課程名稱︰微積分乙下 課程性質︰共同必修 課程教師︰張樹城 開課學院：管院 開課系所︰會計系/財金系/工管系企管組/國企系/地理系/經濟系 考試日期（年月日）︰2011/4/28 考試時限（分鐘）：110 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.(25%) Compute the followings(5 out of 7) t (a)lim(∫(1+x)^100 dx)^(1/t) t→0 0 ∞ (b)∫ e^(-x^2) dx 0 dx (c)∫------------ 9x^2+6x+10 (d)∫x^(-1) lnxdx (e)∫sin^3x cos^2x dx (f)∫e^x cosx dx ∞ 1 (g)∫ ------- dx ∞ 1+x^2 2.(10%) Suppose that f(x) is a continuous function difined by x f(t) f(x)=2+∫------------ dt, x≧0 0 (t+3)(t+2) Find (a)f'(x) (b)f(1) 3.(15%) (a) Find the area of the region between the curves x=y^2 and x+y+2 in the first quadrant. (b) Find the length of the unit circle x^2+y^2=1 (c) Find surface area of the sphere of radius 1 which is generated by x^2+y^2=1 about the x-axis. 4.沒有第四題 5.(20%) Determine which series converge and which series diverge? Give reasons for your answers. ∞ 2n (a)Σ --------- n=1 n^2-n+1 ∞ ln n (b)Σ (-1)^(n+1) ---- n=1 n ∞ ln n (c)Σ ----- n=1 n ∞ n^n (d)Σ ----- n=1 n! 6.(10%) (a) Find the Maclaurin series for e^x (b) Estimate e with an error less than 10^(-2) 7.(10%) Consider the sequence {a } with n a =2^(1/2), a =(2+a )^(1/2) 1 n+1 n (a) Find an upper bound of {a } n (b) Find lim　a n→∞ n 8.(10%) Let Σ be the plane : 3x+2y+6z=6 (a) Find a vector N normal to the plane Σ (b) Find the distance from the point S(1,1,3) to the plane Σ 9.(10%, Bonus) (a) Find the Maclaurin series for tan^(-1) x (b) Show that --

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