113年 - 113 國立臺北大學_日間學士班轉學生招生考試試題_經濟學系日間學士班2年級：微積分#147888

(a) \( (5\%) \) Find \( \lim_{x\to 0}\frac{x}{\sqrt{x + 1} - 1} \)

(b) \( (5\%) \) Find \( \lim_{x\to -\infty}\frac{2^x + 5(3^x)}{3(2^x) - 3^x} \).

(c) \( (5\%) \) Given \( z = ay^2 \) and \( y = bx^2 + cx \), find \( dz / dx \).

(d) \( (5\%) \) Given \( y = x^{5} + x \), find \( dx / dy \).

(e) \( (5\%) \) Given \( y = (x^{2} + 3)e^{x^{2} + 1} \), find \( dy / dx \).

(f) \( (5\%) \) Given \( y = \frac{x_1 + x_2}{2x_1^2} \), find the total differential of \( y \).

(g) \( (10\%) \) Given \( F(x,y,z) = x^{2}y^{3} + z^{3} + xyz - 3 = 0 \), find \( \partial y / \partial x \) and \( \partial z / \partial x \) at the point \( (x,y,z) = (1,1,1) \).

(h) \( (10\%) \) Approximate \( e^x \) around \( x = 1 \) with a Taylor polynomial of degree four.

(a) \( (6\%)\int_{0}^{10}4xe^{-2x}dx. \)

(b) \( (6\%)\int_{-1}^{3}\int_{0}^{1}(x^{2}y + xy^{2} + 2x)dxdy. \)

(c) \( (8\%)\int_{0}^{1 / 2}\int_{0}^{2\pi}y^{3}\sin (xy^{2})dxdy. \)

3. \( (10\%) \) Sketch the function \( f(x) = x^{3} - 12x^{2} + 36x + 8 \) and find the critical values.

4. (10%) Find the absolute maximum and minimum points of \( f(x, y) = x^2 + 3y - 3xy \) over the region bounded by
$y = x, y = 0, \text{ and } x = 2.$

5. (10%) Find the points on the surface $x^2 - yz = 5$ that are closest to the origin.