Q: A population grows according to $P(t) = 1000e^{0.05t}$. What is the instantaneous rate of growth at $t = 10$?

$P'(10) = 0.05 \times 1000e^{0.5} \approx 82$

Q: A medication decays in the bloodstream according to $C(t) = 50e^{-0.2t}$. What is the rate of change of concentration at $t = 5$?

$C'(5) = -0.2 \times 50e^{-1} \approx -3.68$

Q: A company’s cost function is $C(x) = 200 + 5x$. What is the average cost per item when $x = 20$?

$C_{avg} = \frac{200 + 5 \times 20}{20} = 15$

Q: A savings account earns interest compounded continuously at a rate of $6\%$. If the initial deposit is $\$500$, what is the amount after $5$ years?

$A = 500e^{0.06 \times 5} \approx 674$

Question 1

Which of the following functions is NOT linear?

Accepted Answer

$f(x) = x^2 + 3$

Question 2

A company’s profit $P$ (in thousands of dollars) is modeled by $P(x) = 5x^2 - 20x + 100$, where $x$ is the number of units sold (in thousands). What is the minimum profit, and at how many units sold does it occur?

Accepted Answer

Minimum profit is $80$ at $x = 2$

Question 3

Which of the following limits does NOT exist?

Accepted Answer

$\lim_{x 	o 0} \frac{1}{x}$

Question 4

Find the derivative of $f(x) = 3x^4 - 2x^2 + 7$.

Accepted Answer

$f'(x) = 12x^3 - 4x$

Question 5

A business’s revenue $R(x)$ is given by $R(x) = 100x - x^2$, where $x$ is the number of items sold. What is the marginal revenue when $x = 10$?

Accepted Answer

$R'(10) = 80$

Question 6

Which rule should be used to find the derivative of $f(x) = (2x^3 + 1)(x^2 - 5)$?

Accepted Answer

Product Rule

Question 7

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

Accepted Answer

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

Question 8

The price $p$ and quantity $q$ of a product are related by the demand function $p = 50 - 2q$. What is the elasticity of demand when $q = 10$?

Accepted Answer

Elasticity $E = \frac{2q}{50 - 2q}$, so $E = \frac{20}{30} = 0.67$

Question 9

Which of the following is the correct application of the Chain Rule for $f(x) = \sin(3x^2)$?

Accepted Answer

$f'(x) = \cos(3x^2) \cdot 6x$

Question 10

A population grows according to $P(t) = 1000e^{0.05t}$. What is the instantaneous rate of growth at $t = 10$?

Accepted Answer

$P'(10) = 0.05 	imes 1000e^{0.5} \approx 82$

Question 11

Find the second derivative of $f(x) = x^3 - 4x^2 + 2x$.

Accepted Answer

$f''(x) = 6x - 8$

Question 12

Which of the following is the correct derivative of $f(x) = \ln(x^2 + 1)$?

Accepted Answer

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

Question 13

A medication decays in the bloodstream according to $C(t) = 50e^{-0.2t}$. What is the rate of change of concentration at $t = 5$?

Accepted Answer

$C'(5) = -0.2 	imes 50e^{-1} \approx -3.68$

Question 14

Evaluate the definite integral $\int_0^2 (3x^2 - 2x + 1) dx$.

Accepted Answer

$8$

Question 15

Which of the following is the correct antiderivative of $f(x) = 4x^3$?

Accepted Answer

$F(x) = x^4 + C$

Question 16

A company’s cost function is $C(x) = 200 + 5x$. What is the average cost per item when $x = 20$?

Accepted Answer

$C_{avg} = \frac{200 + 5 	imes 20}{20} = 15$

Question 17

Which of the following is the correct application of the Quotient Rule for $f(x) = \frac{x^2}{x + 1}$?

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

A savings account earns interest compounded continuously at a rate of $6\%$. If the initial deposit is $\$500$, what is the amount after $5$ years?

Accepted Answer

$A = 500e^{0.06 	imes 5} \approx 674$

Business Calculus: Course Syllabus and Topic Overview

Study Guide - Practice Questions

Study Guide - Flashcards