(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.

(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.

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