109年 - 109 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#146412

1. Find the vertical asymptotes (if any) of the graph of the function \( f(x) = \frac{x - 1}{x² - 1} \). (10 分)

2. Find \( f'(x) \) where \( f(x) = \sin^4(3x^2) \). (10 分)

3. Evaluate \( \displaystyle \frac{d}{dx} \left[ \int_{x}^{x^2+2} \frac{t}{1+t^2} dt \right] \). (10 分)

4. Find the volume of the solid of revolution formed by revolving the region bounded by \( y = \sqrt{x} \) and \( y = x^2 \) about the x-axis. (10 分)

5. Find the indefinite integral \( \int \frac{1}{\sqrt{1 + x^2}} dx \). (10 分)

6. Prove that \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^2} \) converges absolutely. (10 分)

7. Prove that both of \( \int_{1}^{\infty} \frac{1}{\sqrt{x}} dx \) and \( \int_{0}^{1} \frac{1}{\sqrt{x}} dx \) diverge. (10 分)

8. Compute the integral \( \int_{R} (6x^{2} + 4xy^{3})dA \) where \( R = \{(x,y) | 0 \leq x \leq 2, 1 \leq y \leq 4\} \). (10分)

9. Find the interval of convergence of the series \( \sum_{k=1}^{\infty} \frac{\sin(k^3 x)}{k^2} \). (10 分)

10. Given \( x > 0 \). Compute the value of \( \int_{0}^{1/x} \frac{1}{t^2 + 1} dt + \int_{0}^{x} \frac{1}{t^2 + 1} dt \). (10 分)