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台灣大學 · 數學系 · 轉學考考古題 · 民國112年（2023年）

112 年度 微積分(A)

台灣大學 · 數學系 · 轉學考

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第 1 題10 分
Let f:[a,b]→Rf : [a, b] \to \mathbb{R} be a one-to-one and continuous function. If f(a)<f(b)f(a) < f(b), show that ff is increasing.
第 2 題20 分
Let Ω\Omega be the domain in the xyxy-plane bounded by the xx-axis and one arc of the cycloid (x,y)=(θ−sin⁡θ,1−cos⁡θ)(x,y) = (\theta - \sin \theta, 1 - \cos \theta), θ∈[0,2π]\theta \in [0, 2\pi]. Find the volume of the solid {(x,y,z)∈R3:0<z<y2,(x,y)∈Ω}\{(x, y, z) \in \mathbb{R}^3 : 0 < z < y^2, (x,y) \in \Omega\}.
第 3 題20 分
Suppose that the cubic equation a0+b0x+c0x2+d0x3=0a_0 + b_0 x + c_0 x^2 + d_0 x^3 = 0 has three distinct roots α0<β0<γ0\alpha_0 < \beta_0 < \gamma_0. Given ϵ>0\epsilon > 0, prove that there exists δ>0\delta > 0 such that for every (a,b,c,d)∈R4(a, b, c, d) \in \mathbb{R}^4 with (a−a0)2+(b−b0)2+(c−c0)2+(d−d0)2<δ2(a - a_0)^2 + (b - b_0)^2 + (c - c_0)^2 + (d - d_0)^2 < \delta^2, the equation a+bx+cx2+dx3=0a + bx + cx^2 + dx^3 = 0 has three roots α\alpha, β\beta, and γ\gamma satisfying (α−α0)2+(β−β0)2+(γ−γ0)2<ϵ2(\alpha - \alpha_0)^2 + (\beta - \beta_0)^2 + (\gamma - \gamma_0)^2 < \epsilon^2.
第 4 題20 分
For each ϵ∈(0,1)\epsilon \in (0, 1), let Sϵ={(x,y,z)∈R3:ϵ<x2+y2<1,−1+ϵ<z<1}S_\epsilon = \left\{(x, y, z) \in \mathbb{R}^3 : \epsilon < \sqrt{x^2 + y^2} < 1, -1 + \epsilon < z < 1\right\}. Evaluate lim⁡ϵ→0+∭Sϵzx2+y2ln⁡(1+x2+y2)dxdydz\lim\limits_{\epsilon \to 0^+} \iiint_{S_\epsilon} \frac{z}{\sqrt{x^2 + y^2} \ln(1 + x^2 + y^2)} dx dy dz.
第 5 題30 分
Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a bounded and continuous function. For every k∈Nk \in \mathbb{N} and x∈Rx \in \mathbb{R}, define fk(x)=min⁡y∈R(f(y)+k∣y−x∣)f_k(x) = \min_{y \in \mathbb{R}} (f(y) + k|y - x|).
(a)10 分
Show that fk(x)f_k(x) is well-defined (i.e., the minimum does exist).
(b)10 分
Show that fk:R→Rf_k : \mathbb{R} \to \mathbb{R} is Lipschitz continuous.
(c)10 分
Prove that on every finite interval, fkf_k converges to ff uniformly as k→∞k \to \infty.
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