) Hence, f(x) is continuous but not differentiable at x = 3. 2. Question Show that f(x) = is not differentiable at x = 0. Answer For differentiability, LHD(at x = 0) = RHD (at x = 0) (LHD at x = 0) = = = = = = = Not defined (RHD at x = 3) = = = = = = Not defined Since, LHD and RHD does not exist at x = 0 Hence, f(x) is not differentiable at x = 0 3. Question Show that is differentiable at x = 3. Also, find f’(3). Answer For differentiability, LHD(at x = 3) = RHD (at x = 3) (LHD at x = 3) = = = = = = 12 (RHD at x = 3) = = = = = = = 12 Since, (LHD at x = 3) = (RHD at x = 3) Hence, f(x) is differentiable at x = 3. 4. Question Show that the function f defined as follows, Is continuous at x = 2, but not differentiable there at x = 2. Answer For continuity, LHl(at x = 2) = RHL (at x = 2) f(2) = 2(2) 2 - 2 = 8 - 2 = 6 LHL = = = = 8 - 2 = 6 RHL = = = = 6 Since, LHL = RHL = f(2) Hence, F(x) is continuous at x = 2 For differentiability, LHD(at x = 2) = RHD (at x = 2) (LHD at x = 2) = = = = = = = = 6 (RHD at x = 2) = = = = = 5 Since, (RHD at x = 2) (LHD at x = 2) Hence, f(2) is not differentiable at x = 2. 5. Question Discuss the continuity and differentiability of f(x) = |x| + |x - 1| in the interval ( - 1,2). Answer f(x) = f(x) = We know that a polynomial and a constant function is continuous and differentiable every where. So, f(x) is continuous and differentiable for x ( - 1,0) and x (0,1) and (1,2). We need to check continuity and differentiability at x = 0 and x = 1. Continuity at x = 0 = 1 = 1 F(0) = 1 Since, f(x) is continuous at x = 0 Continuity at x = 1 = 1 = 1 F(1) = 1 = 1 Since, f(x) is continuous at x = 1 For differentiability, LHD(at x = 0) = RHD (at x = 0) Differentiability at x = 0 (LHD at x = 0) = = = = 2 (RHD at x = 0) = = = = 0 Since,(LHD at x = 0) (RHD at x = 0) So, f(x) is differentiable at x = 0. For differentiability, LHD(at x = 1) = RHD (at x = 1) Differentiability at x = 1 (LHD at x = 1) = = = 0Read More
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