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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