110年 - 110 國立臺北大學_碩士班一般入學考試試題_財政學系：微積分#146910

1. (10%) Find the limit: \( \lim_{x\to 1}\frac{x^{1 / 3} - 1}{\sqrt{x} - 1} \).

2. (10%) Find the limit of \( g(x) \) as \( x \) approaches the indicated values:
$\lim_{x \to \sqrt{5}} \frac{1}{x+g(x)} = 2.$

3. (10%) Find \( dy / dx \) by implicit differentiation: \( y^2 = \sqrt{\frac{1 + x}{1 - x}} \).

4. (10%) Show that the function \( f(x) = x + \sin^2\left(\frac{x}{3}\right) - 8 \) has exactly one zero in the interval \( (-\infty, \infty) \).

5. (15%) Graph the function: \( y = \frac{\sqrt{1 - x^2}}{2x + 1} \). (Find where the curve is increasing and where it is decreasing. Find the local extreme points. Find the points of inflection, if any occur, and determine the concavity of the curve. Identify any asymptotes.)

6. (10%) Find the area of the triangular region bounded on the left by \( x + y = 2 \), on the right by \( y = x^2 \), and above by \( y = 2 \).

7. (10%) Find the derivative of \( y \) with respect to \( x \) if \( y = 2(\ln x)^{x/2} \).

8. (10%) Evaluate \( \int x^{2} \sin(1 - x) dx \).

9. (15%) Sketch the region of integration and write an equivalent integral with the order of integration reversed. Then evaluate both integrals. $$\int_0^{3/2} \int_{-\sqrt{9-4y^2}}^{\sqrt{9-4y^2}} y \, dx \, dy.$$