课程名称：微积分甲上 课程性质：物理系必带 课程教师：康明昌 开课学院：理学院 开课系所；物理系 考试日期：2010/11/16 考试时间：120分钟 是否发放奖励金：是 1.Define <an>=[((n^2)-1)^-0.5]+[((n^2)-4)^-0.5]+...+[((n^2)+(n-1)^2)^-0.5] Find lim<an>. (10%) n→∞ 2.Find y=y(x) where y satisfies xy'=3y+x^3 and y(-exp(0.5))=exp(0.5) (10%) 3.Evaluate the integral ∫x*cosx*exp(x)dx (10%) (Hint: If you have a smart idea, go ahead. Otherwise, find ∫x*cosx*exp(x)dx+∫x*sinx*exp(x)dx and ∫x*cosx*exp(x)dx-∫x*sinx*exp(x)dx.) 4.Let y=y(x) (1)Find the general solution of y"+4y'+3y=0 (5%) (2)Find y where satisfies y(0)=0,y'(0)=1/2, and y"+4y'+3y=x*cosx+exp(x) (15%) (Hint: Even if you can not find ʃx*cosx*exp(x)dx, you may write something till the last step.) 5.Define a function f:R→R by f(x)=0 if x belongs to R\Q, and f(x)=1/p if x=q/p where p,q belong to Z, p≧1 and gcd{p,q}=1. Show that lim f(x)=0 (15%) x→1/2 6.Let f,g:[0,1]→R, be continuous functions. Define h(x)=max{f(x),g(x)} for all x belong to [0,1]. Show that h(x) is also a continuous function. (15%) (Hint: If you have no smart idea, how about considering the cases f(a)=g(a) and f(a)≠g(a) separately.) 7.Find all the values of c such that f(x)=(3/2)x^4-2x^3-6x^2+c=0 has four distinct real roots. (15%) (Hint: Sketch the graph y=(3/2)x^4-2x^3-6x^2+c) 8. Suppose x^(2/3)+y^(2/3)=1. Find y"(1/8) if y(1/8)=((27)^0.5)/8 (15%) 9.Discuss the convergence or divergence of the integrals ∞ ∞ ∫(sinx/x)dx and ∫(|sinx|/x)dx (20%) 0 0 --

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