課程名稱︰微積分甲下
課程性質︰必修
課程教師︰薛克民
開課學院：電資學院、工學院、管理學院
開課系所︰電機系、資工系、材料系、資管系
考試日期（年月日）︰100/4/11
考試時限（分鐘）：40分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
˙每題十分
˙請詳述計算過程，無計算過程的答案不予計分
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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※ 編輯: rod24574575 來自: 218.167.197.181 (05/01 20:17)