Q: Suppose the population of bacteria in a petri dish at time $$t (in $$hours) is given by $$P(t) = 500e$$^{0.2t}$$. What is the instantaneous rate of change of the population at t = 5$ hours?

$$P'(5) = 100e$$^{0.2 \times 5}$$

Q: Which of the following is the correct definition of the derivative of a function $$f(x) at a $$point $$x=a$$?

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following best describes the concept of a derivative in the context of life sciences applications?

Accepted Answer

The derivative represents the rate of change of a quantity with respect to another, such as population growth over time.

Question 2

Suppose the population of bacteria in a petri dish at time $$t (in $$hours) is given by $$P(t) = 500e$$^{0.2t}$$. What is the instantaneous rate of change of the population at t = 5$ hours?

Accepted Answer

$$P'(5) = 100e$$^{0.2 	imes 5}$$

Question 3

Which of the following is the correct definition of the derivative of a function $$f(x) at a $$point $$x=a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 4

A scientist models the concentration of a drug in the bloodstream by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate at which the concentration is changing at t = 2$ hours?

Accepted Answer

$$C'(2) = -5e$$^{-1}$$

Question 5

Which of the following functions is NOT a type of life science-related function commonly explored in Business Calculus?

Accepted Answer

$$f(x) = \sin(x) ($$Trigonometric)

Question 6

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 7

A biologist observes that the growth of a plant height $$h (in cm) $$over time $$t (in $$days) is modeled by $$h(t) = 3t^2$ + 2t + 5. $$What is the average rate of change in height from day 2 to day 5?

Accepted Answer

$$\frac{h(5) - h(2)}{5 - 2}$$

Question 8

Which of the following best describes a Riemann sum?

Accepted Answer

A method for approximating the area under a curve by summing the areas of rectangles.

Question 9

Given the function $$f(x) = 2x^3$ - 5x^2$ + 4x - 7$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 6x^2$ - 10x + 4$$

Question 10

A population of fish in a lake is modeled by $$P(t) = 1000(1 - e$$^{-0.3t}$$). $$What is the initial rate of change of the population at $$t = 0$$?

Accepted Answer

$$P'(0) = 0$$

Question 11

Which of the following statements about exponential functions is TRUE in the context of life sciences?

Accepted Answer

Exponential functions can model population growth when resources are unlimited.

Question 12

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

Accepted Answer

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

Question 13

A researcher is studying the decay of a chemical substance, modeled by $$Q(t) = 50e$$^{-0.1t}$$. What is the half-life of the substance?

Accepted Answer

$$t = \frac{\ln(2)}{0.1}$$

Question 14

Which of the following is the correct formula for the sum of a geometric series with first term $$a$$ and common ratio $$r$$ ($$|r| \u003c 1) $$over $$n$$ terms?

Accepted Answer

$$S_n = a \frac{1 - r^n}{1 - r}$$

Question 15

Suppose a population doubles every 3 years. Which equation best models the population $$P(t) at $$time $$t (in $$years), if the initial population is $$P_0$$?

Accepted Answer

$$P(t) = P_0$ \cdot 2$$^{t/3}$$

Question 16

Which of the following is the correct derivative of $$f(x) = \ln(5x)$$?

Accepted Answer

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

Question 17

A medical researcher models the spread of a virus by $$N(t) = 200e$$^{0.4t}$$. How long will it take for the number of cases to reach 1000?

Accepted Answer

$$t = \frac{\ln(5)}{0.4}$$

Question 18

Which of the following is the correct definition of a limit?

Accepted Answer

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

Question 19

A scientist uses the function $$f(x) = 4x^2$ - 3x + 2 to $$model the growth of a cell culture. What is the instantaneous rate of growth at $$x = 2$$?

Accepted Answer

$$f'(2) = 13$$

Business Calculus for Life Sciences: Syllabus and Course Overview

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