Question 1

Given the price-demand equation $$x = 10{,}000 - 1{,}000p$$, where $$x is $$the number of headphones and $$p is $$the price per set, solve for $$p as a $$function of $$x.$$

Accepted Answer

$$p(x) = 10 - 0.001x$$

Question 2

What is the domain of the function $$p(x) = 10 - 0.001x in $$the context of the price-demand equation?

Accepted Answer

$$0 \leq x \leq 10{,}000$$

Question 3

Given the cost function $$C(x) = 7{,}000 + 2x$$, what is the marginal cost function $$C'(x)$$?

Accepted Answer

$$C'(x) = 2$$

Question 4

Interpret the marginal cost $$C'(x)$$ for the cost function $$C(x) = 7{,}000 + 2x in a $$real-world context.

Accepted Answer

It is the approximate cost to produce one additional headphone, which is $2 per unit.

Question 5

Find the revenue function $$R(x) in $$terms of $$x$$ using the price-demand equation.

Accepted Answer

$$R(x) = x(10 - 0.001x)$$

Question 6

What is the domain of the revenue function $$R(x) = x(10 - 0.001x)$$?

Accepted Answer

$$0 \leq x \leq 10{,}000$$

Question 7

At what value(s) of $$x do $$the cost and revenue functions intersect? (Use $$C(x) = 7{,}000 + 2x$$ and $$R(x) = x(10 - 0.001x).)$$

Accepted Answer

$$x = 0$$ and $$x = 7{,}000$$

Question 8

What does the intersection point $$x = 7{,}000$$ between the cost and revenue functions represent in a business context?

Accepted Answer

It is the break-even point where total revenue equals total cost.

Question 9

Find the profit function $$P(x) in $$terms of $$x$$ using the given cost and revenue functions.

Accepted Answer

$$P(x) = x(10 - 0.001x) - (7{,}000 + 2x)$$

Question 10

What is the domain of the profit function $$P(x) = x(10 - 0.001x) - (7{,}000 + 2x)$$?

Accepted Answer

$$0 \leq x \leq 10{,}000$$

Question 11

Use marginal analysis to estimate the profit from producing the 1,000th unit. (Recall: $$P'(x) = R'(x) - C'(x).)$$

Accepted Answer

$$P'(1{,}000) = 8$$

Question 12

If the price per headphone is set at $$p = 5$$, how many headphones will be sold according to the price-demand equation?

Accepted Answer

$$x = 5{,}000$$

Question 13

Suppose the company wants to maximize revenue. At what value of $$x$$ does the maximum revenue occur?

Accepted Answer

$$x = 5{,}000$$

Question 14

What is the maximum possible revenue the company can earn?

Accepted Answer

$$25{,}000$$

Question 15

If the company produces 8,000 headphones, what is the profit or loss?

Accepted Answer

$$-10{,}000 (a $$loss)

Question 16

If the fixed cost increases to $$10,000$$, how does this affect the break-even quantity?

Accepted Answer

The break-even quantity increases.

Question 17

If the marginal cost increases to $$3$$ per unit, what is the new cost function?

Accepted Answer

$$C(x) = 7{,}000 + 3x$$

Question 18

If the company wants to maximize profit, at what value of $$x$$ should it produce? (Hint: Set $$P'(x) = 0.)$$

Accepted Answer

$$x = 4{,}000$$

Question 19

What is the economic interpretation of the marginal revenue at $$x = 2{,}000$$?

Accepted Answer

It is the approximate additional revenue from selling one more headphone when 2,000 have already been sold.

Calculus Applications: Cost, Revenue, and Profit Functions

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