课程名称︰微积分甲上 课程性质︰暑修 课程教师︰周青松 开课学院： 开课系所︰数学系 考试日期（年月日）︰2013/7/3 考试时限（分钟）：50分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1) Decide whether or not the indicated limits exist. Evaluate the limits that do exit. (a) lim (x^2+x-12)/(x-3) x→3 (b) lim (f(x)-f(1))/(x-1), f(x)=x^3 x→1 (do not use differentiation) (20%) 2) (a)Let f(x)=｛A^2*x^2, x≦2 ｛(1-A)x, x＞2 Find the value for A such that f is continuous at x=2. (b)Give the sufficient and necessary conditions on A and B such that the function f(x)=｛Ax-B, x≦1 ｛3x, 1＜x＜2 ｛Bx^2-A, 2≦x is continuous at x=1 but discontinuous at x=2 (30%) 3) Let f(x)=sinx. Evaluate lim(f(c+h)-f(c))/h at c=π/4 h→0 (do not use differentiation) (20%) 4) Let f(x)=｛x^2-x, x≦2 ｛2x-2, x＞2 (a)Show that f is continuous at x=2 (b)Prove or disprove: f is differentiable at x=2 (30%) -- --

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