(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.

(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 $f(x)=\left(\frac{x+5}{x-1}\right)(2x+1)$

1. (10 points) Upon retiring, you anticipate living for another 25 years. You'd like to make \$75,000 a year in income. If the annual interest rate is 9%, how much should you have saved in your retirement account to obtain this income? (Presume the payments begin a year after your retirement.)

2. (10 points) A giant crane is about to be purchased by Mass Company. It is anticipated to require a $6 million initial investment and to produce a $3 million end-of-year cash flow per year for three years. There will be a disposal fee of $1 million at the conclusion of the fourth year. If the project's cost of capital is 12 percent, determine the project's MIRR (modified internal rate of return).

3. (15 points) A project needs a $200,000 initial expenditure and anticipates generating $120,000 in annual cash flow before taxes for two years (cash flows will occur at t = 1 and t = 2). The rate of corporation taxation is 21%. Using the MACRS 3-year plan, the assets will depreciate as follows: (t = 1: 33%); (t = 2: 45%); (t = 3: 15%); (t = 4: 7%). The company's tax condition enables it to utilize all available tax shields. The opportunity cost of capital is 12 percent. Assume the asset can be sold for book value when the project is finished. Determine the project's NPV.

4. (15 points) An investment of $900 is needed for a project today. Sales of $1,100 year can be produced forever. The first year's costs are $600, and after that they rise by 20% annually. (Assume all the sales and costs occur at year's end; in this case, expenses are $600 at time t = 1). Anytime and without penalty, the project can be terminated. Ignore taxes and calculate the NPV of the project at a 12 percent discount rate.

5. (10 points) The common stock of XYZ Corporation is currently worth $40 million, and its risk-free debt is currently worth $60 million. The estimated market risk premium is 10%, and the common stock of the corporation has a beta of 0.8. What is the firm's cost of capital if the Treasury bill rate is 6%? (Assume no taxes.)

115年 - 115 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#147568

115年 - 115 國立臺北大學_日間學士班轉學生招生考試試題_資訊工程學系日間學士班2年級：微積分#147413

115年 - 115 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：微積分#143923

115年 - 115 國立嘉義大學_轉學生招生考試試題_理工學院：微積分#143894

115年 - 115 中央警察大學_碩士班入學考試試題_海洋科技組：微積分#139855

115年 - 115 中央警察大學_碩士班入學考試試題_消防科學研究所、交通管理研究所：微積分#139760

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

114年 - 114 中央警察大學_碩士班入學考試試題_消防科學研究所、交通管理研究所：微積分(同等學力加考)#146806

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

114年 - 114 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：微積分#144160