107年 - 107 國立臺北大學_日間學士班曁進修學士班轉學生招生考試試題_資訊工程學系日間學士班2年級：微積分#147397

一、 1. (12%) Compute \( \int_0^2 tan^{-1}xdx \)

2. (12%) Find the area enclosed by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \).

3. (12%) Compute \( \int_0^2\ln x dx \)

4. (14%) Use Simpson's rule with \( n = 10 \) to approximately compute \( \int_{1}^{2\frac{1}{x}} dx \).

二、 1. \( (8\%) \) Prove that \( \lim_{x\to 2}(3x - 1) = 5 \)

2. \( (8\%) \) Show that the function \( f(x) = \sqrt{4 - x^2} \) is continuous on the interval \( [-2, 2] \).

3. \( (8\%) \) Show that the function \( f(x) = x^{2} - x + 1 \) is differentiable for \( x \in \mathbb{R} \).

4. (8%) Suppose that \( f(0) = 0 \) and \( f'(x) \geq 6 \) for \( x \in \mathbb{R} \). How small can \( f(100) \) possibly be? (Hint: the Mean Value Theorem)

5. \( (6\%) \) Find \( y^{\prime} \) if \( x^4 +y^4 = 8xy^2 \)

6. \( (6\%) \) Find \( y^{\prime} \) if \( y = \cos (x\cos x) \)

7. \( (6\%) \) Find \( y^{\prime} \) if \( y = \frac{x}{x + \frac{a}{x}} \)