課程名稱︰微積分甲下 課程性質︰暑修 課程教師︰朱樺 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017/8/21 考試時限（分鐘）：150 試題 : (1) (30%) Find the limit. -1 n^n (a) lim sin(tan (-----)) n→∞ n! 5 1 4 n^2 (b) lim (n sin--- - n + -----) n→∞ n 6 xy+yz^2+xz^2 (c) lim -------------- (x,y,z)→(0,0,0) x^2+y^2+z^4 x^3+3y^3 (d) lim ---------- (x,y)→(0,0) 2x^2+y^2 e^(x^2)+y^2-1 (e) lim ------------ (x,y)→(0,0) x^2+y^2 (2) (24%) Determine whether the series is absolutely convergent, conditionally convergent or divergent. ∞ (-1)^n (a) Σ -------- n=2 n ln n ∞ n 1 (b) Σ (-1) sin n sin(-----) n=1 n^2 ∞ n 1‧4‧7...(3n-2) (c) Σ (-1) ------------------ n=1 3‧5‧7...(2n+1) ∞ n (n!)^n (n^n)^2 (d) Σ (-1) ---------------- n=1 n^(n^2) (3) (12%) Find the sum of the series. ∞ -1 -1 (a) Σ (tan (n+1)-tan n) n=1 ∞ n^2 (b) Σ ----- n=1 3^n (4) (18%) Find the Maclaurin series of the function. 1 (a) f(x) = -------- e^(2x) 3 (b) f(x) = cos x x^2 (c) f(x) = ------- √(x+2) (5) (6%) Find the interval of convergence of the power series ∞ n (2x-1)^(2n+1) Σ (-1) --------------- . n=1 4^n n^5 0.1 -1 (6) (8%) Approximate the difinite intergral ∫ x tan (3x^2)dx so that it is 0 accurate to within 10^(-10). (7) (6%) Find the equation of the surface generated by rotating the curve C. (a) C：x^2-y^2 = 1; about the x-axis. (b) C：xy = 1; about the y-axis. (8) (8%) Two particles travel along the space curves r1(t) =〈t^2,7t-12,t^2〉and r2(t) =〈4t-3,t^2,5t-6〉, t≧0. (a) Do their paths intersect? What is the angle between then at the intersection point? (b) Do the particles collide? (9) (8%) What point at the curve y = ln x has the largest curvative? (10) (10%) Find the unit tangent vector T(t), the unit normal vector N(t) and the unit binormal vector B(t) for the curve r(t) =〈t^2, sin t - t cos t, cos t + t sin t〉, t ∈ R. --

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