107年 - 107 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#145645

1. Describe Part I and Part II of the Fundamental Theorem of Calculus in detail.

2. Show that $f(x) = \frac{|x^2 - 1|}{x - 1}$ is not differentiable at x = 1.

3. Evaluate $\frac{d}{dx} \left[ \int_0^{x^3} \sqrt{\ln(t^2 + 3)} dt \right]$

4. Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{4n^2 - 1}$.

5. Evaluate $\iint_R (2x + y) dA$, where $R = \{(x,y) : 1 \le x \le 5, 2 \le y \le 6\}$

6. Suppose that $4x^2 + 3\sin(x + y) + 6\ln(y^2 + 1) = 1$. Find $\frac{dy}{dx}$.

7. Find the arc length of the graph of $y = \frac{1}{6}x^3 + \frac{1}{2x}$ on the interval $\left[\frac{1}{2}, 2\right]$

8. Decide whether the series $\sum_{n=1}^{\infty} \frac{e^{2n}}{n^n}$ converges or diverges.

9. If $F(x) = f(3f(4f(x)))$, where $f(0) = 0$ and $f'(0) = 2$, find $F'(0)$.

10. Let $f(x, y) = y^2 e^{\sin x} \ln(x + y)$. Find $\nabla f(x, y)$.