113年 - 113 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#147587

科目：研究所、轉學考(插大)-微積分 | 年份：113年 | 選擇題數：0 | 申論題數：8

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 (8)

1. Evaluate the following integrals.
(a) $\int\frac{xe^{2x}}{(2x+1)^{2}}dx$
(b) $\int_{1}^{3}\frac{3}{x^{2}-3x}dx$
(c) $\int_{-\infty}^{\infty}(x-u)\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-u)^{2}}{2\sigma^{2}}}dx$
2. Evaluate the double integral $\int{0}^{1}\int{y/2}^{1/2}e^{-x^{2}}dx~dy$
Note that it is necessary to change the order of integration.
3. A conservation organization releases 100 animals of an endangered species into a game preserve. During the first 2 years, the population increases to 103 animals. The organization believes that the preserve has a capacity of 200 animals and that the herd will grow according to a logistic growth model. That is, model assumes that the rate of growth is proportional to both the quantity y and the difference between the quantity and the limit L. (30%)
(a) Write a differential equation where the size y of the herd will follow the equation and t is measured in years. (10%)
(b) Then derive this logistic curve, find / solve for the constant of integration and the proportionality constant. (10%)
(c) How long will it take to establish a population of 150? (10%)
4. Find an equation of the tangent line to the graph of the function given below at the given point, x=0, and
$f(x)=\left(\frac{x+5}{x-1}\right)(2x+1)$