105年 - [非官方正解]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

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, 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

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

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 

1. (10%) Find if F(X) =. 

3. (10%) Find [In|cos(x)|]. dx

4.(10%) If w = x³y⁴ and x = t² - 2, y= 5t -3, findat t = 1.

5. (10%) Find ∫Inx dx.