111年 - 111 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#146745

1. (10%) Suppose the function $f(x) = 3x^3 - ax^2 + bx + c$ has extreme values at $x = 1, \frac{1}{3}$ and f(2) = 0. Find a, b, c.

A. \( \lim_{x\to 0}\left(\frac{1}{x^2}\right)^x \)

B. \( lim_{y\to \pi /2}\left(\frac{\pi}{2} -y\right)\tan y \)

C. \( lim_{t\to 1}\frac{t - 1}{\ln t - \sin\pi t} \)

D. \( lim_{x\to 2}\frac{x - 2}{x^2 - 4} \)

A. \( \int_0^2 x\sqrt{16 - 3x^2} dx \)

B. \( \int x\cos 2xdx \)

C. \( \int \frac{e^x + e^{-x}}{e^x - e^{-x}} dx \)

D. \( \int \frac{2x + 1}{3x^2 + x - 2} dx \)

4. (10%) Find the local extrema of \( f(x) = \frac{12}{5} x^5 + \frac{5}{4} x^4 - \frac{14}{3} x^3 - \frac{5}{2} x^2 + 2x + 3 \), where \( -1 \leq x \leq 1 \).

5. (10%) Find the tangent line of \( x^4 - x^2y + 2xy^4 = 4 \) passing through (1,1).

6. (10%) Find \( \iint_{R} x + y \, dA \), where \( R = \{(x, y) | 0 \leq 2y \leq x + 4, x \leq y \leq x - 2\} \).

A. \( \sum_{n=0}^{\infty} x^n \)

B. \( \sum_{n=0}^{\infty} \frac{x^{2n+1}}{n!} \)

C. \( \sum_{n=0}^{\infty} \frac{(4x - 5)^{2n+1}}{n^{3/2}} \)

D. \( \sum_{n=0}^{\infty} (-2)^{n} (n + 1)(x - 1)^{n} \)