107年 - 107 國立臺北大學_碩士班一般入學考試試題_都市計畫研究所甲組：微積分#146754

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：15

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (15)

1. (10%) Find the domain and range of the following function.

A. \(f(x) = \sqrt{4 - 3x^2}\)

B. \(y = 1 + \sin x\)

2. (10%) Use the following graph of $f(x)$ to state the value of each quantity, if it exists. If it does not exist, explain why.

A. \(\lim_{x\to 1^{-}}f(x)\)

B. \(\lim_{x\to 1^{+}}f(x)\)

C. \(\lim_{x\to 1}f(x)\)

D. \(\lim_{x\to 5}f(x)\)

E. \(f(S)\)

3. (20%) Evaluate the following limit.

A. \(\lim_{x\to 0}\left(\frac{3x^3 - 2x}{3x^2 + x - 2} -x\right)\)

B. \(\lim_{x\to 1}\frac{\sin(x - 1)}{x^2 + x - 2}\)

4. (20%) Find $dy/dx$.

A. \(y = \sin x + 10\tan x\)

B. \(\sqrt{x} +\sqrt{y} = 1\)

5. (20%) Evaluate the following integrals.

A. \(\int x^{2} \ln |x| \, dx\)

B. \(\int_{3}^{6} x e^{4x^2 + 1} dx\)

6. (10%) Find the local extrema of \( f(x) = x^{2}e^{-x} \).

7. (10%) Find the volume of the solid that is bounded by the elliptic paraboloid \( x^{2} + 2y^{2} + z = 16 \), the planes \( x = 2 \), \( y = 2 \), and the three coordinate planes.