Question

The function gives the cost to manufacture x items.C(x) = 10,000 + 70x + 1100/XLevel of production x= 100Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of x, where x is as indicated and h = 10 and 1. (Use smaller values of h to check your estimates.) HINT (See Example 1.] (Round your answers to two decimal places.) h 10 1 Cave 70.1 x 7.01 X Estimate the instantaneous rate of change of the total cost at the given production level x, specifying the units of measurement. C'(100) = 69.89 $/item

Answer 1

We have to calculate the average cost per unit of manufacturing h more items.

We first calculate the total additional cost:

$\begin{gathered} C(x+h)-C(x) \\ 10000+70(x+h)+\frac{1100}{x+h}-(10000+70x+\frac{1100}{x}) \\ (10000-10000+70(x+h-x)+\frac{1100}{x+h}-\frac{1100}{x}) \\ 0+70h+1100(\frac{x-(x+h)}{x(x+h)}) \\ 70h+1100(\frac{h}{x^2+h}) \end{gathered}$

To calculate the average cost, we divide the total additional cost by the number of additional units h:

$\begin{gathered} C_{av}=\frac{C(x+h)-C(x)}{h} \\ C_{av}=\frac{70h+1100(\frac{h}{x^2+h})}{h}=70+1100(\frac{1}{x^2+h}) \end{gathered}$

If the level of production is x=100, we can write:

$C_{av}=70+1100(\frac{1}{100^2+h})=70+1100(\frac{1}{10000+h})$

Then, for h=10 we have:

$C_{av}=70+1100(\frac{1}{10000+10})=70+\frac{1100}{10010}=70+0.10989=70.10989$

When h=1, we have:

$C_{av}=70+1100(\frac{1}{10000+1})=70+\frac{1100}{10001}=70+0.10999=70.10999$

When h tends to 0, we can calculate:

$C_{av}=70+1100(\frac{1}{10000+0})=70+0.11=70.11$

Answer:

The average unit cost of h=10 more units is 70.10989.

When h=1, the average unit cost is 70.10999.

The average unit cost of an additional unit, at the level of production x=100 (also known as "marginal unit cost" C'(x)) is 70.11.