112年 - 112 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#144163

1. Find the limit if it exists. \[\lim_{x\to 0} \frac{\sin x - \sin x \cos x}{x^2}\]

2. Find \(\frac{d}{dx}\int_{x}^{1}\sqrt{1+t^4}dt\).

3. Evaluate \[\int_{1}^{2e} \frac{(1+\ln x)^2}{x} dx.\]

4. Find \(\frac{dy}{dx}\) for \(x^2 - 3\ln y + y^2 = 10\) by implicit differentiation.

5. Find the indefinite integral. \(\int \frac{1}{1+e^{-x}} dx\)

6. Determine the convergence or divergence of the series \(\sum_{n=0}^{\infty} \frac{2^n}{n!}\).

7. Find the second derivative of \( f(x) = 5e^{-x} - 2e^{-5x} + x \ln \sqrt{x} \).

8. Evaluate the definite integral \(\int_{2}^{8} |3x - 9| dx\).

9. Find the area of the region bounded by the graphs of equations \(f(x) = x^2 - 4x + 3\) and \(g(x) = 3 + 4x - x^2\).

10. Evaluate the double integral \[\int_{-1}^{2}\int_{-3}^{3} (x^2 - xy^2) dy dx.\]