课程名称︰微积分乙上 课程性质︰必修 课程教师︰王振男 开课学院：医学院 开课系所︰医学系 考试日期（年月日）︰2009.11.XX 考试时限（分钟）：120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题: 1. (10%) Suppose that the derivative of the function y=f(x) is y'=[(x-1)^2](x-2)(x-4). At what points, if any, does the graph of f have a local minimum, local maximum, or point of inflection? 2. (20%) Find the centroid of the region bounded by the curve y=4-x^2 and the lines x=-1 and y=1. 3. (20%) Find the limit: ε/(1+ε) lim ∫ [e^(-x^2)]/(x^2) dx ε→0- ε lim Hint: the limit ε→0 [e^(-x^2)-1]/(x^2) may be useful. 4. (20%) Let f:Ｒ→Ｒ be a differentiable and bounded function, i.e., there exists a positive constant M such that |f(x)|≦M for all x属於Ｒ. Given a属於Ｒ with a≠0. Assume that f'(x)≠a for all x属於Ｒ. Show that there exists one and only one x_0属於Ｒ such that f(x_0)=ax_0. 5. (10%) At points on the curve y=2[x^(1/2)], line segments of length h=y are drawn perpendicular to the xy-plane. (图略) Find the area of the surface formed by these perpendiculars from (0,0) to (3,2[(3)^(1/2)]). 6. (20%) A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge and held there as the paper is smoothed flat. The problem is to make the length of crease as small as possible. Call the length L. a. Show that L^2=(2x^3)/(2x-8.5) b. What value of x minimizes L^2? c. What is the minimum value of L? ＿＿＿＿＿＿＿＿ |＼ | |\ ＼ | | \ ＼ | | \ 　＼ | | \ 　＼ | | \ 　　＼ | 11 in. √(L^2-x^2)| \ L 　＼ | |　　　\ ／＼| | \　 ＼／| |_ \　 ／ | |_|＿x＿＿\／＿＿| 　 　 　 　　 8.5 in. 　　　　　　　　　　　　 --

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