Q: Suppose the cost function for producing x units of a product is $$C(x) = 0.5x^2$ + 120x + 1200. $$What is the marginal cost when producing 20 units?

$$C'(20) = 0.5 \times 2 \times 20 + 120 = 20 + 120 = 140$$

Q: Given the revenue function $$R(x) = 60x - 5x^2$, what is the marginal revenue when x = 6$?

$$R'(6) = 60 - 10 \times 6 = 60 - 60 = 0$$

Q: A factory's cost function is $$C(x) = 3x^2$ + 14x + 108. $$What is the exact cost of producing the 16th unit?

$$C(16) - C(15) = [3 \times 16^2$ + 14 \times 16 + 108] - [3 \times 15^2$ + 14 \times 15 + 108] = [768 + 224 + 108] - [675 + 210 + 108] = 1100 - 993 = 107$$

Q: If the price-demand equation is $$p = 60 - 5x$$, what is the price when 8 units are sold?

$$p = 60 - 5 \times 8 = 60 - 40 = 20$$

Q: Given $$C(x) = 200 + 20x$$, what is the average cost per unit when producing 10 units?

$$\frac{C(10)}{10} = \frac{200 + 20 \times 10}{10} = \frac{200 + 200}{10} = \frac{400}{10} = 40$$

Q: A company’s revenue function is $$R(x) = 60x - 3x^2$. If production increases from 150 to 151 units, what is the approximate change in revenue?

$$R'(150) = 60 - 6 \times 150 = 60 - 900 = -840$$

Question 1

Suppose the cost function for producing x units of a product is $$C(x) = 0.5x^2$ + 120x + 1200. $$What is the marginal cost when producing 20 units?

Accepted Answer

$$C'(20) = 0.5 	imes 2 	imes 20 + 120 = 20 + 120 = 140$$

Question 2

Given the revenue function $$R(x) = 60x - 5x^2$, what is the marginal revenue when x = 6$?

Accepted Answer

$$R'(6) = 60 - 10 	imes 6 = 60 - 60 = 0$$

Question 3

A factory's cost function is $$C(x) = 3x^2$ + 14x + 108. $$What is the exact cost of producing the 16th unit?

Accepted Answer

$$C(16) - C(15) = [3 	imes 16^2$ + 14 	imes 16 + 108] - [3 	imes 15^2$ + 14 	imes 15 + 108] = [768 + 224 + 108] - [675 + 210 + 108] = 1100 - 993 = 107$$

Question 4

If the price-demand equation is $$p = 60 - 5x$$, what is the price when 8 units are sold?

Accepted Answer

$$p = 60 - 5 	imes 8 = 60 - 40 = 20$$

Question 5

Given $$C(x) = 200 + 20x$$, what is the average cost per unit when producing 10 units?

Accepted Answer

$$\frac{C(10)}{10} = \frac{200 + 20 	imes 10}{10} = \frac{200 + 200}{10} = \frac{400}{10} = 40$$

Question 6

A company finds its break-even point where profit $$P(x) = 0. If$$ $$R(x) = 60x - 5x^2$ and C(x) = 200 + 20x$, what is the break-even quantity?

Accepted Answer

$$60x - 5x^2$ = 200 + 20x \Rightarrow 60x - 5x^2$ - 20x - 200 = 0 \Rightarrow 40x - 5x^2$ - 200 = 0 \Rightarrow 5x^2$ - 40x + 200 = 0 \Rightarrow x = 4, x = 10$$

Question 7

If the marginal profit at $$x = 150 is$$ $$P'(150) = 10$$, what does this value represent in a business context?

Accepted Answer

Producing one more unit when $$x = 150$$ will increase profit by approximately $10

Question 8

Given $$f(x) = \frac{1}{x-2}$$, which of the following statements is true about its asymptotes?

Accepted Answer

It has a vertical asymptote at $$x = 2$$ and a horizontal asymptote at $$y = 0$$

Question 9

If $$C(x) = 3x^2$ + 14x + 108$$, what is the marginal cost function?

Accepted Answer

$$C'(x) = 6x + 14$$

Question 10

A company’s revenue function is $$R(x) = 60x - 3x^2$. If production increases from 150 to 151 units, what is the approximate change in revenue?

Accepted Answer

$$R'(150) = 60 - 6 	imes 150 = 60 - 900 = -840$$

Question 11

If $$C(x) = 200 + 20x$$ and $$R(x) = 60x - 5x^2$, what is the profit function P(x$$)$$?

Accepted Answer

$$P(x) = R(x) - C(x) = (60x - 5x^2$) - (200 + 20x) = 40x - 5x^2$ - 200$$

Question 12

If the cost of producing 15 units is $$C(15) = 393$$ and the cost of producing 16 units is $$C(16) = 524$$, what is the exact cost of producing the 16th unit?

Accepted Answer

$$C(16) - C(15) = 524 - 393 = 131$$

Question 13

Given the price-demand equation $$p = 60 - 5x$$, what is the domain of $$x$$ for which the price is non-negative?

Accepted Answer

$$0 \leq x \leq 12$$

Question 14

If $$C(x) = 0.5x^2$ + 120x + 1200$$, what is the added cost of increasing production from 20 to 21 units?

Accepted Answer

$$C(21) - C(20) = [0.5 	imes 21^2$ + 120 	imes 21 + 1200] - [0.5 	imes 20^2$ + 120 	imes 20 + 1200] = [220.5 + 2520 + 1200] - [200 + 2400 + 1200] = 3940.5 - 3800 = 140.5$$

Question 15

If $$f(x)$$ has a horizontal asymptote at $$y = L$$, which of the following is true?

Accepted Answer

$$\lim_{x 	o \infty} f(x) = L$$

Question 16

If $$C(x) = 3x^2$ + 14x + 108$$, what is the average cost per unit when producing 6 units?

Accepted Answer

$$\frac{C(6)}{6} = \frac{3 	imes 36 + 14 	imes 6 + 108}{6} = \frac{108 + 84 + 108}{6} = \frac{300}{6} = 50$$

Question 17

If $$R(x) = 60x - 5x^2$ and C(x) = 200 + 20x$, what is the marginal profit function P$$'(x)$$?

Accepted Answer

$$P'(x) = R'(x) - C'(x) = (60 - 10x) - 20 = 40 - 10x$$

Business Calculus: Marginal Analysis, Cost, Revenue, Profit, and Asymptotes

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