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, X_i, and the weighting coefficients, W_i, 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

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

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

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

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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