114年 - 114 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#144160

1. Determine the vertical and slant (oblique) asymptote of \(f(x) = \frac{x^2-2x+4}{x-2}\).

2. Differentiate both forms of \( y = \frac{1-\cos x}{\sin x} = \csc x - \cot x \).

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

4. Given \(x^2 + y^2 = 25\), find \(\frac{d^2y}{dx^2}\).

5. Find the derivative of \(F(x) = \int_{\pi/2}^{x^2} \cos t \, dt\).

6. Find the indefinite integral \(\int \frac{1}{x\sqrt{4-(\ln(x))^2}}dx.\)

7. Find the indefinite integral \(\int \frac{1}{x^2(1+x)} dx\).

8. Find the improper integral \(\int_{0}^{1} x^3 (\ln(x))^2 dx\).

9. Find the sum of the series \(\sum_{n=1}^{\infty} \frac{1}{(3n-2)(3n+1)}\).

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