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

科目：研究所、轉學考(插大)-微積分 | 年份：107年 | 選擇題數：0 | 申論題數：10

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所屬科目：研究所、轉學考(插大)-微積分

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申論題 (10)

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 $$\nabla f(x, y)$$.