Q: Which of the following is the correct formula for the derivative of a function $$f(x)$$ using the definition of the derivative?

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Q: A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = 100x - 2x^2$, where x$ is the number of units sold (in thousands). What is the marginal profit when x = 10$?

$$P'(10) = 100 - 4 \times 10 = 60$$

Q: If the cost function for producing $$x$$ units is $$C(x) = 500 + 20x + 0.5x^2$, what is the average cost per unit when x = 10$?

$$\frac{C(10)}{10} = \frac{500 + 20 \times 10 + 0.5 \times 10^2$}{10} = \frac{500 + 200 + 50}{10} = 75$$

Q: A company’s revenue function is $$R(x) = 80x - x^2$. What is the marginal revenue when x = 20$?

$$R'(20) = 80 - 2 \times 20 = 40$$

Q: A business’s revenue function is $$R(x) = 120x - 3x^2$. What is the maximum revenue and at what value of x$ does it occur?

Maximum at $$x = 20$$, $$R(20) = 120 \times 20 - 3 \times 20^2$ = 2400 - 1200 = 1200$$

Question 1

Which of the following is the correct formula for the derivative of a function $$f(x)$$ using the definition of the derivative?

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 2

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = 100x - 2x^2$, where x$ is the number of units sold (in thousands). What is the marginal profit when x = 10$?

Accepted Answer

$$P'(10) = 100 - 4 	imes 10 = 60$$

Question 3

If the cost function for producing $$x$$ units is $$C(x) = 500 + 20x + 0.5x^2$, what is the average cost per unit when x = 10$?

Accepted Answer

$$\frac{C(10)}{10} = \frac{500 + 20 	imes 10 + 0.5 	imes 10^2$}{10} = \frac{500 + 200 + 50}{10} = 75$$

Question 4

Which of the following best describes the economic interpretation of the derivative of a cost function $$C(x)$$?

Accepted Answer

It represents the marginal cost, or the cost of producing one additional unit.

Question 5

Suppose the demand function for a product is $$p(x) = 50 - 0.5x$$, where $$p is $$the price and $$x is $$the quantity sold. What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 50x - 0.5x^2$

Question 6

A manufacturer finds that the marginal cost of producing $$x$$ units is $$MC(x) = 10 + 0.2x. $$What is the total cost function $$C(x) if $$the fixed cost is $$500$$?

Accepted Answer

$$C(x) = 500 + 10x + 0.1x^2$

Question 7

If $$f(x) = x^3$ - 5x^2$ + 6x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 3x^2$ - 10x + 6$$

Question 8

A business wants to maximize its profit, given by $$P(x) = -2x^2$ + 40x - 100. $$What is the number of units $$x$$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 9

Which of the following is the correct application of the product rule for differentiation for $$f(x) = u(x)v(x)$$?

Accepted Answer

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

Question 10

A company’s revenue function is $$R(x) = 80x - x^2$. What is the marginal revenue when x = 20$?

Accepted Answer

$$R'(20) = 80 - 2 	imes 20 = 40$$

Question 11

If the supply function is $$S(p) = 2p^2$ + 5p$$, where $$p is $$the price, what is the rate of change of supply with respect to price?

Accepted Answer

$$S'(p) = 4p + 5$$

Question 12

Which of the following is NOT a valid reason for an excused absence according to the syllabus?

Accepted Answer

Personal travel plans

Question 13

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 14

A business calculates its cost function as $$C(x) = 2000 + 15x. $$What is the marginal cost?

Accepted Answer

$$15$$

Question 15

If the profit function is $$P(x) = x^2$ - 10x + 25$$, what is the minimum profit and at what value of $$x$$ does it occur?

Accepted Answer

Minimum at $$x = 5$$, $$P(5) = 0$$

Question 16

Which of the following best describes the chain rule for differentiation?

Accepted Answer

If $$y = f(g(x))$$, then $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

Question 17

A company’s cost function is $$C(x) = 1000 + 25x - 0.5x^2$. For what value of x$ is the marginal cost zero?

Accepted Answer

$$x = 50$$

Question 18

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{1}{x}$$

Question 19

A business’s revenue function is $$R(x) = 120x - 3x^2$. What is the maximum revenue and at what value of x$ does it occur?

Accepted Answer

Maximum at $$x = 20$$, $$R(20) = 120 	imes 20 - 3 	imes 20^2$ = 2400 - 1200 = 1200$$

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