Question

C(x)=0.01x³-0.6x²+17x gives the cost, in thousands of​ dollars, to produce x items.

​(a) Find a formula for the marginal cost.

​(b) Find C'(0). Give units.

​(c) Graph the marginal cost function. Use your graph to find the minimum marginal​ cost, the production level for which the marginal cost is the smallest.

​(d) For what value of x does the marginal cost return to C'(0)?

Answer 1

To find the formula for the marginal cost, differentiate the given cost function C(x). To find C'(0), substitute x = 0 into the derivative function. To graph the marginal cost function, plot points by substituting different values of x into the derivative function. The minimum marginal cost corresponds to the lowest point on the graph. The marginal cost function does not return to C'(0) as it is not periodic.

Step-by-step explanation:

1. To find the formula for the marginal cost, we need to find the derivative of the cost function. Differentiating the given function C(x) = 0.01x³ - 0.6x² + 17x, we get C'(x) = 0.03x² - 1.2x + 17.
2. To find C'(0), substitute x = 0 into the derivative function: C'(0) = 0.03(0)² - 1.2(0) + 17 = 17. Therefore, C'(0) = 17, and the units are thousands of dollars per item.
3. To graph the marginal cost function, plot points by substituting different values of x into the derivative function. These points will form the graph of the marginal cost function.
4. To find the minimum marginal cost, locate the lowest point on the graph. The x-coordinate of this point represents the production level for which the marginal cost is the smallest.
5. The marginal cost function won't return to C'(0) as it is a continuous function and not periodic in nature.