7. Find the Taylor series at x =1 for In x, and find its interval of convergence.

1. Find the valu of lim

2. Find the value of the constant c such that g(x) is continuous on all real numbers, where 

3. Sketch the graph of the function . Indicate (if any) the asymptotes of the graph and  the coordinates of the relative extremes.

4. Find (the integral is fiom 0 to x) if it exists, otherwise indicate "does not exist."

5. Find , if it exists or please indicate "it does not exist."

6. Evaluate the improperintegral de if it converges, othervis inticate divergence.

8. Compute the double integral dA where R is the rectangular region 0≤x≤2, and O≤y≤1.Also, dA indicates either dydx or dxdy may be used.

9. Suppose that a company's output is given by the Cobb-Douglas production function , where L and K are the numbers of units of labor and capital that are used. To stay within its budget of $2520, these amounts must satisfy 35L +140K = 2520 . Find the amounts of labor and capital that maximize production subject to the budget constraint.

10. The 72 rule says that money invested at 6% per year doubles in approximately 12 years. Also, money invested at 8 % per ycar doubles in approximately 9 years. Suppose that the number of years n is required for an investment at interest rate r to double in valuc, and i is the interest rate percentage (that is, i =100r). Using In2 = 0.693 and the approximation valid for small , show that it holds .

(14) Find the extreme values of f(x, y, z) = z subject to the constraints x² + y² = z² and x + 2y + z =   (16) 

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