Q: A manufacturer’s cost function is $$C(x) = 500 + 20x + 0.1x^2$. What is the marginal cost when producing 50 units?

$$C'(x) = 20 + 0.2x$$,\ C'(50) = 20 + 0.2 \times 50 = 30$

Q: 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?

$$x = \frac{-b}{2a} = \frac{-40}{2 \times -2} = 10$$

Q: A company’s profit function is $$P(x) = 120x - 3x^2$ - 200. $$What is the maximum profit?

Maximum at $$x = \frac{120}{6} = 20$$,\ $$P(20) = 120 \times 20 - 3 \times 400 - 200 = 2400 - 1200 - 200 = 1000$$

Q: A business’s cost function is $$C(x) = 1000 + 15x. $$What is the average cost per unit when producing 40 units?

$$\frac{C(40)}{40} = \frac{1000 + 15 \times 40}{40} = \frac{1000 + 600}{40} = \frac{1600}{40} = 40$$

Question 1

A company’s revenue $$R(x)$$ from selling $$x$$ units of a product is given by $$R(x) = 100x - 0.5x^2$. What is the marginal revenue when x = 20$ units?

Accepted Answer

$$R'(x) = 100 - x$$,\ R'(20) = 100 - 20 = 80$

Question 2

Which of the following best describes the concept of the derivative in a business context?

Accepted Answer

It gives the instantaneous rate of change of a function, such as marginal cost or marginal revenue.

Question 3

A manufacturer’s cost function is $$C(x) = 500 + 20x + 0.1x^2$. What is the marginal cost when producing 50 units?

Accepted Answer

$$C'(x) = 20 + 0.2x$$,\ C'(50) = 20 + 0.2 	imes 50 = 30$

Question 4

If the demand function for a product is $$p(x) = 200 - 4x$$, what is the elasticity of demand when $$x = 30$$?

Accepted Answer

$$E = \frac{p}{x} \cdot \frac{dp}{dx} = \frac{200 - 4x}{x} \cdot (-4) at$$ $$x=30$$;\ $$E = \frac{200 - 120}{30} \cdot (-4) = \frac{80}{30} \cdot (-4) = -10.67$$

Question 5

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

$$x = \frac{-b}{2a} = \frac{-40}{2 	imes -2} = 10$$

Question 6

Which of the following is NOT a requirement for a function to be continuous at a point $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 7

A company’s profit function is $$P(x) = 120x - 3x^2$ - 200. $$What is the maximum profit?

Accepted Answer

Maximum at $$x = \frac{120}{6} = 20$$,\ $$P(20) = 120 	imes 20 - 3 	imes 400 - 200 = 2400 - 1200 - 200 = 1000$$

Question 8

If $$f(x) = e$$^{0.05x}$$ models the value of an investment over time, what is the instantaneous rate of growth at x = 10$?

Accepted Answer

$$f'(x) = 0.05e$$^{0.05x}$$,\ f'(10) = 0.05e$$^{0.5}$$ \approx 0.05 	imes 1.6487 \approx 0.0824$

Question 9

A business’s cost function is $$C(x) = 1000 + 15x. $$What is the average cost per unit when producing 40 units?

Accepted Answer

$$\frac{C(40)}{40} = \frac{1000 + 15 	imes 40}{40} = \frac{1000 + 600}{40} = \frac{1600}{40} = 40$$

Question 10

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It states that the integral of a function over an interval is equal to the difference in its values at the endpoints.

Question 11

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

Accepted Answer

Maximum at $$x = \frac{80}{2} = 40$$,\ $$R(40) = 80 	imes 40 - 40^2$ = 3200 - 1600 = 1600$$

Question 12

If the supply function is $$S(p) = 2p^2$ + 5p$$, what is the rate of change of supply with respect to price when $$p = 3$$?

Accepted Answer

$$S'(p) = 4p + 5$$,\ S'(3) = 4 	imes 3 + 5 = 12 + 5 = 17$

Question 13

A company’s cost function is $$C(x) = 2000 + 10x + 0.5x^2$. Find the production level x$ that minimizes average cost.

Accepted Answer

Set $$\frac{d}{dx} \left( \frac{C(x)}{x} \right) = 0$$;\ $$\frac{C(x)}{x} = \frac{2000}{x} + 10 + 0.5x$$;\ $$-\frac{2000}{x^2$} + 0.5 = 0 \Rightarrow x^2$ = 4000 \Rightarrow x = 63.25$$

Question 14

If $$f(x) = \ln(x)$$, what is $$f'(x)$$ and what does it represent in a business context?

Accepted Answer

$$f'(x) = \frac{1}{x}$$; it represents the marginal change in a logarithmic growth process, such as diminishing returns.

Question 15

A company’s profit function is $$P(x) = 60x - x^3$. What is the inflection point of the profit function?

Accepted Answer

Set $$P''(x) = -6x = 0 \Rightarrow x = 0$$

Question 16

If the demand for a product is $$q = 500 - 2p$$, what is the price elasticity of demand at $$p = 100$$?

Accepted Answer

$$E = \frac{p}{q} \cdot \frac{dq}{dp} = \frac{100}{500 - 200} \cdot (-2) = \frac{100}{300} \cdot (-2) = -0.67$$

Question 17

A company’s revenue function is $$R(x) = 150x - 2x^2$. What is the maximum revenue and at what production level does it occur?

Accepted Answer

Maximum at $$x = \frac{150}{4} = 37.5$$,\ $$R(37.5) = 150 	imes 37.5 - 2 	imes (37.5)^2$ = 5625 - 2812.5 = 2812.5$$

Question 18

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

Accepted Answer

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

Syllabus Summary: Calculus for Business and Social Sciences (MATH-1325)

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