103年 - 103 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)#118499

1. (5%) Suppose that the relationship between the tax rate t on imported shoes and the total sales S (in millions of dollars) is given by . Find the tax rate t that maximizes revenue for the government. (A)3.6% (B) 6.4% (C) 9.2% (D) None of the above

2. (5%) The capital value of an asset (such as an oil well) that produces a continuous streamn of income is the sum of the present values of all future earnings from the asset. Therefore, the capital value of an asset that produces income of r(t) dollars at time t (discounted at a continuous interest rate t) is

where T is the expected life (in years) of the asset. For an oil well that produces income of r(t) = 10000t² for the following 10 years and the interest rate i = 5%, what is the capital value of this oil well? (A) 2,302,028 dollars (B) 99,346,934 dollars (C) 26,563,254 dollars (D) None of the above

6. What is the value of?(5%) (A) 2 (B) 1 (C)0 (D) None of the above

7. What is the value of (5%) (A)∞ (B) -In1 (C) -In2 (D) -∞

8. What is the value of dxdy? (5%) (A) e²-3 (B) (C) e²+2 (D) None of the above

9. If a variable x is exponentially distributed with the following probability density function, what is the expected value of x?( 5%) (A) e² (B) 2 (C) 0.5 (D) None of the above

10.(5%) Find the maximum value of f(x,y)=cosx+cosy subject to the constraint condition (A) (B) (C) (D) None of the above 

12.(10%) Compute
  

13. (10%) Calculate cos mx cos nx dx and  sin mx sin nx dx, where m and n are different positive integers.