1. (10%) Find the limit: \( \lim_{x\to 1}\frac{x^{1 / 3} - 1}{\sqrt{x} - 1} \).

2. (10%) Find the limit of \( g(x) \) as \( x \) approaches the indicated values: $\lim_{x \to \sqrt{5}} \frac{1}{x+g(x)} = 2.$

3. (10%) Find \( dy / dx \) by implicit differentiation: \( y^2 = \sqrt{\frac{1 + x}{1 - x}} \).

4. (10%) Show that the function \( f(x) = x + \sin^2\left(\frac{x}{3}\right) - 8 \) has exactly one zero in the interval \( (-\infty, \infty) \).

5. (15%) Graph the function: \( y = \frac{\sqrt{1 - x^2}}{2x + 1} \). (Find where the curve is increasing and where it is decreasing. Find the local extreme points. Find the points of inflection, if any occur, and determine the concavity of the curve. Identify any asymptotes.)

6. (10%) Find the area of the triangular region bounded on the left by \( x + y = 2 \), on the right by \( y = x^2 \), and above by \( y = 2 \).

7. (10%) Find the derivative of \( y \) with respect to \( x \) if \( y = 2(\ln x)^{x/2} \).

8. (10%) Evaluate \( \int x^{2} \sin(1 - x) dx \).

9. (15%) Sketch the region of integration and write an equivalent integral with the order of integration reversed. Then evaluate both integrals. $$\int_0^{3/2} \int_{-\sqrt{9-4y^2}}^{\sqrt{9-4y^2}} y \, dx \, dy.$$

I. (10%) A stock portfolio is composed of two stocks: \( A_{1} \) and \( B_{1} \). Another stock portfolio also has two investments: stocks \( A_{2} \) and \( B_{2} \). Suppose the mean prices \( (\mu) \) and the standard deviations \( (\sigma) \) of \( A_{1} \) and \( A_{2} \) are equal, and the mean price and standard deviation of \( B_{1} \) are equal to those of \( B_{2} \) (i.e., \( \mu_{A_1} = \mu_{A_2}, \mu_{B_1} = \mu_{B_2}, \sigma_{A_1} = \sigma_{A_2}, \sigma_{B_1} = \sigma_{B_2} \)). Suppose each stock comprises 50% of the portfolio. However, the correlation between stocks is higher in the first portfolio (i.e., \( \rho_1 > \rho_2 \)). Explain the difference in expected returns and risks between the two portfolios.

(a) Describe the probability distribution of \( P_A \), the proportion of respondents who was identified with Party A in the sample. Explain whether normal distribution is appropriate to approximate the distribution of \( P_A \).

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