Question 1

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

Accepted Answer

$$p(x) = \frac{10{,}000 - x}{1{,}000}$$

Question 2

What is the domain of the function $$p(x) = \frac{10{,}000 - x}{1{,}000}$$, given that price $$p$$ and demand $$x$$ must be non-negative?

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 $$C(x) = 7{,}000 + 2x in $$the context of headphone production.

Accepted Answer

It is the 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 $$p(x) = \frac{10{,}000 - x}{1{,}000}.$$

Accepted Answer

$$R(x) = x \cdot \frac{10{,}000 - x}{1{,}000}$$

Question 6

What is the domain of the revenue function $$R(x) = x \cdot \frac{10{,}000 - x}{1{,}000}$$?

Accepted Answer

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

Question 7

Find the intersection points of the cost and revenue functions, $$C(x) = 7{,}000 + 2x$$ and $$R(x) = x \cdot \frac{10{,}000 - x}{1{,}000}$$, and interpret their meaning.

Accepted Answer

They represent the break-even points where profit is zero.

Question 8

Given $$C(x) = 7{,}000 + 2x$$ and $$R(x) = x \cdot \frac{10{,}000 - x}{1{,}000}$$, find the profit function $$P(x).$$

Accepted Answer

$$P(x) = x \cdot \frac{10{,}000 - x}{1{,}000} - (7{,}000 + 2x)$$

Question 9

What is the domain of the profit function $$P(x) = x \cdot \frac{10{,}000 - x}{1{,}000} - (7{,}000 + 2x)$$?

Accepted Answer

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

Question 10

Using marginal analysis, estimate the profit from selling the 1,000th headphone. Use $$P'(x)$$, the derivative of the profit function.

Accepted Answer

It is approximately equal to $$P'(1{,}000)$$, the rate of change of profit at $$x = 1{,}000.$$

Question 11

If the price per headphone is set at $$p = 5$$, how many headphones will be sold according to the price-demand equation $$x = 10{,}000 - 1{,}000p$$?

Accepted Answer

Question 12

Suppose the company wants to maximize profit. Which value of $$x$$ should they choose, given the profit function $$P(x)$$?

Accepted Answer

The value of $$x$$ where $$P'(x) = 0$$ and $$P''(x) \u003c 0.$$

Question 13

If the fixed cost increases from $$7{,}000 to$$ $$10{,}000$$, how does this affect the break-even points?

Accepted Answer

The break-even points move further apart, requiring more units to be sold to break even.

Question 14

If the marginal cost increases to $$C'(x) = 5$$, what is the new cost function assuming the fixed cost remains $$7{,}000$$?

Accepted Answer

$$C(x) = 7{,}000 + 5x$$

Question 15

If the company sells 8,000 headphones, what is the total revenue?

Accepted Answer

$$R(8{,}000) = 8{,}000 \cdot \frac{10{,}000 - 8{,}000}{1{,}000} = 8{,}000 \cdot 2 = 16{,}000$$

Question 16

For which value of $$x$$ does the revenue function $$R(x)$$ reach its maximum?

Accepted Answer

$$x = 5{,}000$$

Question 17

If the price is set too high, what happens to the demand according to the price-demand equation?

Accepted Answer

Demand decreases linearly as price increases.

Question 18

If the company wants to maximize revenue, what price should they set per headphone?

Accepted Answer

$$p = 5$$

Calculus Applications: Cost, Revenue, and Profit Functions

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