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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