课程名称︰微积分乙 课程性质︰共同必修 课程教师︰张秀瑜 开课学院：管院 考试日期（年月日）︰100年1月11日 考试时限（分钟）：120分钟 是否需发放奖励金：是，谢谢 （如未明确表示，则不予发放） 试题 : 1.(15%)Find the following derivative. d 2 sinx (a)──( x + 1 ) dx 2 d -1 x (b)──sin (──) dx a d 2x _______ (c)── ∫ √1 + t^6 dt dx x+1 2.(20%)Find the limit： 1/x (a) lim (1-3x) x→0 (b) lim (sinx)(㏑x) x→0+ f(2+3x) + f(2+5x) (c) lim ────────── , where f' is continuous, f(2)=0, and f'(2)=7. x→0 x 3.(24%)Evaluate the integral 1 (a) ∫ ──── dx 3 x(㏑x) x 4 3 (b) ∫ ───── dx 0 x 3 + 1 2 _____ (c) ∫ √x^2+4 dx 0 2x (d) ∫ e sin3x dx 1 1 4.(10%)If f is continuous on [0,1], prove that ∫ f(x)dx =∫ f(1-x)dx 0 0 5.(10%)Find the volume common to two spheres, each with radius R, if the center of each sphere lies on the surface of the other sphere. 6.(10%)Find the volume of the solid obtained by rotating about the y-axis the 2 -x region between y= e , y=0 , x=0 , x=1 7.(15%)Solve the differential equation y 2 dy e sin θ ── = ───── dθ y secθ --

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