110年 - 110 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：微積分#146593

1. Find the derivative of \( y = \left( \frac{3x - 1}{x^2 + 3} \right)^2 \). (10%)

2. Determine the slope of the graph of \( 3(x^2 + y^2)^2 = 100xy \) at the point (3,1). (10%)

3. Evaluate \( \int_{0}^{1} \arcsin(x) \, dx \). (10%)

4. Find the volume of the solid formed by revolving the region bounded by the graphs of \( y = x^2 + 1 \), \( y = 0 \), \( x = 0 \) and \( x = 1 \) about the \( y \)-axis. (10%)

5. Find \( \lim_{x \to 0^+} (\sin x)^x \). (10%)

6. Find the area of the region bounded by the graphs \( f(x) = 2 - x^{2} \) and \( g(x) = x \). (10%)

7. Determine the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{e^{2n}}{n^n} \). (10%)

Find the indefinite integral \( \int \sqrt{16 - 4x^2} \, dx \). (10%)

9. Find the arc length of the graph of the function \( y = \frac{x^3}{6} + \frac{1}{2x} \) over the interval \( [\frac{1}{2}, 2] \). (10%)

10. Determine the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n}n}{\ln(n+1)} \). (10%)