Q: 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?

$R'(20) = 100 - 1 \times 20 = 80$

Q: A bank offers continuous compounding on savings at an annual rate of $6\%$. If you deposit $\$2000$, how much will you have after 5 years? (Use $A = Pe^{rt}$)

$A = 2000e^{0.06 \times 5} \approx 2699.72$

Q: A company’s profit function is $P(x) = -2x^2 + 40x - 100$. What is the production level $x$ that maximizes profit?

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

Q: A business’s cost function is $C(x) = 1000 + 15x + 0.05x^2$. What is the marginal cost when $x = 40$?

$C'(40) = 15 + 0.1 \times 40 = 19$

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

$R'(20) = 100 - 1 	imes 20 = 80$

Question 2

If the cost function for producing $x$ items is $C(x) = 500 + 20x + 0.1x^2$, what is the average cost per item when $x = 50$?

Accepted Answer

$\frac{C(50)}{50} = \frac{500 + 1000 + 250}{50} = 35$

Question 3

A bank offers continuous compounding on savings at an annual rate of $6\%$. If you deposit $\$2000$, how much will you have after 5 years? (Use $A = Pe^{rt}$)

Accepted Answer

$A = 2000e^{0.06 	imes 5} \approx 2699.72$

Question 4

Given the demand function $p = 50 - 2x$, where $p$ is the price and $x$ is the quantity sold, what is the elasticity of demand when $x = 10$?

Accepted Answer

$E = \left| \frac{p}{x} \cdot \frac{dx}{dp} \right| = \left| \frac{30}{10} \cdot \frac{1}{-2} \right| = 1.5$

Question 5

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. What is the production level $x$ that maximizes profit?

Accepted Answer

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

Question 6

If $f(x) = e^{3x}$, what is $f''(x)$?

Accepted Answer

$f''(x) = 9e^{3x}$

Question 7

A pharmaceutical company models the spread of a drug in the bloodstream by $C(t) = 10e^{-0.2t}$, where $C$ is the concentration in mg/L and $t$ is time in hours. What is the rate of change of concentration at $t = 3$ hours?

Accepted Answer

$C'(3) = -2e^{-0.6} \approx -1.098$

Question 8

Suppose the supply function for a product is $S(p) = 5p^2 + 10$, where $p$ is the price. What is the rate of change of supply with respect to price when $p = 2$?

Accepted Answer

$S'(2) = 10p = 20$

Question 9

A business’s cost function is $C(x) = 1000 + 15x + 0.05x^2$. What is the marginal cost when $x = 40$?

Accepted Answer

$C'(40) = 15 + 0.1 	imes 40 = 19$

Question 10

If $f(x) = \ln(x^2 + 1)$, what is $f'(x)$?

Accepted Answer

$f'(x) = \frac{2x}{x^2 + 1}$

Question 11

A company’s revenue is $R(x) = 200x - x^2$ and cost is $C(x) = 50x + 1000$. What is the break-even quantity?

Accepted Answer

$x^2 - 150x + 1000 = 0 \implies x = 142.96, 7.04$

Question 12

If $f(x) = 3x^2 - 4x + 7$, what is the equation of the tangent line at $x = 2$?

Accepted Answer

$y = 8x - 9$

Question 13

A company’s profit is maximized when marginal revenue equals marginal cost. If $R(x) = 120x - x^2$ and $C(x) = 40x + 400$, what is the profit-maximizing output?

Accepted Answer

$MR = MC \implies 120 - 2x = 40 \implies x = 40$

Question 14

If $f(x) = \sqrt{4x + 1}$, what is $f'(x)$?

Accepted Answer

$f'(x) = \frac{2}{\sqrt{4x + 1}}$

Question 15

A company’s cost function is $C(x) = 5000e^{0.02x}$. What is the marginal cost when $x = 10$?

Accepted Answer

$C'(10) = 5000 	imes 0.02 	imes e^{0.2} \approx 122.10$

Question 16

If $f(x) = \ln(x)$ and $g(x) = x^2$, what is the derivative of $h(x) = f(g(x))$?

Accepted Answer

$h'(x) = \frac{2x}{x^2}$

Question 17

A company’s revenue is $R(x) = 50x \ln(x)$ for $x > 0$. What is the marginal revenue when $x = 2$?

Accepted Answer

$R'(2) = 50 \ln(2) + 25 \approx 84.66$

Question 18

If $f(x) = x^3 - 6x^2 + 9x + 15$, find all critical points.

Accepted Answer

$f'(x) = 3x^2 - 12x + 9 = 0 \implies x = 1, 3$

Question 19

A company’s cost function is $C(x) = 1000 + 20x$. If the company wants to minimize average cost, what is the optimal production level?

Accepted Answer

There is no minimum; average cost decreases as $x$ increases.

Question 20

If $f(x) = \frac{1}{x}$, what is the equation of the tangent line at $x = 2$?

Accepted Answer

$y = -\frac{1}{4}x + \frac{3}{2}$

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

Study Guide - Practice Questions

Study Guide - Flashcards