Question

the function c gives the cost, in dollars, to produce a particular product, where c(x) is the cost, in dollars, to produce x units of the product. the function m defined by m(x)=c(x + 1)-c(x) gives the marginal cost, in dollars, to produce unit number x + 1. which of the following gives the best estimate for the marginal cost, in dollars, to produce the 57th unit of the product?
a (\frac{c(56)}{56})
b (\frac{c(57)}{57}-\frac{c(56)}{56})
c (c(56))
d (c(57)-c(56))

Explanation:

Step1: Recall the definition of marginal cost

The marginal cost function \(M(x)\) is an approximation of the derivative of the cost function \(C(x)\). The marginal cost to produce the \((x + 1)\) - th unit is approximated by \(C^{\prime}(x)\).

Step2: Substitute \(x = 56\)

We want to find the marginal cost to produce the \(57\) - th unit. Using the approximation \(M(x)=C(x + 1)-C(x)\approx C^{\prime}(x)\), when \(x = 56\) (since we want to find the cost for \(x+1=57\)), the best estimate for the marginal cost of the \(57\) - th unit is \(C^{\prime}(56)\).

Option A: \(\frac{C(56)}{56}\) is the average cost of producing \(56\) units, not the marginal cost of the \(57\) - th unit.
Option B: \(\frac{C(57)}{57}-\frac{C(56)}{56}\) is the difference in average costs, not the marginal cost.
Option D: \(C^{\prime}(57)-C^{\prime}(56)\) is the difference in marginal costs at \(x = 57\) and \(x = 56\), not the marginal cost of the \(57\) - th unit.

Answer:

C. \(C'(56)\)