#Unit 4: Applications of Derivatives - Interpreting Meaning in Context 🚀

Hey there, future calculus master! You've conquered the art of finding derivatives, and now it's time to unleash their power in the real world. Let's dive into how derivatives tell a story about change!

This unit is all about understanding what derivatives mean, not just how to calculate them. Expect to see these concepts frequently on both multiple-choice and free-response questions.

#4.1 Interpreting the Meaning of the Derivative in Context

#🌍 Derivatives in Real-World Contexts

Remember, the derivative, $f'(x)$, represents the instantaneous rate of change of a function $f(x)$ with respect to its independent variable. Think of it as the speed of change at a specific moment. To interpret a derivative, you need to understand the units of the original function and its independent variable.

#✏️ Derivatives in Context Walkthrough

Let's break it down with an example:

Suppose $f(t)$ gives the volume of water (in liters) in a tank $t$ minutes after it starts filling. What does $f'(10)$ mean?

• $f(t)$ is the volume of water in liters at time $t$ in minutes.

• $f(10)$ is the volume of water in liters after 10 minutes.

• $f'(t)$ is the rate of change of the volume of water in liters per minute.

• $f'(10)$ is the rate at which the water is filling the tank, in liters per minute, exactly 10 minutes after the tank started being filled.