Question 1

Which of the following best describes the concept of a 'limit' in calculus, as applied to modeling population growth in biology?

Accepted Answer

The value that a function approaches as the input approaches a certain point.

Question 2

Suppose the cost $C(x)$ to produce $x$ units of a product is given by $C(x) = 500 + 20x$. What is the marginal cost when $x = 100$?

Accepted Answer

$20$

Question 3

A function $f(x)$ is continuous at $x = a$ if which of the following conditions is satisfied?

Accepted Answer

$\lim_{x 	o a} f(x) = f(a)$

Question 4

The derivative of a function at a point represents:

Accepted Answer

The instantaneous rate of change of the function at that point.

Question 5

Given $f(x) = x^2$, what is the average rate of change of $f$ from $x = 2$ to $x = 5$?

Accepted Answer

$\frac{f(5) - f(2)}{5 - 2} = \frac{25 - 4}{3} = 7$

Question 6

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

Accepted Answer

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

Question 7

A hospital tracks the number of patients $P(t)$ admitted per day, modeled by $P(t) = 100e^{0.05t}$, where $t$ is days since the start of the year. What is the instantaneous rate of change of admissions on day $t = 10$?

Accepted Answer

$P'(10) = 100 \cdot 0.05 \cdot e^{0.05 \cdot 10} = 5e^{0.5}$

Question 8

If $f(x)$ is differentiable on $[a, b]$, which theorem guarantees that there exists $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$?

Accepted Answer

The Mean Value Theorem

Question 9

A pharmaceutical company models the concentration $C(t)$ of a drug in the bloodstream by $C(t) = 50e^{-0.1t}$. What is the rate at which the concentration is decreasing at $t = 5$ hours?

Accepted Answer

$C'(5) = -5e^{-0.5}$

Question 10

Which of the following statements about the definite integral $\int_a^b f(x) dx$ is correct in the context of calculating total revenue over a time period?

Accepted Answer

It represents the accumulated total of $f(x)$ from $x = a$ to $x = b$.

Question 11

A population of bacteria grows according to $P(t) = 200e^{0.03t}$. How long will it take for the population to double?

Accepted Answer

$t = \frac{\ln(2)}{0.03}$

Question 12

Which of the following is NOT a requirement for the application of the Mean Value Theorem to a function $f(x)$ on $[a, b]$?

Accepted Answer

$f(x)$ must be increasing on $[a, b]$

Question 13

If $R(x)$ is the revenue from selling $x$ units and $R'(x) = 15$, what does $R'(x)$ represent in a business context?

Accepted Answer

The marginal revenue per unit sold.

Question 14

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

Accepted Answer

$x = 10$

Question 15

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

Accepted Answer

It links the concept of differentiation and integration.

Question 16

Suppose $f(x) = 3x^4 - 2x^2 + x$. What is $f''(x)$, the second derivative?

Accepted Answer

$f''(x) = 36x^2 - 4$

Question 17

A biologist models the rate of change of a population by $\frac{dP}{dt} = 0.2P$. If $P(0) = 100$, what is $P(5)$?

Accepted Answer

$P(5) = 100e^{1}$

Question 18

Which of the following is the correct interpretation of $\int_0^8 v(t) dt$ if $v(t)$ is the velocity of a car in meters per second?

Accepted Answer

The total distance traveled by the car from $t = 0$ to $t = 8$ seconds.

Question 19

If $f(x) = \ln(x)$, what is $\frac{d}{dx}f(x)$?

Accepted Answer

$\frac{1}{x}$

Business Calculus: Core Concepts, Structure, and Success Strategies

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