You need to explain all reasons in detail and show all of your work on the answer sheet; otherwise you will get NO credits. If you used any theorem in textbook or proved in class, state it carefully and explicitly. I. A. (10%) Show that the function f(x) = ｜x｜ is comtinuous at 0 but not differentiable at 0. B. (10%) If f is differentiable at x, show that f is continuous at x. II. A. (10%) Let f and g be differentiable functions such that f'(x) = g(x), and let T(x) = [f(x)]^2 + [g(x)]^2 find T'(x). B. (10%) Let f be a differentiable function. Use the chain rule to show that: (a) if f is even, then f' is odd. (b) if f is odd, then f' is even. III. A. (10%) Set f(x) = x^(-1), a = -1, b = 1. Verify that there is no number c for which f(b) - f(a) f'(c) = ──────── b - a Explain how this does not violate the mean-value theorem. B. (10%) Let f be differentiable on (a,b) and continuous on [a,b]. Prove that if there is a constant M such that f'(x) <= M for all x∈(a,b), then f(b) <= f(a) + M(b-a). IV. A. (10%) Find f given that f'(x) = 6x^2-7x-5 for all real x and f(2) = 1. B. (10%) Given ｜ x+2, x < 0 ｜(x-1)^2, 0 < x < 3 ｜ 8-x, 3 < x < 7 f(x) = ｜ 2x-9, 7 < x ｜ 6, x = 0, 3, 7 Find the intervals on which f increases and the intervals on which f decreases. V. (20%) Sketch the graph of f(x) = (1/4)x^4 - 2x^2 + 7/4 on [-5,5]. --

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