105年 - 105 國立臺北大學_日間學士班曁進修學士班轉學生招生考試試題_資訊工程學系、通訊工程學系日間學士班2年級：微積分#147499

(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%)