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台灣大學 · 醫學檢驗暨生物技術學系 · 轉學考考古題 · 民國112年（2023年）

112 年度 微積分(C)

台灣大學 · 醫學檢驗暨生物技術學系 · 轉學考

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PART 1: Fill in the blanks

Please ensure that each answer is clearly labeled with the corresponding blank number. Please note that only the final answers will be graded, and each blank is worth 5 points.
第 1 題5 分
Suppose that f(x)f(x) is differentiable at x=1x = 1. Evaluate the following limit in terms of f(1)f(1) and f′(1)f'(1).
lim⁡x→0f(e2x)−f(1)log⁡2(1−3x)=(1)‾.\lim\limits_{x \to 0} \frac{\sqrt{f(e^{2x})} - \sqrt{f(1)}}{\log_2(1 - 3x)} = \underline{\quad (1) \quad}.
第 2 題10 分
Suppose that
3+yx3−xy−1=0.\sqrt{3 + y x^3 - xy - 1} = 0.
At (x,y)=(1,1)(x,y) = (1,1), dxdy=(2)‾\frac{dx}{dy} = \underline{\quad (2) \quad}. By the linear approximation, we can approximate the real root of 4.1x2−1.1x−1=0\sqrt{4.1 x^2 - 1.1x - 1} = 0 with (3)‾\underline{\quad (3) \quad}.
第 3 題5 分
Consider the curve satisfying 5x2+2xy+y2=165x^2 + 2xy + y^2 = 16. The higest point of the curve (point with the largest yy coordinate) is (4)‾\underline{\quad (4) \quad}.
第 4 題5 分
Suppose that f(u)>0f(u) > 0. Let
F(x)=∫0x2∫4uf(u) du dt.F(x) = \int_0^{x^2} \int_4^u f(u) \, du \, dt.
On what intervals is F(x)F(x) increasing? (5)‾\underline{\quad (5) \quad}
第 5 題10 分
Let f(x)=x3+2x+1f(x) = x^3+2x+1 and g(x)=f−1(x)g(x) = f^{-1}(x), the inverse function of f(x)f(x). Then g′(4)=(6)‾g'(4) = \underline{\quad (6) \quad} and ∫24g(x) dx=(7)‾\int_2^4 g(x) \, dx = \underline{\quad (7) \quad}.
第 6 題5 分
Use the Maclaurin series of sin⁡(x2)x\frac{\sin(x^2)}{x} to write the integral as the sum of an infinite series.
∫012sin⁡(x2)x dx=(8)‾.\int_0^{\frac{1}{2}} \frac{\sin(x^2)}{x} \, dx = \underline{\quad (8) \quad}.
第 7 題15 分
Assume that
∣f(x,y)−1+2y∣≤sin⁡(x2+y2)for x2+y2≤1.|f(x,y) - 1 + 2y| \leq \sin(x^2 + y^2) \quad \text{for } x^2 + y^2 \leq 1.
Then the tangent plane of y=f(x,y)y = f(x,y) at (0,0,f(0,0))(0,0,f(0,0)) is (9)‾\underline{\quad (9) \quad}. The tangent line of the level curve f(x,y)=f(0,0)f(x,y) = f(0,0) at (0,0)(0,0) is (10)‾\underline{\quad (10) \quad}. The maximum value of directional derivatives of ff at (0,0)(0,0), Duf(0,0)D_u f(0,0), is (11)‾\underline{\quad (11) \quad}.
第 8 題5 分
Find critical points of f(x,y)=−2x4+x2y−y2+7yf(x,y) = -2x^4 + x^2 y - y^2 + 7y and indicate whether they are local maximum, local minimum, or saddle points. (12)‾\underline{\quad (12) \quad}.
第 9 題10 分
a ∫0π4∫sin⁡−1yπ4cos⁡x1+cos⁡2x dxdy=(13)‾\int_0^{\frac{\pi}{4}} \int_{\sin^{-1} y}^{\frac{\pi}{4}} \frac{\cos x}{\sqrt{1 + \cos^2 x}} \, dx dy = \underline{\quad (13) \quad}. b ∫02∫02x−x2x2+y2 dydx=(14)‾\int_0^2 \int_0^{\sqrt{2x-x^2}} \sqrt{x^2 + y^2} \, dy dx = \underline{\quad (14) \quad}.

PART 2:

Please solve the following problems and provide explanations and computations. Partial credits are allocated according to the level of completeness in your work.
第 1 題20 分
f(x)={2e−x(1+e−x)2,if x≥0,0,if x<0.f(x) = \begin{cases} \frac{2e^{-x}}{(1+e^{-x})^2}, & \text{if } x \geq 0, \\ 0, & \text{if } x < 0. \end{cases} is the probability density function of a random variable XX.
(a)10 分
Sketch the graph of f(x)f(x), indicating intervals of increase/decrease, inflection point(s), and the horizontal asymptote.
(b)10 分
Evaluate the expected value of XX which is ∫−∞∞xf(x) dx\int_{-\infty}^{\infty} x f(x) \, dx.
第 2 題10 分
The plane x+2y+z=2x + 2y + z = 2 intersects the cone y=x2+z2y = x^2 + z^2 in an ellipse. Find the points on the ellipse that are nearest and farthest from the origin.
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