(4)dx

(4) F(x) =dt 

(1) dx

(2)(cos3 x + 1)tan x) dx 

(3) ∫2x dx

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.

(a) (15 分)

(b) (15 分)