(a) (6 分) Use the integral test to determine if the seriesconverges or diverges.

8. We say that the two commodities are substitute commodities if a decrease in the demand for one results in an increase in the demand for the other. Con- versely, two commodities are referred to as complementary commodities if a decrease in the demand for one results in a decrease in the demand for the other as well. Suppose that the demand equations that relate the quantities demanded x and y to the unit prices p and q of the commodities A and B respectively are  given by . Are A and B substitute, complementary or neither? 

(a) (4 分) Show that f is continuous at x = 0.

(b)(4分)Find f'(x) for x≠0.

(c) (4 分) Use the limit definition of the derivative to show that f is not differ- entiable at x = 0.

(b) (6 分) Find all values of x for which converges absolutely. 

3. Suppose a units of labor and y units of capital are required to produce units of a certain product. If each unit of labor costs $200, each unit of capi- tal costs $300, and a total of $60,000 is available for production, determine how many units of Iabor and how many units of capital should be used to maximize production.

1. Find the limit

2. Find f'(-1) if f(x)=e g(x) and g(x)=tan-1(x2)+dt.

3. Find the length of the curve x = In(sect + tant) - sint, y = cost, 0≤t≤π/3.

4.Consider the region bounded by the graphs of y = In x, y = 0, and x = e. Find the volume of the solid formed by revolving the region about the x-axis.