109年 - 109 國立臺灣大學轉學生招生考試試題_理工科聯招：微積分(B)#110852

科目：研究所、轉學考(插大)-微積分 | 年份：109年 | 選擇題數：0 | 申論題數：23

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所屬科目：研究所、轉學考(插大)-微積分

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申論題 (23)

(a)

=__(1)__ , where [x] is the greatest integer which is less than or equal to x.

(b) =__(2)__，where a, b are constants and ab ≠ 0.

(c)=___(3)____.

(a)

(b) x³ -y²+y³ =x. At (x, y) = (0, 1).=__(5)__

(c) f(x，y，z)==__(6)__.

(d)

f(x,y)=

for (x,y)≠(0,0) and f(0,0) = 0. The directional derivative of f

along u = (cosθ,sinθ) at (0,0) is__(7)__.

(a) Choose correct statement(a) about. g(x): __(8)__
i. g(x) is discountinuous at x = 3.
ii. g(x) is not differentiable at x = 3.
iii. g(x) is differentiable at x = 0 and g'(0) = -1.
iv. g(x) ＜ 0for -1 ≤x≤0.

(b) The local minimum values of g(x) occur at x =__(9)__

(c) The t-coordinates of points of inflection for y = g(x) are __(10)__.

4. A vertical fence 2 m high is located 1 m away from a wall. The length of the shortest ladder that can extend from the wall over the fence to a poiut on the ground is __(11)__.

(a)=____ (12)____.

(b)=__(13)__.

(c) =__(14)__.

(a) dA = __(15)__ , where D is the region in the first quedrant that lies betwoen the circles x² + y² = 4and (x-1)²+y²=1.

(b) The volume of the solid R lying below the plane z= 3 -2y and above the paraboloid z = x²+ y² is __(16)__.

(a) C is a smooth curve in the upper half plane going from (1,1) to . =__(17)__.

(b) S is the part of the spherex² + y² + z² = 9 that lies above the cone z = with upward orientation. F = __(18)__.

(a)=__ (19)__.

(b) The Maclaurin series (the 'Taylor series at a = 0) for x is__(20)__.

1. Find the equation of the curve on thexy-plane that passes through (3,1) such that for any point P on the curve, the midpoint of the tangent line at P that lies in the first quadrant is P itself.

(e) Find.

(b) Assume that on another curve , f obtains local naximumn at (Jc0,yo) which is near

. Use linear epproimation to estimate