113年 - 113 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#146749

1. (10%) Suppose the function $f(x) = \frac{1}{3}x^3 - ax^2 + bx + c$ has extreme values at $x = 1, \frac{1}{3}$ and f(3) = 5. Find a, b, c.

2. (10%) Find the local extrema of $f(x) = \frac{12}{5}x^5 + \frac{23}{4}x^4 - 4x^3 - \frac{17}{2}x^2 + 6x + 3$.

A. $\lim_{x \to 0} \left(\frac{1}{x^3}\right)^x$

B. $\lim_{y \to \pi/2} \left(\frac{\pi}{3} - y\right) \cot(y)$

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

D. $\lim_{x \to 3} \frac{x-3}{x^2-9}$

A. $\int_0^2 x \sqrt{16 - x^2} \, dx$

B. $\int x \cos x \, dx$

C. $\int \frac{e^x + e^{-x}}{e^x - e^{-x}} \, dx$

D. $\int \frac{3x-2}{3x^2+x-2} \, dx$

5. (10%) Find the tangent line of $x^3 - 2x^2y + 2xy^3 = 1$ passing through (1,1).

6. (15%) Find $\iint_R x + y \, dA$, where $R = \{(x, y) | 0 \le 2y \le x + 4, x \le y \le x - 1\}$.

A. $\sum_{n=0}^{\infty} (x+1)^n$

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

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