View an example | All parts showing X 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) =60x - 0.5x", C(x) = 4x + 20, when x =50 and dx/dt = 40 units per day Use implicit differentiation to write an expression for dR/dt in terms of x. Differentiate each term of the equation with respect to t, treating x as a differentiable function of t. Calculate the rate of change dR/dt when x =50 and dx/dt = 40 units per day. Differentiate the equation, R(x) =60x -0.5x , term by term with respect to t. Because x is a function of t, it is critical that dx/dt is included as a factor in the result any time a term involving x is differentiated. dt (R(X)) = dt (60x - 0.5x dR d dt It (60x) - (0.5x2 )Use the extended power rule to find dt (60x). Recall that the extended power rule states that if F(x) =[g(x)]", then F'(x) =k[g(x)]" k - g'(x). dx dt (60x) = 60 dt Next find dt (0.5x2) dx dt (0.5x = X- dt Substituting the results from above gives the following.Find the rate of change of total revenue, cost, and profit with respect to time. Assume that R(x) and dR d E dt dt (60x) - dt (0.5x? ) dR dx dx = 60- dt dt - X- dt Now calculate dR/dt when x =50 and dx/dt =40 units per day. Substitute these values into the expression for dR/dt and simplify. dR dx dx = 60 dt dt - X dt = 60(40) - 50(40) = 400 Thus, the rate of change of total revenue when x = 50 and dx/dt = 40 units per day is $400 per day.Use the same procedure as above to find the rate of change of total cost when x =50 and dx/dt = 40 units per day. First use implicit differentiation to find dC/dt. dt (4x + 20) dC d d (4x) + dt dt di (20) dC dx = 4- + 0 dt dt Now calculate dC/dt when x =50 and dx/dt = 40 units per day. Substitute these values into the expression for dC/dt and simplify. dR dx = 4 dt dt= 4(40) = 160 Thus, the rate of change of total cost when x = 50 and dx/dt = 40 units per day is $160 per day. Profit is the amount of money a company gains in excess of expenditures. That is, total profit is the difference between total revenue and total cost. P(x) = R(x) - C(x) dP dR do Since P(x) = R(x) - C(x). dt dt dt Now calculate dP/dt when x = 50 and dx/dt = 40 units per day. Substitute values for dR/dt and dC/dt into the expression for dP /dt and simplify. dP dR dC dt dt dtdP dR do dt dt dt = 400 - 160 = 240 Thus, the rate of change of total profit when x = 50 and dx/dt = 40 units per day is $240 per day.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) = 45x - 0.5x , C(x) = 4x + 20, when x =30 and dx/dt = 15 units per day