Interpreting the Meaning of the Derivative in Context

Benjamin Wright

8 min read

Next Topic - Straight-Line Motion: Connecting Position, Velocity, and Acceleration

Listen to this study note

Study Guide Overview

This unit covers interpreting the meaning of derivatives in context. It focuses on understanding the derivative as an instantaneous rate of change, including real-world examples and practice problems. Key concepts include relating the derivative's units to the original function and variable, and using the derivative to describe how a quantity is changing at a specific moment. Practice questions and solutions reinforce these concepts.

#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.

Key Concept

The derivative $f'(x)$ tells you how fast $f(x)$ is changing at a particular value of $x$ . The units of $f'(x)$ are always the units of $f(x)$ divided by the units of $x$ .

#✏️ 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.

Quick Fact

Units of the Derivative: If $f(x)$ is measured in 'A' units and $x$ is measured in 'B' units, then $f'(x)$ is measured in 'A per B' units. This is a quick way to check your interpretation.

#📝 Interpreting Derivatives: Practice Problems

Let's test your understanding with a few practice problems. Remember to focus on the units and what they represent in each context!

#❓Interpreting Derivatives: Questions

Question 1: Michael has an ant farm. The function $A(t)$ gives the number of ants on the farm after $t$ days. What is the best interpretation of $A'(5) = 12$ ?

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, Michael’s ant farm is decreasing by 12 ants per day.

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

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.

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 of $P'(3) = 200$ ?

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

Question 1: The correct answer is C). $A'(5) = 12$ means that after 5 days, the number of ants is increasing at a rate of 12 ants per day.

Question 2: The correct answer is A). $F'(2) = -300$ means that after 2 months, Anna's follower count is decreasing at a rate of 300 followers per month. The negative sign indicates a decrease.

Question 3: The correct answer is B). $P'(3) = 200$ means that after 3 days, Daniel's business is earning money at a rate of 200 dollars per day.

Common Mistake

Common Mistake: Confusing the value of the function with the value of the derivative. $f(x)$ gives a quantity, while $f'(x)$ gives the rate of change of that quantity. Always pay attention to the units!

#⭐ Closing

You're doing fantastic! You've now got a solid grasp on interpreting derivatives in context. Remember to always think about the units and what they represent. Keep up the great work, and you'll be ready to ace those AP Calculus questions!

Encouraging GIF with ice cream

Practice Question

#Practice Questions

Multiple Choice Questions

The function $v(t)$ represents the velocity of a particle at time $t$ . If $v(3) = 5$ and $v'(3) = -2$ , which of the following is true? (A) At $t=3$ , the particle is moving at a speed of 5 and accelerating at a rate of -2 (B) At $t=3$ , the particle is moving at a speed of 5 and decelerating at a rate of 2 (C) At $t=3$ , the particle’s position is 5 and its velocity is decreasing at a rate of 2 (D) At $t=3$ , the particle’s velocity is 5 and its acceleration is -2
A spherical balloon is being inflated. The function $V(r)$ gives the volume of the balloon in cubic centimeters when the radius is $r$ centimeters. What does $V'(5)$ represent? (A) The volume of the balloon when the radius is 5 cm. (B) The rate of change of the volume with respect to the radius when the radius is 5 cm. (C) The rate of change of the radius with respect to the volume when the radius is 5 cm. (D) The surface area of the balloon when the radius is 5 cm.

Free Response Question

The temperature of a room, in degrees Fahrenheit, is modeled by the function $T(h)$ , where $h$ is the number of hours since midnight.

(a) What are the units of $T'(h)$ ?

(b) Explain the meaning of $T'(6) = 2$ in the context of the problem.

Solutions:

Multiple Choice Answers

(D) $v(3) = 5$ gives the velocity at $t=3$ and $v'(3) = -2$ gives the acceleration at $t=3$ .
(B) $V'(r)$ represents the rate of change of volume with respect to the radius. $V'(5)$ is the rate of change of the volume with respect to the radius when the radius is 5 cm.

Free Response Solution

(a) The units of $T'(h)$ are degrees Fahrenheit per hour (\textdegree F/hour).

(b) $T'(6) = 2$ means that at 6 hours after midnight, the temperature of the room is increasing at a rate of 2 degrees Fahrenheit per hour.

(c) The tangent line approximation at $h=6$ is given by $L(h) = T(6) + T'(6)(h-6)$ . We know $T(6) = 70$ and $T'(6) = 2$ . Thus, $L(h) = 70 + 2(h-6)$ . To estimate the temperature at $h = 6.5$ , we find $L(6.5) = 70 + 2(6.5 - 6) = 70 + 2(0.5) = 70 + 1 = 71$ . Therefore, the estimated temperature at $h=6.5$ is 71 degrees Fahrenheit.

Scoring Breakdown for FRQ

(a) 1 point: Correct units (degrees Fahrenheit per hour) (b) 2 points: 1 point for correct interpretation of rate of change, 1 point for correct time context (c) 3 points: 1 point for tangent line equation, 1 point for substitution, 1 point for correct approximation