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

Key Concept

Remember, the derivative $f'(x)$ always represents the instantaneous rate of change of the function $f(x)$ . The units of $f'(x)$ are always the units of $f(x)$ divided by the units of $x$ .

🚶‍♀️ 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.

Exam Tip

Always include units in your answers! This is a common place where students lose points on the AP exam. Make sure to show all your work, including the derivative and the evaluation.

📝 Practice Problems

Time to try it yourself! Remember to think about what the function represents and what the derivative tells you.

❓ Problems

Question 1:

Thomas's Instagram likes are modeled by:

$L(t) = 200e^{0.1t}$

Find the instantaneous rate of change of likes 5 days after posting.

Question 2:

Jen's gas tank volume is modeled by:

$G(t) = 300 + 4t$

Find the instantaneous rate of change of the gas volume 4 minutes after she starts filling up.

✅ Answers and Solutions

Question 1:

Derivative: $L'(t) = 200 \cdot 0.1e^{0.1t} = 20e^{0.1t}$
Evaluate at $t = 5$ : $L'(5) = 20e^{0.5} \approx 32.97$
Answer: The rate of change is approximately 32.97 likes per day.

Question 2:

Derivative: $G'(t) = 4$
Evaluate at $t = 4$ : $G'(4) = 4$
Answer: The rate of change is 4 liters per minute. Notice that the rate of change is constant in this case.

Memory Aid

Units Matter! Always include units in your answers. The units of the derivative are the units of the function divided by the units of the variable. For example, if $f(t)$ is in meters and $t$ is in seconds, then $f'(t)$ is in meters per second.

Practice Question

 {
  "multiple_choice": [
    {
      "question": "The function <math-inline>P(t)</math-inline> models the population of a city at time <math-inline>t</math-inline> years. What does <math-inline>P'(5)</math-inline> represent?",
      "options": [
        "The population of the city in 5 years.",
        "The average population of the city over the first 5 years.",
        "The rate of change of the population at <math-inline>t=5</math-inline> years.",
        "The total population increase over the first 5 years."
      ],
      "answer": "The rate of change of the population at <math-inline>t=5</math-inline> years."
    },
    {
      "question": "A spherical balloon is being inflated. The function <math-inline>V(r)</math-inline> gives the volume of the balloon in cubic centimeters as a function of its radius <math-inline>r</math-inline> in centimeters. What are the units of <math-inline>V'(r)</math-inline>?",
      "options": [
        "cm",
        "cm^2",
        "cm^3",
        "cm^3/cm"
      ],
      "answer": "cm^3/cm"
    },
   {
      "question": "The temperature of a room, in degrees Celsius, is given by the function <math-inline>T(h)</math-inline>, where <math-inline>h</math-inline> is the number of hours since noon. What does <math-inline>T'(3)</math-inline> represent?",
      "options": [
        "The temperature of the room at 3 PM.",
        "The average temperature of the room between noon and 3 PM.",
        "The rate at which the temperature is changing at 3 PM.",
       "The total change in temperature between noon and 3 PM."
      ],
      "answer": "The rate at which the temperature is changing at 3 PM."
    }
  ],
  "free_response": {
    "question": "The amount of water in a tank, in gallons, is modeled by the function <math-inline>W(t) = 100 + 20t - t^2</math-inline>, where <math-inline>t</math-inline> is measured in minutes for $0 \le t \le 10$.\n(a) Find <math-inline>W'(t)</math-inline>. What does <math-inline>W'(t)</math-inline> represent in the context of this problem?\n(b) Find <math-inline>W'(5)</math-inline>. What does this value represent in the context of this problem?\n(c) At what time <math-inline>t</math-inline> is the water level in the tank increasing most rapidly? Justify your answer.\n(d) At what time <math-inline>t</math-inline> is the water level in the tank decreasing most rapidly? Justify your answer.",
    "scoring_guidelines": {
      "(a)": {
        "points": 2,
        "description": "1 point for the correct derivative, 1 point for the correct interpretation"
      },
      "(b)": {
        "points": 2,
        "description": "1 point for the correct value, 1 point for the correct interpretation"
      },
      "(c)": {
        "points": 2,
        "description": "1 point for finding the critical point, 1 point for justification"
      },
       "(d)": {
        "points": 3,
        "description": "1 point for finding the critical point, 1 point for justification, 1 point for correct answer"
      }
    },
    "solution": {
      "(a)": "<math-inline>W'(t) = 20 - 2t</math-inline>. <math-inline>W'(t)</math-inline> represents the rate of change of the amount of water in the tank in gallons per minute at time <math-inline>t</math-inline>.",
      "(b)": "<math-inline>W'(5) = 20 - 2(5) = 10</math-inline>. This means that at <math-inline>t=5</math-inline> minutes, the amount of water in the tank is increasing at a rate of 10 gallons per minute.",
      "(c)": "To find when the water level is increasing most rapidly, we need to find when <math-inline>W'(t)</math-inline> is maximized. Since <math-inline>W'(t)</math-inline> is a linear function with a negative slope, it is maximized at the smallest value of <math-inline>t</math-inline>, which is <math-inline>t=0</math-inline>. Thus the water level is increasing most rapidly at <math-inline>t=0</math-inline>.",
       "(d)": "To find when the water level is decreasing most rapidly, we need to find when <math-inline>W'(t)</math-inline> is minimized. Since <math-inline>W'(t)</math-inline> is a linear function with a negative slope, it is minimized at the largest value of <math-inline>t</math-inline>, which is <math-inline>t=10</math-inline>. However, <math-inline>W'(10) = 20-2(10)=0</math-inline>. The water level is decreasing when <math-inline>W'(t)<0</math-inline>, which is when $20-2t<0$ or <math-inline>t>10</math-inline>. However, since the domain is $0 \le t \le 10$, we need to check the endpoints. At <math-inline>t=10</math-inline>, the rate of change is zero. Therefore, the water level is decreasing most rapidly at <math-inline>t=10</math-inline>."
    }
  }
}

Final Exam Focus 🎯

High-Priority Topics: * Interpreting derivatives in context (especially non-motion). * Understanding units of derivatives. * Applying derivatives to solve real-world problems.
Common Question Types: * Multiple-choice questions asking for the interpretation of a derivative. * Free-response questions involving finding and interpreting derivatives in applied contexts.
Last-Minute Tips: * Time Management: Don't spend too much time on one question. If you're stuck, move on and come back later. * Common Pitfalls: Forgetting units, not showing work, and misinterpreting the question are common mistakes. Double-check your work! * Strategies: Read the question carefully, identify what the function represents, and then think about what the derivative means in that context.

You've got this! You're well-prepared, and now it's time to show off your calculus skills. Go get that 5! 🌟

Rates of Change in Applied Contexts other than Motion

Benjamin Wright

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

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

🚶‍♀️ Walkthrough: Pogo Stick Jumping

📝 Practice Problems

❓ Problems

Question 1:

Question 2:

✅ Answers and Solutions

Question 1:

Question 2:

Final Exam Focus 🎯

How are we doing?