Change Lab Slides THE CHANGE LAB
Beyond the Physics of Motion
Modeling Real-World Instantaneous Rates of Change
Warm-Up: The Ripple Effect
Brainstorm Challenge
The video ends by saying derivatives are used for "basically any real life scenario."
What else changes over time besides position?
Population Growth
Market Fluctuations
Thermal Cooling
Key Takeaways
TIMESTAMP: 10:49 - 12:44
Embedded media
Definition
"The slope of the tangent line is the instantaneous rate of change."
The Mission
Today, we move from falling rockets to bacteria, stocks, and social hype.
Lab Procedure
01
Analyze Your Scenario
Review your group's dossier. Identify your function \(f(t)\) and the 'critical moment' \(a\).
02
Calculate \(f'(a)\)
Use the limit definition formula to find the instantaneous rate of change at your assigned time.
03
Interpret & Report
What does the number mean? Is it growing? Shrinking? How fast? Prepare your statement.
Formula: \[f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]
Reporting Findings
Each group must present one finding using this script:
"At time T, our VARIABLE was changing at an instantaneous rate of R per UNIT."
Scenario Dossier Handout The Change Lab
Scenario Dossiers & Instructions
Document Ref:
CL-2026-B
Lab Instructions
Identify your assigned Scenario from the dossiers below.
Locate your Modeling Function \(f(t)\) and your Critical Moment \(a\).
Using the Limit Definition of the Derivative, calculate the instantaneous rate of change at exactly time \(t = a\).
Interpret your result: Is the system growing or shrinking? What are the units?
Complete the Rate of Change Record worksheet for presentation.
Dossier Alpha: The Petri Dish
A culture of bioluminescent bacteria is growing in a nutrient-rich agar. Biologists need to know how fast the population is exploding right before they introduce an antibiotic.
Population Function (millions):
B(t) = 2t² + 10t + 500
Critical Moment:
t = 5 hours
Dossier Bravo: The Bull Market
An investment firm is tracking a high-volatility tech stock. The stock price has been rising, but analysts suspect a momentum shift. They need the exact rate of change at high noon.
Stock Price ($):
S(t) = -t² + 20t + 100
Critical Moment:
t = 12 days (since open)
Dossier Charlie: The Social Wave
A viral video is sweeping through a new social media platform. Marketing experts want to know the speed of follower acquisition at the peak of the 72-hour mark.
Follower Count:
V(t) = 10t² + 5t
Critical Moment:
t = 3 days
Dossier Delta: The Arctic Melt
Climate scientists are monitoring the thickness of a critical ice shelf. As summer approaches, the melting speed is accelerating. Calculate the rate of loss at the solstice.
Ice Thickness (meters):
I(t) = 200 - 4t²
Critical Moment:
t = 6 months
REMINDER: \(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\). Do not use power rule shortcuts. Show all expansion steps.
Rate of Change Worksheet Researcher Name:
Lab Group:
Rate of Change Record
Field Analysis Report
Scenario Assigned
Function \(f(t)\)
Critical Moment \(a\)
The Derivative Calculation
Show all limit steps: \(f(a+h)\), expansion, simplification, and substitution.
Final Derivative Value
\(f'(a) =\)
Interpret Your Result
Describe what this number means for your scenario. Is it increasing/decreasing? Include units.
Presentation Script Preview
"At time , our was changing at a rate of per ."
Change Lab Teacher Guide Instructor's Manual
The Change Lab: Real-World Derivatives
Grade Level
11th Grade Precalculus
Total Duration
45-50 Minutes
Core Competency
Limit Definition of Derivative
Pacing & Facilitation
05m
Warm-Up Brainstorm
Encourage students to look past velocity. Focus on variables like volume (filling a pool), temperature (coffee cooling), or followers (viral growth).
05m
Video Rewatch (10:49-12:44)
Reinforce the terminology transition: "Slope of Tangent" \(\to\) "Derivative" \(\to\) "Instantaneous Rate of Change".
30m
Main Activity: Rate of Change Research
Assign groups one of the four dossiers. Circulate and check for common algebra errors in the \((a+h)^2\) expansion.
Activity Master Key
Dossier Alpha: The Petri Dish
B(t) = 2t² + 10t + 500 | a = 5
\(B(5+h) = 2(5+h)^2 + 10(5+h) + 500 = 2(25 + 10h + h^2) + 50 + 10h + 500 = 50 + 20h + 2h^2 + 50 + 10h + 500 = 2h^2 + 30h + 600\)
Interpretation: The population is growing at 30 million bacteria per hour at t = 5.
Dossier Bravo: The Bull Market
S(t) = -t² + 20t + 100 | a = 12
\(S(12+h) = -(144 + 24h + h^2) + 20(12+h) + 100 = -144 - 24h - h^2 + 240 + 20h + 100 = -h^2 - 4h + 196\)
Interpretation: The stock price is decreasing at $4 per day at t = 12.
Dossier Charlie: The Social Wave
V(t) = 10t² + 5t | a = 3
\(V(3+h) = 10(9 + 6h + h^2) + 5(3+h) = 90 + 60h + 10h^2 + 15 + 5h = 10h^2 + 65h + 105\)
Interpretation: The follower count is increasing at 65 followers per day at t = 3.
Dossier Delta: The Arctic Melt
I(t) = 200 - 4t² | a = 6
\(I(6+h) = 200 - 4(36 + 12h + h^2) = 200 - 144 - 48h - 4h^2 = 56 - 48h - 4h^2\)
Interpretation: The ice thickness is shrinking at 48 meters per month at t = 6.
