Q: If the total cost function is $$C(q) = 0.001q^2$ - 0.05q^2$ + 40q + 4,000$$, estimate the cost of producing the 251st unit using marginal analysis.

$$C'(250) = 0.002 \times 250 - 0.1 \times 250 + 40 = 0.002 \times 250 - 25 + 40 = 0.5 - 25 + 40 = 15.5$$

Question 1

Given the total cost function $$C(x) = \frac{1}{4}x^2$ + 3x + 67$$, what is the marginal cost function?

Accepted Answer

$$C'(x) = \frac{1}{2}x + 3$$

Question 2

If the price function for a commodity is $$p(x) = \frac{5}{3}(55 - x)$$, what is the total revenue function $$R(x)$$?

Accepted Answer

$$R(x) = \frac{5}{3}(55x - x^2$)$$

Question 3

A manufacturer’s total cost is $$C(q) = 0.001q^2$ - 0.05q^2$ + 40q + 4,000. $$What is the marginal cost function?

Accepted Answer

$$C'(q) = 0.002q - 0.1q + 40$$

Question 4

Suppose the total cost of printing $$x$$ dictionaries is $$C(x) = 20,000 + 10x. $$What is the average cost function?

Accepted Answer

$$AC(x) = \frac{20,000}{x} + 10$$

Question 5

Given $$C(x) = \frac{1}{4}x^2$ + 3x + 67$$ and $$p(x) = \frac{5}{3}(55 - x)$$, find the marginal profit function.

Accepted Answer

$$P'(x) = R'(x) - C'(x) = \frac{5}{3}(55 - 2x) - (\frac{1}{2}x + 3)$$

Question 6

If the total cost of producing $$x$$ units is $$C(x) = 20,000 + 10x$$, what is the marginal cost of producing the 1,001st dictionary?

Accepted Answer

$$MC = 10$$

Question 7

A company finds that the marginal revenue function is $$MR(x) = 100 - 2x. $$What is the economic interpretation of $$MR(30)$$?

Accepted Answer

The approximate additional revenue from selling the 31st unit is $$MR(30) = 100 - 2 	imes 30 = 40$$

Question 8

Given $$C(x) = \frac{1}{4}x^2$ + 3x + 67$$ and $$p(x) = \frac{5}{3}(55 - x)$$, what is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = \frac{5}{3}(55x - x^2$) - (\frac{1}{4}x^2$ + 3x + 67)$$

Question 9

If the total cost function is $$C(q) = 0.001q^2$ - 0.05q^2$ + 40q + 4,000$$, estimate the cost of producing the 251st unit using marginal analysis.

Accepted Answer

$$C'(250) = 0.002 	imes 250 - 0.1 	imes 250 + 40 = 0.002 	imes 250 - 25 + 40 = 0.5 - 25 + 40 = 15.5$$

Question 10

For the cost function $$C(x) = 20,000 + 10x$$, what happens to the average cost per unit as $$x$$ increases?

Accepted Answer

It decreases and approaches $$10 as$$ $$x$$ becomes very large.

Question 11

If $$C(x) is $$the total cost function, which of the following best describes the marginal cost?

Accepted Answer

The derivative of the total cost function, $$C'(x)$$, representing the approximate cost of producing one more unit.

Question 12

Suppose the total cost of producing $$x$$ units is $$C(x) = 20,000 + 10x. $$What is the average cost per unit if 1,000 units are produced?

Accepted Answer

$$AC(1000) = \frac{20,000}{1000} + 10 = 20 + 10 = 30$$

Question 13

Given $$C(x) = \frac{1}{4}x^2$ + 3x + 67$$, what is the actual cost of producing the 251st unit?

Accepted Answer

$$C(251) - C(250)$$

Question 14

If $$C(x) is $$the total cost and $$R(x) is $$the total revenue, what is the marginal profit function?

Accepted Answer

$$P'(x) = R'(x) - C'(x)$$

Question 15

A company’s total cost function is $$C(x) = 20,000 + 10x. If 1$$,001 dictionaries are produced, estimate the average cost per dictionary using marginal analysis.

Accepted Answer

$$AC(1001) = \frac{20,000}{1001} + 10$$

Question 16

If the average cost function is $$AC(x) = \frac{C(x)}{x}$$, what is the marginal average cost?

Accepted Answer

$$\frac{d}{dx} \left( \frac{C(x)}{x} \right)$$

Question 17

Suppose the price function is $$p(x) = \frac{5}{3}(55 - x)$$ and all $$x$$ units are sold. What is the marginal revenue function?

Accepted Answer

$$R'(x) = \frac{5}{3}(55 - 2x)$$

Question 18

A company’s total cost function is $$C(x) = 20,000 + 10x. If $$the company wants to minimize average cost, what should they do?

Accepted Answer

Increase production, since average cost decreases as $$x$$ increases.

Marginal Cost, Revenue, and Profit Functions in Calculus

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