the cost (in dollars) of producing x units of a certain commodity is C(x)=5579+13x+0.03x^2. Find the instantaneous rate of change with respect to x when x=103.
To find the instantaneous rate of change with respect to x, we need to take the derivative of the cost function C(x) with respect to x.
The cost function is given by C(x) = 5579 + 13x + 0.03x^2.
To find the derivative of C(x), we can use the power rule for derivatives and the constant rule:
dC/dx = d/dx (5579 + 13x + 0.03x^2)
= 0 + 13 + 0.06x
Now, we need to find the instantaneous rate of change when x = 103.
Plug in x = 103 into the derivative:
dC/dx = 13 + 0.06(103)
= 13 + 6.18
= 19.18
Therefore, when x = 103, the instantaneous rate of change is 19.18 dollars per unit.