PastExamLabPastExamLab

台灣大學 · 數學系 · 轉學考考古題 · 民國113年（2024年）

113 年度 微積分(A)

台灣大學 · 數學系 · 轉學考

PDF
第 1 題15 分
Consider the sequence of functions {fn}n=1∞\{f_n\}_{n=1}^{\infty}, where fn(x)=x1+nx2f_n(x) = \frac{x}{1+nx^2}. Does fnf_n converge uniformly on R\mathbb{R}? Justify your result.
第 2 題15 分
Show that ∑m,n=1∞(m+n)−p\sum_{m,n=1}^{\infty}(m+n)^{-p} converges if and only if p>2p > 2.
第 3 題20 分
Let f(x,y):R2→Rf(x,y): \mathbb{R}^2 \to \mathbb{R} be a C1C^1 function. Show that we can find two points p,q∈R2p,q \in \mathbb{R}^2 so that p≠qp \neq q and f(p)=f(q)f(p) = f(q).
第 4 題15 分
Assume {an}n=1∞\{a_n\}_{n=1}^{\infty} are positive, and lim⁡n→∞an+1an=L\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} = L. Show that lim⁡n→∞(an)1/n=L\lim\limits_{n\to\infty}(a_n)^{1/n} = L.
第 5 題15 分
Let SS be the surface formed by paraboloid z=1−x2−y2,z≥0z = 1 - x^2 - y^2, z \geq 0, and the unit disk centered at the origin in xyxy plane. Let F=(0,0,z)F = (0,0,z). Compute ∬SF⋅nds\iint_S F \cdot n ds, where nn is the unit outward normal vector on SS.
第 6 題20 分
Consider the integral g(x)=∫0∞sin⁡tte−txdtg(x) = \int_0^{\infty} \frac{\sin t}{t} e^{-tx} dt. Show that the integral converges uniformly for x≥0x \geq 0.
廣告區域 (Google AdSense)