13.(5%) Let f(x, y) = 2x² + y² + 2xy + 4x + 2y+ 7. Find the minimum value of the function f(x,y) subject to the constraint 4x² + 4xy = 1.
(A) 4
(B) 5
(C) 6
(D) None of the above

6. (5%) What is the value of x + 22x2+ ...+ n2xn if x=3 and n = 12? (A)106022478 (B)108070187 (C)108080187 (D) None of the above

7. (5%) For the summation equation in Question 6, what is its value if x = 0.5 and n →∞? (A) 5 (B) 6 (C) 7 (D) None of the above

8. (5%) Determine whether the following statement is true or false. (A) False (B) True (C) Uncertain (need more information) (D) None of the above n.-)00

9.(5%) The Gaussian Quadrature method is a numerical way to generate the approximation of the definite integral of any function f(x) as follows.Both the abscissas, Xi, and the weighting coefficients, Wi, are free parameters and they can be solved based on the following 2n equations If n = 2, what is the value of w1 + w2 + x1+x2? (A) 0 (B) 1 (C) 2 (D) None of the above

10.(5%) Given the solutions of w1, w2, x1, and x2 in Question 9, use the Gaussian Quadrature method (with n= 2) to approximatedx. What is the absolute value of the approximation error? (A) In(1/3) (B)(C) 0 (D) None of the above

11. (5%) Suppose that a random variable S follows a lognormal distribution and its probability density function is  where μ and σrepresent the mean and standard deviation of the lognormal distribution, respectively. Define N(d) as the cumulative probability of the standard normal distribution from -∞ to a constant d. What is the expected value of S conditional on S≥ K, where K is a nonnegative constant? (A) (B)(C) (D) None of the above

12. (5%) A company grows in value by 10% each year, and also gains 20% of a growing market estimated at 100e0.1t million dollars, where t is the number of years that the company has been in business. Therefore, the value y(t) of the company (in millions of dollars) satisfies y' = 0.1y+20e0.1t and y(0)= 1 (million dollars).What is the value of this company after 5 years? (A) 173 million dollars (B) 167 million dollars (C) 180 million dollars (D) None of the above

14. (5%) Evaluate the following double integral for a specified region A.where A is the region bounded by y = , y = 1,and x = 0. (A) (B) (C) 1 (D) None of the above

15. (5%) What is the value of dx?(A)0 (B)∞ (C)1 (D) None of the above 

10.(5%) The death rate and birth rate of many animal and plant specie periodically with the seasons. The population P(t) of such a species at time t changes at a rate that may be modeled by a differential equation of the form Given the initial population P(0) to be 1000, what is the population (A) 1482 (B) 2482 (C) 3482 (D) None of the above 

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