108年 - 108 國立臺北大學_日間學士班轉學生招生考試試題_通訊工程學系、資訊工程學系日間學士班2年級：微積分#147501

科目：研究所、轉學考(插大)-微積分 | 年份：108年 | 選擇題數：0 | 申論題數：8

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (8)

1. Please evaluate the integrals:

(a) \(\int x^{2}e^{x}dx\) (10%)
(b) \(\int x e^{x^2} dx\) (10%)
(c) \(\int \frac{1}{x\sqrt{x^2 + 4}} dx\) (15%)
(d) \(\iint_{D} x + 2ydA\), where \(D\) is the region bounded by the parabolas \(y = 2x^{2}\) and \(y = 1 + x^{2}\). (15%)
2. Please sketch the curve \( y = \frac{2x^2}{x^2 - 1} \). (You must describe the domain, intercepts, symmetry, asymptotes, interval of increasing or decreasing, local maximum and minimum values, and concavity and points of inflection.) (15%)
3. If \( f(x, y) = \frac{xy}{x^2 + y^2} \), does \( \lim_{(x, y) \to (0, 0)} f(x, y) \) exists? Why? (10%)
4. A rectangular box without a lid is to be made from \(12\mathrm{m}^2\) of cardboard. Please use the method of Lagrange multiplier to find the maximum volume of such a box.(10%)
5. Find the radius of convergence and interval of convergence of the series \(\sum_{n=0}^{\infty} \frac{(-3)^n x^n}{\sqrt{n+1}}\). (15%)