114年 - 114 國立臺北大學_日間學士班轉學生招生考試試題_統計學系日間學士班2、3年級：微積分#147569

(a) $\int \ln \left(\frac{x}{x + 1}\right)dx$

(b) $\int_{1}^{3}\frac{3}{x^{2} - 3x} dx$

(c) $\int x^{n} \ln x \, dx, n \neq -1$

(b) Write down the differential equation. (5%)

(c) Solve it to derive the general form of the logistic growth model. (10%)

(d) The proportion of U.S. households that own a cell phone relates to the year, with a limit of 0.95. Let \( t = 0 \) represent 2000, \( t = 1 \) represent 2001, and so on. If the proportion of U.S. households that own a cell phone in 2000 is \( 23.75\% \), and in 2005 the proportion is \( 59.16\% \), find a logistic growth model for this data. (10%)

3. Analyze the graph of this model \( f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{\frac{(x - \mu)^2}{2\sigma^2}}, - \infty \leq x \leq \infty \), for inflection points. (25%)

4. Find the volume of the solid region bounded in the first octant by the plane \( z = 2 - x - 2y \). (20%)