课程名称︰微积分乙上 课程性质︰必修 课程教师︰李聪成 开课学院：社科院、管院、理学院 开课系所︰经济、财金、国企、工管、会计、地理 考试日期（年月日）︰2012年1月12日 考试时限（分钟）：120分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 请将每一步骤表达清楚,不可以只写答案。 1.Evaluate the indefinite integral. 1 ∫ ──── dx 1+e^x 2.Find the area (面积) of the region (区域) bounded by the graphs of x 1 y= ────, y= - ──x^2, and x=1 x^2+1 2 3.Evaluate the definite integral. 1 x^3 ∫ ──── dx 0 1+x^8 4.Evaluate the limit. 1 x lim ── ∫ sin(t^2)dt x→0 x^3 0 5.Evaluate the limit. 1 1 1 lim ( ── + ── + ... + ── ) n→+∞ n+1 n+2 n+n 6.Use logarithmic differentiation to find the derivative of the function. x^3cos^2(πx) 1 y= ─────── where x∈(0, ──) √(x^2+1) 2 7.Show that f is one-to-one on (1,+∞) and find (f^-1)(0), where x f(x) =∫ √(1+t^3)dt 1 and f^-1 is the inverse function (反函数) of f. 8.The disk (圆板) x^2+y^2≦a^2 is revolved (旋转) about the line x=b (with b>a>0) to generate a solid called a tortus. Find its volume (体积). 9.Prove that 1 x^5 1 0 ≦ ∫ ─────── dx ≦ ── 0 (1+x^4)^(1/3) 6 10.Find an equation for the line through the origin tangent to the graph of y=㏑x. 参考答案： 1.-㏑[e^(-x)+1]+C 2.㏑2 3.π/16 4.1/3 5.㏑2 6.(答案较丑，老师说微出即可) 7.√2/2 8.2π^2a^2b 9.略 10.y=(1/e)x 用书：Soo T. Tan, Calculus, 2010, Cengage Learning 范围： Ch4 Integration Ch5 Applications of the Definite Integral Ch6 The Transcendental Functions ※ 编辑: mickeyhp 来自: 220.136.82.27 (01/22 14:37)