4. Evaluate the integral dydx.

5. 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.

6. Let y = (sin a)x, sin x ＞ 0. Find

7. Assume that constants a and b are positive. Find equations for all horizontal asymptotes for the graph of .

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.

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

(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.

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