4.1 Interpreting the Meaning of the Derivative in Context

Congratulations! Over the past two units, you have become a master at calculating derivatives! But, now what? In Unit 4, we will delve into how derivatives apply to real-world contexts.

🌍 Derivatives in Real-World Contexts

To understand how to interpret the meaning of derivatives in context, we must think back to the definition of a derivative—the derivative of a function measures the instantaneous rate of change, or rate of change at a specific point, with respect to its independent variable.

So, if we understand the context of the given function, then we can easily determine the meaning of the function’s derivative in context as well.

Let’s walk through an example together.

✏️ Derivatives in Context Walkthrough

Let $f(x)$ give the volume, in liters, of water in a tank $t$ minutes after it starts being filled. What does $f'(10)$ mean?

The function $f(x)$ models volume in liters with respect to time in minutes. This means that, for example, $f(10)$ models the volume of water in the tank $10$ minutes after beginning to fill it.

Now, after understanding what the function itself models, we can interpret what its derivative models.

Since the derivative is the instantaneous rate of change, the derivative of $f(x)$ or $f'(x)$ is, therefore, the volume of water, in liters, filling into the tank per minute at a specific point in time. $f'(10)$ can thus be interpreted as “The amount of water filling into the tank per minute after $10$ minutes.”

📝 Interpreting Derivatives: Practice Problems

Give a couple of questions a try yourself!

❓Interpreting Derivatives: Questions

Interpreting Derivatives: Question 1

Michael has an ant farm. The function $A(t)$ gives the amount of ants on the farm after $t$ days. What is the best interpretation $A'(5)=12$?

A) After $5$ hours, Michael’s ant farm is increasing by $12$ ants per hour.

B) After $12$ days, Michael’s ant farm is increasing by $5$ ants per day.

C) After $5$ days, Michael’s ant farm is increasing by $12$ ants per day.

D) After $5$ days, Michaels’ ant farm is decreasing by $12$ ants per day.

Interpreting Derivatives: Question 2

Anna has an Instagram account. The function $F(t)$ gives the amount of followers she has after $t$ months. What is the best interpretation $F'(2)=-300$?

A) After $2$ months, Anna’s account is losing $300$ followers per month.

B) After $2$ months, Anna’s account is gaining $300$ followers per month.

C) After $2$ weeks, Anna’s account is losing $300$ followers per week.

D) After $2$ weeks, Anna’s account is gaining $300$ followers per week.

Interpreting Derivatives: Question 3

Daniel owns a business. The function $P(t)$ gives the amount of money in dollars his business has made after $t$ days. What is the best interpretation $P'(3)=200$?

A) After $3$ months, Daniel’s business is losing $200$ dollars per month.

B) After $3$ days, Daniel’s business is earning $200$ dollars per day.

C) After $3$ days, Daniel’s business has made $200$ dollars.

D) After $3$ days, Daniel’s business has lost $200$ dollars.

✅ Interpreting Derivatives: Answers and Solutions

Interpreting Derivatives: Question 1

$A(t)$ gives the number of ants in Michael’s ant farm after a given time in days, so $A'(t)$ gives the instantaneous rate of change of $A(t)$ in ants per day. Specifically, $A'(5)$ is the rate at which the amount of ants changes at $t=5$ days.

Therefore, the is the best interpretation of $A'(5)=12$ is C) “After $5$ days, Michael’s ant farm is increasing by $12$ ants per day.”

Interpreting Derivatives: Question 2

$F(t)$ gives the number of followers Anna has after $t$ months, so $F'(t)$ gives the instantaneous rate of change of $F(t)$ in followers per month. Specifically, $F'(2)$ is the rate at which the number of followers changes at $t=2$ months. A negative value means that the number of followers is decreasing or the account is losing followers.

Therefore, the is the best interpretation of $F'(2)=-300$ is A) “After $2$ months, Anna’s account is losing $300$ followers per month.”

Interpreting Derivatives: Question 3

$P(t)$ gives the amount of money in dollars Daniel’s business makes after $t$ days, so $P'(t)$ gives the instantaneous rate of change of $P(t)$ in dollars per day. Specifically, $P'(3)$ is the rate at which the amount of money the business has changes at $t=3$ days.

Therefore, the is the best interpretation of $P'(3)=200$ is B) “After $3$ days, Daniel’s business is earning $200$ dollars per day.”

⭐ Closing

Great work! You now have a better idea of how to interpret the meaning of a derivative in the context of a given problem. You got this!

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