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台灣大學 · 農業經濟學系 · 轉學考考古題 · 民國105年（2016年）

105 年度 微積分(C)

台灣大學 · 農業經濟學系 · 轉學考

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第 1 題15 分
Assume a (real valued) function y=f(x)y = f(x) satisfy y5+9y=x3+xy^5 + 9y = x^3 + x. Suppose g(x)g(x) is a (real valued) function satisfying f(g(x))=xf(g(x)) = x.
(a)5 分
The value g(1)g(1) must be 2. True or false? Answer: __(1)__.
(b)10 分
The derivative g′(1)=g'(1) = __(2)__.
第 2 題15 分
An experiment detects that a particle at (1,0)(1, 0) (on the xyxy-plane) is moving towards the north at the speed 3 meters per minute. At the same time another particle at (−2,1)(-2, 1) is moving towards the east at the speed 4 meters per minute.
(a)5 分
Let s(t)s(t) be the distance (in meters) between the two particles in tt minutes. s(t)=s(t) = __(3)__.
(b)10 分
Suppose at t=t0t = t_0 the two particles are closest to each other. Then t0=t_0 = __(4)__.
第 3 題10 分
Compute the indefinite integral ∫dx(2x−1)(x+1)=\int \frac{dx}{(2x-1)(x+1)} = __(5)__.
第 4 題10 分
Let Ω\Omega be a region (on the xyxy-plane) enclosed by x=lnyx = \sqrt{ln y}, x=0x = 0 and y=ey = e. Let SS be the solid obtained by revolving Ω\Omega about the yy-axis. Let VV be the volume of SS. V=V = __(6)__.
第 5 題20 分
Let f(x,y)=x2−e3yf(x,y) = x^2 - e^{3y} and the surface SS be the graph of the function z=f(x,y)z = f(x,y). Let P=(1,0,0)∈SP = (1,0,0) \in S and p=(1,0)p = (1,0) in the xyxy-plane.
(a)10 分
If the unit vector uu (in the xyxy-plane) at pp is the direction (among all directions at pp) along which the height (i.e. the value of zz) of SS increases most rapidly, then u=u = __(7)__.
(b)10 分
Write HH for the plane x+2y+3z=1x + 2y + 3z = 1 and the curve CC for the intersection S∩HS \cap H. Let LL be the tangent line to CC at PP and NN be the plane perpendicular to LL at PP. Then the equation of NN is __(8)__.
第 6 題15 分
Let RR be the region enclosed by y=xy = x, y=x−2y = x - 2, y=1y = 1 and y=0y = 0.
(a)10 分
Let AA be the area of RR. A=A = __(9)__.
(b)5 分
The double integral ∬Rx−y dxdy=\iint_R \sqrt{x - y} \, dx dy = __(10)__.
第 7 題15 分
Let Ω\Omega be the region {(x,y)∣(x−1)2+y2<1}\{(x,y) | (x-1)^2 + y^2 < 1\}. Evaluate the double integral ∬Ω1x dxdy\iint_\Omega \frac{1}{x} \, dx dy.
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