课程名称︰微积分乙上 课程性质︰必带 课程教师︰张树城 开课学院：生科院 开课系所︰生科系.生技系 考试日期（年月日）︰2016/11/10 考试时限（分钟）：110 试题 : I.(25%) Compute the following: (i) lim (√( x + √x) - √x ) x→∞ (ii) d^2 y /dx^2, where x = t - t^2, y = t - t^3. (iii)d^2 y /dx^2, where 2x^3 - 3y^2 = 8. (iv)d^101 y /dx^101, where y = sin x. (v)f'(0), where cos x <= f(x) <= x^2 +cos x, |x| <= 1/8. II.(15%) Let x^2 sin x, x ≠ 0 f(x) = 0 , x = 0 (i) Find f'(x) for x≠0. (ii) Find f'(0). (iii)Is f'(x) continuous at x=0? III.(15%) Prove or disprove the followings: (i) If |f(x)| is continuous at c, then f(x) is continuous at c. (ii)If f(x) is differentable at c, then f(x) is continuous at c. (iii)There is exact one real root for x^3 + 3x + 1 = 0. IV.(15%) Let the function f(x) = x^3 - 2x^2 + x + 1 (i)Indicate where f(x) increases and decreases. (ii)Indicate the critical points and points of inflection and the concavity. (iii)Sketch the graph. V.(10%) Suppose that f(x) = x^(2/3), x ε[-2,3]. Find the maximum and minimum values of f(x) and indicate where they are taken on. VI.(10%) Define the function f(x) = x^(1/3). (i)Find the tangent line to the curve y = f(x) at (1000,10). (ii) Use the linearization method (i) to esimate 三次根号(1001). VII.(10%) Let f(x), g(x): [a,b] →R be differentable function with g'(x)≠0. Define G(x) by G(x) = f(x) - f(a) - [f(b)-f(a)]/[g(b)-g(a)] (g(x)-g(a)) Show that (i) G(a) = G(b) (ii)there exist c ε (a,b) such that f'(c) f(b) - f(a) ──── = ─────── g'(c) g(b) - g(a) --

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