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

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 

112年 - 112 國立臺灣大學_碩士班招生考試:微積分(C)#114173

110年 - 110 國立臺灣大學_碩士班招生考試_國際企業研究所乙組:微積分(C)#101835

109年 - 109 國立臺灣大學_碩士班招生考試_國際企業研究所:微積分(C)#106068

108年 - 108 國立臺灣大學_碩士班招生考試:微積分(C)#114100

107年 - 107 國立臺灣大學_碩士班招生考試_國際企業研究所乙組：微積分(C)#114199

106年 - 106 年 - 106 國立臺灣大學_碩士班招生考試_國際企業研究所乙組:微積分(C)#114292

105年 - 105 年 - 105 國立臺灣大學_碩士班招生考試_國際企業研究所乙組:微積分(C)#114296

104年 - 104 國立臺灣大學_碩士班招生考試:微積分(C)#118498

102年 - 102國立臺灣大學_碩士班招生考試_國際企業研究所乙組：微積分(C)#118500