(1) Find the margin of error for a \( 95\% \) confidence interval. \( (10\%) \)

(2) We expect that the margin of error is not greater than 0.02. How many people should be surveyed? (95% confidence interval) (10%)

4. The employee with low productivity in KID company is \( 5\% \). For the employee marked as low productivity, the attendance system showed that the probability of personal leave twice in a month is \( 100\% \). In contrast, the probability of personal leave twice in a month for other employees is \( 20\% \). Mary took the personal leave twice in the last month. What is the probability that Mary is an employee with low productivity? (10%)

5. One survey of education level in Taipei showed that 33 persons are elementary school, 44 persons are junior high school, 89 persons are senior high school, 14 persons are undergraduate and 20 persons are graduate. In Taiwan, the survey of education level showed that \( 10.2\% \) of population is elementary school, \( 12.3\% \) of population is junior high school, \( 57.9\% \) of population is senior high school, \( 7\% \) of population is undergraduate and \( 12.6\% \) of population is graduate. Is the education level in Taipei significantly different from the education level in Taiwan? (95% confidence interval) (20%)

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 \( y = 2(\ln x)^{x/2} \).

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