Shrinking Gap Slides Pre-Calculus
THE SHRINKING
GAP
Average vs. Instantaneous Rate of Change
The Speed Paradox
HOOK (5 MIN)
The Map
Total Distance: 120 miles
60 mph
The Speedometer
Looking at the needle at
72 mph
Discussion: How can both numbers be "correct"? Which one tells you if you'll get a speeding ticket?
Visualizing Change
ANALYSIS (15 MIN)
Embedded media
Key Question 1:
The average was 4,000/day. Was there any single day where it was exactly 4,000?
Key Question 2:
How does the "secant line" simplify the complex curve of the COVID graph?
Math Checkpoints
1:55
Definition Check
"What variable usually represents vertical change? Horizontal?"
5:12
Prediction Check
"Look at the first interval of \(y = x^2 + 2x\). Predict: will the average rate be positive or negative?"
7:20
Zero Slope Check
"Without doing any math, what is the slope of a horizontal line? Why does this happen on a curve?"
Shrinking the Interval
INVESTIGATION (20 MIN)
The Lab Task
We are going to find the rate of change for \(y = x^2 + 2x\) at the exact point \(x = 1\). To do this, we'll calculate the average rate of change over smaller and smaller gaps.
GAP 1
x = 1 to 3
GAP 2
x = 1 to 2
GAP 3
x = 1 to 1.1
GAP 4
x = 1 to 1.01
Use your worksheet table to record the y-values and slopes!
The Result
As the second point got closer and closer to \(x = 1\), what happened to the slope? Did it start to "level off" or approach a specific number?
TANGENT LINE
When the gap is zero, we aren't finding the average between two points anymore.
Shrinking Gap Worksheet The Shrinking Gap
Pre-Calculus: Average vs. Instantaneous Rate of Change
Name:
Date:
1
Video Case Study: COVID-19 Trends
Based on the video's analysis of August to November 2020:
Average Rate Calculation:
Vertical Change (\(\Delta y\)): ________________________ cases
Horizontal Change (\(\Delta x\)): ________________________ days
Final Average Rate: ________________________ cases/day
Reflect:
The average was 4,000 cases per day. Look at the curve in your mind. Is it likely that the increase was exactly 4,000 on every single day? Why or why not?
2
The Investigation: Shrinking the Interval
Let's look at the function \(f(x) = x^2 + 2x\). We want to find the rate of change exactly at \(x = 1\) . Complete the table below to see what happens to the average rate of change as the second point approaches \(x = 1\).
Interval \([1, x_2]\) Value of \(f(x_1)\) Value of \(f(x_2)\) Slope (\(\Delta y / \Delta x\)) \([1, 3]\) \(f(1) = 3\) Show work: \([1, 2]\) \(f(1) = 3\) \([1, 1.1]\) \(f(1) = 3\) \([1, 1.01]\) \(f(1) = 3\)
3
Analysis: Closing the Gap
1. Observing the Trend
Based on your table, what specific number does the slope seem to be approaching as the interval gets smaller?
2. The "Limit" of Average
If you could make the interval infinitely small (approaching a gap of 0), what do you think the slope would be at exactly \(x = 1\)?
3. Definition Deep Dive
Explain the difference between a secant line and a tangent line using the investigation you just performed.
Exit Ticket
In your own words: Why is it impossible to calculate the slope at a single point using the standard formula \(\frac{y_2 - y_1}{x_2 - x_1}\)?
Shrinking Gap Answer Key Teacher Answer Key
The Shrinking Gap: Average vs. Instantaneous Rate of Change
Pre-Calculus
1. Video Case Study (COVID-19 Trends)
Vertical Change (\(\Delta y\)): 645,000 - 285,000 = 360,000 cases
Horizontal Change (\(\Delta x\)): ~3 months = 90 days
Final Average Rate: 4,000 cases/day
Sample Reflection Answer:
No, the increase was not exactly 4,000 every day. The 4,000 is an average. On some days (like weekends), reporting might be lower, and on other days, spikes might reach much higher than 4,000. The average "smooths out" these fluctuations with a straight line.
2. Investigation: Shrinking the Interval
Function: \(f(x) = x^2 + 2x\) | Target Point: \(x = 1\) (where \(f(1) = 3\))
Interval \([1, x_2]\) \(f(x_2)\) Calculation Value of \(f(x_2)\) Slope (\(\Delta y / \Delta x\)) \([1, 3]\) \(3^2 + 2(3) = 9 + 6\) 15 \((15-3)/(3-1) = 12/2 = 6\) \([1, 2]\) \(2^2 + 2(2) = 4 + 4\) 8 \((8-3)/(2-1) = 5/1 = 5\) \([1, 1.1]\) \(1.1^2 + 2(1.1) = 1.21 + 2.2\) 3.41 \((3.41-3)/0.1 = 0.41/0.1 = 4.1\) \([1, 1.01]\) \(1.01^2 + 2(1.01) = 1.0201 + 2.02\) 3.0401 \((3.0401-3)/0.01 = 4.01\)
3. Analysis & Key Insights
1. Observing the Trend
The slope is approaching 4.
2. The "Limit" of Average
Expected answer: The slope at exactly \(x=1\) is likely 4. (Note: Teachers can point out that \(f'(x) = 2x + 2\), and at \(x=1\), \(f'(1) = 4\).)
3. Secant vs. Tangent
A secant line crosses the curve at two distinct points and represents the average rate of change between them. A tangent line touches the curve at exactly one point and represents the instantaneous rate of change at that moment.
Exit Ticket Answer Guidance
The formula \(\frac{y_2 - y_1}{x_2 - x_1}\) requires two distinct points. If we try to find the slope at a single point (where \(x_1 = x_2\)), the denominator becomes \(1 - 1 = 0\). Division by zero is undefined. This is why we must "approach" the limit rather than calculating it directly with the basic formula.
