Prove that the necessary conditionm for integrability of the total dif j equation X · dr = Pdx + Qdy + Qdz = 0 is X · curl X = (

Suppose that the output Q (in units) of a certain company is Q = 75K^1/3L^2/3, where K is the capital expenditures in thousands of dollars and L is the number of labor hours.  Find ∂Q/∂K when capital expenditures are $2,744,000 and the labor hours total 2744. Interpret ∂Q/∂K. If labor hours remain at 2744 and K increases by $1000, Q will increase about how much??                                                                                                                                                                                                       Find ∂Q/∂L when capital expenditures are $2,744,000 and the labor hours total 2744. (Round your answer to the nearest whole number.)                                                                                                                                                                                                                                                                                          Interpret ∂Q/∂L. If…

sketch the solid and a washer for the graph of these two functions f(x) =2 -(x-2)^2, g(x)=-x after rotating the enclosed region around the line y=1

A charged particle on a flat surface in a magnetic field travels along the curve y = In(sin x) over the time span of 1 millisecond (ms), where the curve is measured in centimeters (cm). At time t = O ms , the x- 5л coordinate of the particle is x = 5 cm . At time t 1 ms, the x-coordinate of the particle is x = ст 6. Find the total distance travelled by the particle over this one millisecond.