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台灣大學 · 數學系 · 轉學考考古題 · 民國114年（2025年）

114 年度 微積分(A)

台灣大學 · 數學系 · 轉學考

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第 1 題15 分
Determine whether
∫0∞(sin⁡2(π(x+1x))−12)dx\int_0^\infty \left(\sin^2\left(\pi\left(x + \frac{1}{x}\right)\right) - \frac{1}{2}\right) dx
converges or diverges.
第 2 題15 分
Find the least value aa such that
(1+1x)x+a>e\left(1 + \frac{1}{x}\right)^{x+a} > e
for every x>0x > 0.
第 3 題20 分
Let f0(x)f_0(x) be a continuous function on [0,2π][0, 2\pi]. Define a sequence of functions on [0,2π][0, 2\pi] by
fn(x)=∫0xfn−1(s)sin⁡(s)ds.f_n(x) = \int_0^x f_{n-1}(s) \sin(s) ds.
(a)10 分
For every x∈[0,2π]x \in [0, 2\pi], determine lim⁡n→∞fn(x)\lim\limits_{n \to \infty} f_n(x).
(b)10 分
Is the convergence in part (a) uniform on [0,2π][0, 2\pi]? Justify your answer.
第 4 題20 分
Fix positive numbers a,b,ca, b, c. For any (x,y,z)(x, y, z) on the ellipsoid
x2a2+y2b2+z2c2=1 with x>0,y>0,z>0,\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \text{ with } x > 0, y > 0, z > 0,
let A(x,y,z)A(x, y, z), B(x,y,z)B(x, y, z) and C(x,y,z)C(x, y, z) be the intercept that the tangent plane of the ellipsoid at (x,y,z)(x, y, z) makes on the xx-axis, the yy-axis and the zz-axis, respectively. Find the minimum of A(x,y,z)+B(x,y,z)+C(x,y,z)A(x, y, z) + B(x, y, z) + C(x, y, z).
第 5 題15 分
Let UU be the region in R2\mathbb{R}^2 defined by r≤1+cos⁡θr \leq 1 + \cos \theta in polar coordinate. Let γ\gamma be the boundary of UU; namely, γ\gamma is given by r=1+cos⁡θr = 1 + \cos \theta in polar coordinate. Does there exist a second continuously differentiable function f(x,y)f(x, y) on R2\mathbb{R}^2 such that
∂2f∂x2+∂2f∂y2=2x+3y2 at every (x,y)∈U, and\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 2x + 3y^2 \text{ at every } (x, y) \in U, \text{ and}
(∂f∂x,∂f∂y)=(4x2−3y2+x,y3−6y) at every (x,y)∈γ?\left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) = (4x^2 - 3y^2 + x, y^3 - 6y) \text{ at every } (x, y) \in \gamma ?
If your answer is YES, construct such a function. If your answer is NO, give a proof.
第 6 題15 分
Let Σ\Sigma be the part of the paraboloid z=x2+y2z = x^2 + y^2 that lies within the cylinder x2+y2=1x^2 + y^2 = 1. Let n\mathbf{n} be its upward unit normal field. For the vector field
F=(2x+ey+z2,ln⁡(z+2)−y,2),\mathbf{F} = (2x + e^y + z^2, \ln(z + 2) - y, 2),
compute the flux integral ∬ΣF⋅ndS\iint_\Sigma \mathbf{F} \cdot \mathbf{n} dS.
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