(a) (15 分)

3. (15%) Suppose that the source of current in an electric circuit is a battery. Then the power output P (in watts) obtained if the circuit has a resistance of R ohms is given by where E is the electromotive force in volts and r is the internal resistance of the battery in ohms. If E and r are constant, find the value of R that will result in the greatest power output. What is the maximum power output?

4. (15%) A solid ball can be obtained by revolving the semi-circular region x² + y² ≤ R², x ≥ 0, about the y-axis. Use the method of disks or cylindrical shells to find the volume of such a solid ball.

5. (15%) Let M, N, k be three positive constants, with M> N. Solve the differential equation = k(M - y)(N - y),under the initial condition y(0) = 0.

6. (15%) Suppose x units of labor and y units of capital are required to producef(x, y) = 100x3/4y1/4units of a certain product. If each unit of labor costs $200 and each unit of capital costs $300 and a total of $60,000 is available for production, determine how many units of labor and how many units of capital should be used to maximize production.

(b) (15 分)

2. Evaluate the limit of the following expression. (15 分)

3. Find the sum of the following series. (15 分)

4. Find the 3th Taylor polynomial (centered at 1) of the function f(x) = ln(x+1). (10 分)

5. Find the equation of a plane that is perpendicular to the plane 2x-3y+z=3 and passes through (-1,1,-1) and (2,2,1). (10 分)

6. Find the maxima and minima of the function f(x, y) = x² + 3xy + y² subject to the constraint x² + y² ≤ 1. (20 分)