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

2. (5%) What is the value ofdx? (A) 0.713 (B) 0.865 (C) 0.918 (D) None of the above

4. (5%) Consider a cash-or-nothing call option which pays the option holder an amount K at the maturity T only if the asset price at T is larger than K. The pricing formula for this option is as follows.
and S is the current asset price, r is a constant risk-free interest rate, σ is the volatility of the asset price, and N(●) is the cumulative distribution function of the standard normal distribution defined as

 Derive the Delta (i.e., ) of this option,.(A) (B)(C) 0 (D) None of the above

5. (5%) For the pricing formula of the cash-or-nothing call option in Question 4, derive the vega of this option. (A)(B)(C)(D) None of the above

6. (5%) What is the value of? (A) -∞(B) 1/3 (C) 1/6 (D)∞

7. (5%) Determine whether the following statement is true or false. "Suppose a_n is positive for all n. If converges, then also converges." (A) False (B) True (C) Uncertain (need more information) (D) None of the above

8. (5%) A community is laid out as a rectangular grid in relation to two main streets that intersect at the city center. Each point in the community has coordinates (x,y) in this grid, for -10≤x≤ 10, -8 ≤y≤ 8 with x and y measured in kilometers. Suppose the value of the land located at the point (x,y) is V thousand dollars, where

Estimate the value of the block of land occupying the rectangular region1≤ x≤ 3,0≤ y≤ 2. (A) 759 thousand dollars (B) 859 thousand dollars (C) 959 thousand dollars (D) None of the above

9.If a positive variable λ follows a gamma distribution, ie., λ~Gamma(a,β),where a and 8 are positive real numbers. Consequently, the probability density function for λ is as follows.
where Γ(Z) is the gamma function and defined as

if Z is a complex number with a positive real part. What is the var(λ) a = 1 and β = 2? (5%) (A) 1/4 (B) 1/2 (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 

12.(10%) Evaluate

13. (10%) Find the interval of convergence for

14. (10%) Find the surface area of the portion of the plane x+ y+ z = 1 that lies in the first octant (where x ≥0,y ≥0,z≥0).