课程名称︰微积分下 课程性质︰必修 课程教师︰黄维信 开课学院：工学院 开课系所︰工科海洋学系 考试日期（年月日）︰2008/4/22 考试时限（分钟）：120mins 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. State whether the sequence converges and if it does, find the limit.(15%) (a) [ln(1+1/n)]^n (b) 3ln(2n)-ln(n^3+1) n t^2 (c) ne^(-n^2)∫ (e ) dt 0 2. Determine whether the integral converges and, if so, evaluate the integral.(10%) π/2 cosx ∞ dx (a)∫ ──── dx (b) ∫─────── 0 √sinx e {√(x+1)}lnx ∞ 3. Show that Σ ln{(k+1)/k} diverges, although ln{(k+1)/k}→0. (5%) k=1 4. Find the taylor series expansion of f(x)=√(x+1) in powers of x and give the radius of convergence. (15%) 5. Find the unit tangent, the principal normal, and write an equation in x,y,z for the osculating plane at the point on the curve → → → → r(t)=(sint-tcost)i+(cost+tsint)j +t^2/2 k at t=0. Parametrize the curve by its arc length for t≧0. (20%) → → → 6. Suppose the curve r(t)=x(t)i+y(t)j is twice differentiable. If this curve has nonzero tangent vector. Prove its curvature is ' " ' " ' ' │x(t)y(t)-y(t)x(t)│/([x(t)]^2+[y(t)^2])^3/2 .(10%) 7. Set f(x,y)=(x-y^4)/(x^3-y^4). Find the domain and range of f. Determine whether or not f has a limit at (1,1). (10%) 8. Suppose that f is differentiable at (x ,y ) and ▽f(x ,y )≠0. Calculate 0 0 0 0 → → the rate change of f in the direction of f (x ,y )i-f (x ,y )j. y 0 0 x 0 0 Give a geometric interpretation to your answer. (10%) → → → → 9. Set u(x,y,z)=u (x)i+u (x,y)j+u (x,y,z)k with x=x(t),y=y(t),z=z(t). 1 2 3 → Derive a formula for du/dt. (10%) --

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