#AP Calculus AB/BC: Rates of Change in Applied Contexts 🚀

Hey there, future AP Calculus master! Let's dive into rates of change beyond just motion. You've already rocked rectilinear motion; now, let's apply those skills to other real-world scenarios. This is where calculus gets super cool and practical! 😎

This topic is crucial because it connects derivatives to real-world applications, which is a common theme in both multiple-choice and free-response questions. Expect to see these concepts in various forms on the exam.

# 📉 Rates of Change in Applied Contexts (Non-Motion)

The key to understanding rates of change in non-motion problems is to focus on what the given function represents.

• Function Meaning: If $f(x)$ gives the volume (in liters) of water in a tank $t$ minutes after the drain opens, then: * $f(t)$ = Volume of water at time t * $f'(t)$ = Rate of change of volume (liters per minute) at time t

#🚶‍♀️ Walkthrough: Pogo Stick Jumping

Let's break down an example: Karen's height on a pogo stick is given by:

$$H(t) = 3\sin\left(\frac{t}{10}\right) + \frac{1}{2}$$

• Goal: Find the instantaneous rate of change of her height at $t = 10$ seconds. This means we need to find $H'(10)$.

• Step 1: Find the derivative, $H'(t)$:

$$H'(t) = \frac{3}{10}\cos\left(\frac{t}{10}\right)$$

• Step 2: Evaluate at $t = 10$:

$$H'(10) = \frac{3}{10}\cos(1) \approx 0.162$$

• Answer: The instantaneous rate of change of Karen's height is approximately 0.162 feet per second. The units are feet per second because the original function was in feet and time was in seconds.