114年 - 114 國立臺北大學_碩士班一般入學考試試題_都市計畫研究所：微積分#146750

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

2. (10%) Find the local extrema of $f(x) = \frac{1}{5}x^5 - \frac{1}{2}x^4 - x^3 + 4x^2 - 4x + 3$ for $x \in (-3, 3)$.

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

B. \( \lim_{x\to \pi /2}x\tan (x) \)

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

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

A. \( \int_0^{\sqrt[3]{7}}x^2\sqrt{16 - x^3} dx \)

B. \( \int x\sin x dx \)

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

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

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

6. (15%) Find \( \iint_{R} 1 + 2y \, dA \), where \( R = \{(x, y) | 0 \leq 2y \leq x + 2, x \leq y \leq x - 1\} \).

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

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

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