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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

科目：研究所、轉學考(插大)-微積分 | 年份：105年 | 選擇題數：0 | 申論題數：18

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所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (18)

(A). Find the interval on which y =x3 - 3x2 + 2x + 1 is both decreasing and concave downward. Answer. __(1)__

(B). Find632a63d98dc1b.jpg =__(2)__ and632a63f6ec628.jpg=__ (3)__ at the point x = 1, y = 1 of the curve  x3+2y-2y4 =0.

(C)Evaluate632a643d851cb.jpg. -Answers:__ (4)__ .

(D)Find the volume of the solid generated by revolving the following region about the y-axis: {(x,y) :0 ≤x ≤π and 0 ≤y ≤sinx}. Answer: __(5)__ .

(E) Find the arc length of the curve y = 632a64ddc1bbd.jpg from x= 0 to x = π/3.

Answer: __(6)__ .
(F) Find the solution of the differential equation y'=(1+2x)(1+y2) with the initial condition y(0) = 1. Answer: __(7)__.
(G) Find the solution of the differential equation y' + ytan(x) = sec(x) with the initial condition y(0) = 2.
Answer: __(8)__.
(H)Find the first three nonzero terms of the Mclaurin series of tan(x). Answer: __(9)__.

(I) Let u = u(r, y) be a function of t, y. Express 632a65715cfd9.jpg in terms of polar coordinates  r,θ together with 632a65cd4a950.jpgAnswer: __(10)__

(J) Find the directional derivative of f(x, y, 2) =xy z2 at the point (e,e, 1) in the direction u =632a660b40498.jpg Answer: __(11)__.
(K) Find the critical points of f(x, y) = x2 + y4 + 3xy2 - 5x which are saddle points. Answer:__(12)__.
(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)__.

(M) Evaluate632a66bf6a775.jpg dydx. Answer: __(14)__ .

(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. Evaluate632a672797783.jpgdxdy. Answer:____ (15)____.

(O) Evaluate 632a676000226.jpgdxdydz, where Ω is given by Ω  = {(x,y, 2) : 1≤x2+y2+z2≤2}.Ansuer:__(16)__.

(P) Evaluate632a688a73292.jpgdrdydz, whereΩ is the cylinder defined by Ω ={(x,y,z):x2 +y2≤1  and 0 ≤z≤1}. Answer: __(17)__.

(Q) Let S be the surface described by z = x2+632a6a446b4a5.jpg with 4x2+ y2 ≤ 1 oriented with normals with positive k-components. Let F(x,y,z) = xi - yj + k. Evaluate 632a6a9870e93.jpgF.dS. Answer:__ (18)__ Also, evaluate632a6ad6c4ab1.jpgd.S. Answer: __(19)__.

(R) Let C be the counterclockwise oriented boundary of the region in the xy- plane enclosed byx2 + y2 - 20 = 0 andx2 + y2 - 2y = 0. Evaluate the line integral 632a6b5908278.jpg. Answer:__ (20)__.