課程名稱︰微積分3 課程性質︰物理系必帶 課程教師︰李瑩英 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰2019/03/18 考試時限（分鐘）：90分鐘 後來延長到105分鐘 試題 : 1. Assume that f(x,y) = xy^2/(x+y^4) , (x,y) =/= (0,0) 0 , (x,y) = (0,0) (a) (3pts) Determine and explain whether f(x,y) is continuous. (b) (10pts) Compute f_x(x,y) and f_y(x,y) if it exists. (c) (3pts) Is f(x,y) differentiable at (0,0)? Give your reasons. 2. Assume that f(x,y,z) = x^3*(y^2+z^2)^(1/2), (a) (8pts) Determine the level surface of f that pass through (2,3,4) and its tangent plane at that point. (b) (6pts) Use the linear approximation of f at (2,3,4) to estimate (1.98)^3*(3.01^2+3.97^2)^(1/2) 3. (10pts) Find z_x and z_y (z對x和對y的偏導數) for yz+zlny = z^2 4. (12pts) Use the chain rule to compute z_s and z_t , where z = arctan(x^2+y^2) , s=lnt , y=te^s. 5. Assume that f(x,y) = ln(1+x^2+y^2)-xy^2 (a) (8pts) Find the directional derivative of f at point (1,1) in the direction of (1,-1). (b) (7pts) In which direction will f increase the most rapidly at(1,1)? What is the rate? 6. f(x,y) = 4xy^2-x^2y^2-xy^3 , D is the closed triangular region in the xy plane with vertices (0,0) , (0,6) and (6,0). (a) (10pts) Determine the critical points of f in the interior of D and their types. (b) (8pts) Find the absolute maximum and minimum values of f. 7. (15pts) A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon. /\ / \ /ψ__\ | | | | |______| p.s. 示意圖我盡力了QQ 就是一個等腰三角形放在一個長方形上面，給定一個ψ角 --

※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 42.73.242.248 ※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1553050375.A.933.html