The average cost function for the weekly manufacture of portable CD players is given by C(x)=140,000x−1+20+0

From the given information, the cost function is

C(x)=140000x−1+20+0.0002xC(x)=140000x−1+20+0.0002x dollars per year.

And the production for current week is, x=2000x=2000 players.

And the given increasing rate of change for production with respect to t,t, is dxdt=100dxdt=100 players per week. Where tt is time.

Now, the rate of change for average cost is,

dCdt=dCdx⋅dxdt.dCdt=dCdx⋅dxdt.

dCdt=ddx(140000x−1+20+0.0002x)⋅dxdt=ddx(140000x−1⋅dxdt+20⋅dxdt+0.0002x⋅dxdt)=[ddx(140000x)⋅dxdt−ddx(1)⋅dxdt+ddx(20)⋅dxdt+ddx(0.0002x))⋅dxdt]=[140000ddx(x)⋅dxdt−0⋅dxdt+0⋅dxdt+0.0002ddx(x)⋅dxdt]=[140000⋅dxdt−0+0+0.0002⋅dxdt]=[140000(100)+0.0002(100)](?dxdt=100)=[14000000+0.02]dCdt=14000000.02.dCdt=ddx(140000x−1+20+0.0002x)⋅dxdt=ddx(140000x−1⋅dxdt+20⋅dxdt+0.0002x⋅dxdt)=[ddx(140000x)⋅dxdt−ddx(1)⋅dxdt+ddx(20)⋅dxdt+ddx(0.0002x))⋅dxdt]=[140000ddx(x)⋅dxdt−0⋅dxdt+0⋅dxdt+0.0002ddx(x)⋅dxdt]=[140000⋅dxdt−0+0+0.0002⋅dxdt]=[140000(100)+0.0002(100)](?dxdt=100)=[14000000+0.02]dCdt=14000000.02.

Estimating the Rate of Change for Average Cost::

Let's consider the given cost function as, C(x),C(x), where xx is the number of CD players manufactured per week. Now we need to find the rate of change of average cost, which is, dCdt.dCdt.

To find that, the formula is,

dCdt=dCdx⋅dxdt.dCdt=dCdx⋅dxdt. Therefore, we need to find the answer, by using the differential method.