Find the rate of change of total revenue, cost, and profit with respect to time. Assume that R (x) and C (x) are in dollars.

R (x) = 2x, C (x) = 0.01x² + 0.4x + 30, when x=25 and dx/dt=7 units per day.

1. Find the rate of change of total revenue per day.

2. Find the rate of change of total cost per day.

3. Find the rate of change of total profit per day.

1. - dR/dt = 14 $/day

2. - dC/dt = 4.55 $/day

3. - dP/dt = 7,7 $/day

Step-by-step explanation:

By definition Profit is equal to Revenue minus total costs then.

We have

R (x) = 2*x

C (x) = 0,01*x² + 0,4*x + 30

Then Profit P (x) = 2*x - 0,01*x² - 0,4*x - 30

P (x) = - 0.01*x² + 1.6*x - 30

1. - Find rate of change total revenue per day, when

x = 25 and dx/dt = 7 u/day

R (x) = 2*x

dR/dt = 2*dx/dt ⇒ dR/dt = 2 * 7

dR/dt = 14 $/day

2.-

C (x) = 0,01*x² + 0,4*x + 30

dC/dt = 0,01*x * dx/dt + 0,4*dx/dt

dC/dt = 0,01*25*7 + 0.4*7

dC/dt = 1.75 + 2.8

dC/dt = 4.55 $/day

3.-

P (x) = - 0.01*x² + 1.6*x - 30

dP/dt = - 2*0,01*x*dx/dt + 1.6*dx/dt

dP/dt = - 2*0,01*25*7 + 1.6*7

dP/dt = - 3,5 + 11.2

dP/dt = 7,7 $/day