(i) Prove from the definition of differentiability that the function f(x) = x+1 2x 1 is...

 

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(i) Prove from the definition of differentiability that the function f(x) = x+1 2x 1 is differentiable at the point 1, and determine f'(1). (ii) Sketch the graph of the function f(x) = (sin(x), x 1, 1-1, x> 1. Use a result or rule from the module to determine whether this function is differentiable at 1. (b) The function g is continuous on the interval [-3, 1] and differentiable on the interval (-3, 1). Also, g(1) = 1 and -2 < g'(x) < 1 for x (-3, 1). Use the Mean Value Theorem to prove that -3 g(-3) 9. (c) Prove the inequality 2/3 - for r = [0, 1]. 3' (d) Prove that the following limit exists, and evaluate it: lim 6x2 x3 0 cos(2x)+x-1 (i) Prove from the definition of differentiability that the function f(x) = x+1 2x 1 is differentiable at the point 1, and determine f'(1). (ii) Sketch the graph of the function f(x) = (sin(x), x 1, 1-1, x> 1. Use a result or rule from the module to determine whether this function is differentiable at 1. (b) The function g is continuous on the interval [-3, 1] and differentiable on the interval (-3, 1). Also, g(1) = 1 and -2 < g'(x) < 1 for x (-3, 1). Use the Mean Value Theorem to prove that -3 g(-3) 9. (c) Prove the inequality 2/3 - for r = [0, 1]. 3' (d) Prove that the following limit exists, and evaluate it: lim 6x2 x3 0 cos(2x)+x-1