课程名称︰微积分甲下 课程性质︰必修 课程教师︰薛克民 开课学院：电资学院、工学院、管理学院 开课系所︰电机系、资工系、材料系、资管系 考试日期（年月日）︰100/4/11 考试时限（分钟）：40分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : ˙每题十分 ˙请详述计算过程，无计算过程的答案不予计分 1. Find the tangent, normal and curvature of the curve γ(t) = ( cos(t) + t*sin(t) , sin(t) - t*cos(t) , t ). (10%) 2. At the point (1,3), a function f(x,y) has a derivative of -√5 in the direction from (1,3) toward (2,1) and a derivative of √5 in the direction from (1,3) toward (3,2). Find the directions in which the function f increases most rapidly at (1,3), and find the rates of change in this direction. (10%) 3. Determine whether the following limits exist. If it does, find the value. (10%) 5x*√y (a) lim ──────── (x,y)→(0,0) √(x^3 + y^3) 2 2 (x^2 + y^2) x y e (b) lim ───────── (x,y)→(0,0) x^4 + y^4 4. Let u = u(x,y), where x = x(s,t) and y = y(s,t), and assume that all these functions have continuous second derivatives. show that 2 2 2 2 2 2 δ u δ u ┌ δx ┐ δ u δx δy δ u ┌ δy ┐ ─── = ───│ ──│ + 2 * ─── * ──* ── + ───│ ──│ δs^2 δx^2 └ δs ┘ δxδy δs δs δy^2 └ δs ┘ 2 2 δu δ x δu δ y + ──* ─── + ──* ─── δx δs^2 δy δs^2 --

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