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研究所、轉學考(插大)-微積分
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105年 - 105 國立臺灣大學轉學生招生考試試題:微積分(B)#110861
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題組內容
Correctly number each of your answers to indicate which question is answered.
(K) Find the critical points of f(x, y) = x
2
+ y
4
+ 3xy
2
- 5x which are saddle points. Answer:__(12)__.
其他申論題
(G) Find the solution of the differential equation y' + ytan(x) = sec(x) with the initial condition y(0) = 2. Answer: __(8)__.
#474896
(H)Find the first three nonzero terms of the Mclaurin series of tan(x). Answer: __(9)__.
#474897
(I) Let u = u(r, y) be a function of t, y. Express in terms of polar coordinates r,θ together with Answer: __(10)__
#474898
(J) Find the directional derivative of f(x, y, 2) =xy z2 at the point (e,e, 1) in the direction u = Answer: __(11)__.
#474899
(L) Find the minimum of x2+ y2 + z2 for (x,y, 2) on the intersection curve of the two surfaces y + 2z = 1 and 3x2 + y2 -z2 = 1. Answer:__ (13)__.
#474901
(M) Evaluate dydx. Answer: __(14)__ .
#474902
(N) Let R be the region in the first quadrant of the xy-plane bounded by xy = 1, xy = 2, y = 2 and y = 2x. Evaluatedxdy. Answer:____ (15)____.
#474903
(O) Evaluate dxdydz, where Ω is given by Ω = {(x,y, 2) : 1≤x2+y2+z2≤2}.Ansuer:__(16)__.
#474904
(P) Evaluatedrdydz, whereΩ is the cylinder defined by Ω ={(x,y,z):x2 +y2≤1 and 0 ≤z≤1}. Answer: __(17)__.
#474905
(Q) Let S be the surface described by z = x2+ with 4x2+ y2 ≤ 1 oriented with normals with positive k-components. Let F(x,y,z) = xi - yj + k. Evaluate F.dS. Answer:__ (18)__ Also, evaluated.S. Answer: __(19)__.
#474906
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