Q: A hospital’s patient intake is modeled by $$N(t) = 100e$$^{0.03t}$$, where t$ is the number of days since opening. What is the rate of change of patient intake on day 10$?

$$N'(10) = 3 \times 100e$$^{0.03 \times 10}$$ = 3 \times 100e$$^{0.3}$$ \approx 3 \times 134.99 = 404.97$$

Q: A tech startup’s revenue is modeled by $$R(x) = 20x \ln(x)$$, where $$x is $$the number of thousands of users. What is the marginal revenue when $$x = 2$$?

$$R'(2) = 20 \ln(2) + 20 \approx 20 \times 0.6931 + 20 = 13.862 + 20 = 33.862$$

Q: A company’s supply function is $$S(p) = 2p^2$ + 3p$$, where $$p is $$the price per unit. If the market price increases from $$p = 4 to$$ $$p = 5$$, what is the approximate change in supply using the marginal supply?

Marginal supply at $$p = 4 is$$ $$S'(4) = 4 \times 4 + 3 = 35. $$Approximate change is $$35 \times 1 = 35$$ units.

Q: A company’s revenue function is $$R(x) = 100x - x^2$, and its cost function is C(x) = 20x + 200$. What is the maximum profit?

$$P(40) = 100 \times 40 - 40^2$ - 20 \times 40 - 200 = 4000 - 1600 - 800 - 200 = 1400$$

Question 1

A pharmaceutical company models the cost $$C(x) (in $$thousands of dollars) to produce $$x$$ units of a new drug as $$C(x) = 5x + 200. $$What is the marginal cost of producing one additional unit?

Accepted Answer

$$5$$

Question 2

Suppose the revenue $$R(x)$$ from selling $$x$$ units of a product is given by $$R(x) = 100x - 0.5x^2$. What is the maximum revenue, and at how many units is it achieved?

Accepted Answer

Maximum revenue is $$5,000 at$$ $$x = 100$$ units.

Question 3

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the break-even point(s)?

Accepted Answer

$$x = 5$$ and $$x = 10$$

Question 4

If the demand function for a product is $$p = 50 - 2x$$, where $$p is $$the price per unit and $$x is $$the number of units sold, what is the price elasticity of demand when $$x = 10$$?

Accepted Answer

$$-2$$

Question 5

A hospital’s patient intake is modeled by $$N(t) = 100e$$^{0.03t}$$, where t$ is the number of days since opening. What is the rate of change of patient intake on day 10$?

Accepted Answer

$$N'(10) = 3 	imes 100e$$^{0.03 	imes 10}$$ = 3 	imes 100e$$^{0.3}$$ \approx 3 	imes 134.99 = 404.97$$

Question 6

A company’s cost function is $$C(x) = 0.01x^3$ - 0.5x^2$ + 10x + 100. $$What is the average cost per unit when $$x = 20$$?

Accepted Answer

$$C_{avg}(20) = \frac{C(20)}{20} = \frac{0.01(8000) - 0.5(400) + 200 + 100}{20} = \frac{80 - 200 + 200 + 100}{20} = \frac{180}{20} = 9$$

Question 7

A tech startup’s revenue is modeled by $$R(x) = 20x \ln(x)$$, where $$x is $$the number of thousands of users. What is the marginal revenue when $$x = 2$$?

Accepted Answer

$$R'(2) = 20 \ln(2) + 20 \approx 20 	imes 0.6931 + 20 = 13.862 + 20 = 33.862$$

Question 8

A company’s supply function is $$S(p) = 2p^2$ + 3p$$, where $$p is $$the price per unit. If the market price increases from $$p = 4 to$$ $$p = 5$$, what is the approximate change in supply using the marginal supply?

Accepted Answer

Marginal supply at $$p = 4 is$$ $$S'(4) = 4 	imes 4 + 3 = 35. $$Approximate change is $$35 	imes 1 = 35$$ units.

Question 9

A company’s cost function is $$C(x) = 1000 + 20x + 0.1x^2$. What is the minimum average cost, and at what production level does it occur?

Accepted Answer

Minimum average cost is at $$x = 100$$, $$C_{avg}(100) = 1000/100 + 20 + 0.1 	imes 100 = 10 + 20 + 10 = 40$$

Question 10

A company’s profit is maximized when marginal revenue equals marginal cost. If $$R(x) = 50x - x^2$ and C(x) = 10x + 5$, at what value of x$ is profit maximized?

Accepted Answer

$$x = 20$$

Question 11

A population of bacteria in a lab grows according to $$P(t) = 500e$$^{0.04t}$$, where t$ is in hours. How long will it take for the population to double?

Accepted Answer

$$t = \frac{\ln(2)}{0.04} \approx 17.33$$ hours

Question 12

A company’s revenue function is $$R(x) = 80x - 0.4x^2$. What is the price elasticity of demand at x = 50$ if the demand function is p = 80 - 0.8x$?

Accepted Answer

$$-1$$

Question 13

A manufacturer’s cost function is $$C(x) = 500 + 15x$$, and the revenue function is $$R(x) = 25x. $$What is the minimum number of units that must be sold to break even?

Accepted Answer

$$x = 50$$

Question 14

A company’s profit function is $$P(x) = -x^2$ + 12x - 20. $$What is the maximum profit?

Accepted Answer

$$P(6) = -36 + 72 - 20 = 16$$

Question 15

A company’s demand function is $$p = 60 - 3x. $$What is the total revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 60x - 3x^2$

Question 16

A company’s cost function is $$C(x) = 2000 + 8x$$, and the price per unit is fixed at $$p = 20. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 12x - 2000$$

Question 17

A company’s revenue function is $$R(x) = 100x - x^2$, and its cost function is C(x) = 20x + 200$. What is the maximum profit?

Accepted Answer

$$P(40) = 100 	imes 40 - 40^2$ - 20 	imes 40 - 200 = 4000 - 1600 - 800 - 200 = 1400$$

Question 18

A company’s cost function is $$C(x) = 500 + 10x$$, and the revenue function is $$R(x) = 30x. If $$the company wants to make a profit of at least $$1,000$$, what is the minimum number of units it must sell?

Accepted Answer

$$x = 75$$

Question 19

A company’s cost function is $$C(x) = 1000 + 5x^2$, and the revenue function is R(x) = 100x$. For what values of x$ does the company make a profit?

Accepted Answer

$$x \u003e 10$$

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