Final answer:
The instantaneous rate of change of the cost function C(x) for producing x units at x = 100 is found by evaluating the derivative of the function, C'(x) = 5.70 + 1.4x, at x = 100. This gives us an instantaneous rate of $145.70 per unit.
Step-by-step explanation:
The student is asking about the instantaneous rate of change of the cost function C(x) with respect to the number of units x when x = 100. To find this, we need to calculate the derivative of C(x) and evaluate it at x = 100.
The cost function is given by C(x) = 1000 + 5.70x + 0.7x^2. The derivative, which represents the instantaneous rate of change, is C'(x) = 5.70 + 1.4x. We then evaluate this derivative at x = 100.
C'(100) = 5.70 + 1.4(100) = 5.70 + 140 = 145.70.
Therefore, the instantaneous rate of change of the cost when producing 100 units is $145.70 per unit.