3. (5%) Which can be the solution form of A(t) if 

A'(t) = A²(t) + A(t) - 1,and A(0) = 0.
(A) A(t) = a+ be^ct+ d
(B) A(t) = exp(a+be^ct)+d
(C) A(t) = (a+be^ct)^-1+d
(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 an 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, whereEstimate 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 asif 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 

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

3. (5%) What is the solution form of A(t) if A'(t) = A2(t)- 1,and A(0) = 0. (A) A(t) = a + bect (B) A(t) = (a + bect)-1 (C) A(t) = (at bect)-1 + d (D) None of the above

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