Question

1. find a formula for the marginal cost2. find c’(0). give units 3. graph the marginal cost function. Use your graph to find the minimum marginal cost, the production level for which the marginal cost is the smallest4. for what value of X does the marginal cost return to c’(0)

Answer 1

Solution

Step 1:

Write the total cost function

$c(x)\text{ = 0.01x}^3-0.3x^2\text{ + 10x}$

a) Marginal cost is the first derivative of the total cost

$Marginal\text{ cost c'\lparen x\rparen= 0.03x}^2\text{ - 0.6x + 10}$

b)

$\begin{gathered} c^{\prime}(x)\text{ = 0.03x}^2\text{ - 0.6x + 10} \\ c^{\prime}(0)\text{ = 0.03 }\times\text{ 0}^2\text{ - 0.6}\times\text{ 0 + 10} \\ c^{\prime}(0)\text{ = 10 additional dollar per item produced} \end{gathered}$

c)

Graph of the marginal cost

The coordinates of the minimum marginal cost is (10 , 7000)

Minumum marginal cost of 7 additional dollars per item produced occurs when 10 thousand items are produced

d)

$\begin{gathered} 0.03x^2\text{ - 0.6x + 10 = 10} \\ 0.03x^2\text{ - 0.6x = 0} \\ x(0.03x\text{ - 0.6\rparen = 0} \\ \text{x = 0 , x = }\frac{0.6}{0.03} \\ x\text{ = 20} \end{gathered}$

20 thousands items produced