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台灣大學 · 數學系 · 轉學考考古題 · 民國107年（2018年）

107 年度 微積分(A)

台灣大學 · 數學系 · 轉學考

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第 1 題20 分
Find an explicit value of ϵ>0\epsilon > 0 such that for every x∈[0,1]x \in [0,1] ∣x−x+ϵ∣<1200|\sqrt{x} - \sqrt{x+\epsilon}| < \frac{1}{200}. Find an explicit integer NN such that there exists a polynomial PP of degree at most NN such that for every x∈[0,1]x \in [0,1] ∣x−P(x)∣<1100|\sqrt{x} - P(x)| < \frac{1}{100}. Hint: You can use the expansion of x+ϵ\sqrt{x+\epsilon} in power series in (x−1)(x-1).
(a)10 分
Find an explicit value of ϵ>0\epsilon > 0 such that for every x∈[0,1]x \in [0,1] ∣x−x+ϵ∣<1200|\sqrt{x} - \sqrt{x+\epsilon}| < \frac{1}{200}.
(b)10 分
Find an explicit integer NN such that there exists a polynomial PP of degree at most NN such that for every x∈[0,1]x \in [0,1] ∣x−P(x)∣<1100|\sqrt{x} - P(x)| < \frac{1}{100}. Hint: You can use the expansion of x+ϵ\sqrt{x+\epsilon} in power series in (x−1)(x-1).
第 2 題10 分
Let f:[0,1]↦Rf : [0,1] \mapsto \mathbb{R} be monotone increasing, i.e. if c<dc < d implies f(c)≤f(d)f(c) \leq f(d). Use the definition of the Riemann integral (comparing upper and lower sums relative to a partition of [0,1][0,1]) to prove that ∫01f(x)dx\int_0^1 f(x)dx exists.
第 3 題20 分
Evaluate ∫02∫02−xycos⁡(x2)dxdy\int_0^2 \int_0^{2-x} y\cos(x^2)dxdy. Let FF be a vector field F=(x4,y3,z4)F = (x^4, y^3, z^4) and Ω\Omega be the solid region in R3\mathbb{R}^3 bounded by x2+y2≥x4,x2+y2+z4≤9,y≥∣x∣x^2 + y^2 \geq x^4, x^2 + y^2 + z^4 \leq 9, y \geq |x|. Evaluate ∭Ω(x2+y2+z2)dV\iiint_\Omega (x^2+y^2+z^2)dV and find the flux of FF through the boundary surface of Ω\Omega, oriented outwards (i.e. ∬∂ΩF⋅ndS\iint_{\partial\Omega} F \cdot n dS).
(a)10 分
Evaluate ∫02∫02−xycos⁡(x2)dxdy\int_0^2 \int_0^{2-x} y\cos(x^2)dxdy
(b)10 分
Let FF be a vector field F=(x4,y3,z4)F = (x^4, y^3, z^4) and Ω\Omega be the solid region in R3\mathbb{R}^3 bounded by x2+y2≥x4,x2+y2+z4≤9,y≥∣x∣x^2 + y^2 \geq x^4, x^2 + y^2 + z^4 \leq 9, y \geq |x|. Evaluate ∭Ω(x2+y2+z2)dV\iiint_\Omega (x^2+y^2+z^2)dV and find the flux of FF through the boundary surface of Ω\Omega, oriented outwards (i.e. ∬∂ΩF⋅ndS\iint_{\partial\Omega} F \cdot n dS).
第 4 題10 分
Among all planes that are tangent to the surface xy2z2=1xy^2z^2 = 1, find the ones that are farthest from the origin.
第 5 題20 分
Let {an}n=1∞\{a_n\}_{n=1}^\infty be a sequence and bn=1n(∑k=1nk2ak)b_n = \frac{1}{n}(\sum_{k=1}^n k^2a_k). Prove or disprove: If {an}n=1∞\{a_n\}_{n=1}^\infty converges then {bn}n=1∞\{b_n\}_{n=1}^\infty converges. Prove or disprove: If {bn}n=1∞\{b_n\}_{n=1}^\infty converges then {an}n=1∞\{a_n\}_{n=1}^\infty converges.
(a)10 分
Prove or disprove: If {an}n=1∞\{a_n\}_{n=1}^\infty converges then {bn}n=1∞\{b_n\}_{n=1}^\infty converges.
(b)10 分
Prove or disprove: If {bn}n=1∞\{b_n\}_{n=1}^\infty converges then {an}n=1∞\{a_n\}_{n=1}^\infty converges.
第 6 題20 分
Find domain of convergence of ∑n=1∞(n+x)nnn+x\sum_{n=1}^\infty \frac{(n+x)^n}{n^{n+x}}. Does the series ∑n=1∞e−1n(−1)nn\sum_{n=1}^\infty e^{-\frac{1}{n}} \frac{(-1)^n}{n} converge uniformly on [0,∞)[0,\infty)? Prove or disprove.
(a)10 分
Find domain of convergence of ∑n=1∞(n+x)nnn+x\sum_{n=1}^\infty \frac{(n+x)^n}{n^{n+x}}.
(b)10 分
Does the series ∑n=1∞e−1n(−1)nn\sum_{n=1}^\infty e^{-\frac{1}{n}} \frac{(-1)^n}{n} converge uniformly on [0,∞)[0,\infty)? Prove or disprove.
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