Golden Function Teacher Guide The Golden Function
Visualizing the Magic of \( e^x \)
AP Calculus AB/BC
Limits • Derivatives • Integrals
45-50 Minutes
Objective: Verify Slope = Height = Area for \( y = e^x \)
Warm-Up: The Tangent Test (5 min)
Have students sketch \( y = 2^x \) on their worksheet. Ask them to draw a tangent line at \( x = 0 \).
Is the slope of the tangent line steeper or shallower than the y-value?
What about at \( x = 1 \)?
Goal: Intuition that base 2 grows "too slowly" for the slope to match the height.
Direct Instruction: Euler's Number (15 min)
Video Resource
Logarithms - e - Euler's Number
Note: Start the video at 9:29. This section transitions from the compound interest/series derivation to the specific calculus properties (Derivative and Integral).
Key Insight 1
\( \frac{d}{dx}[e^x] = e^x \)
Slope = Height at every point.
Key Insight 2
\( \int_{-\infty}^{x} e^t dt = e^x \)
Accumulated Area = Height.
Main Activity: The Discovery (20 min)
Using Desmos or GeoGebra:
Plot \( f(x) = 2^x \), \( g(x) = 3^x \), and \( h(x) = e^x \).
Create the derivative functions: \( f'(x) \), \( g'(x) \), and \( h'(x) \).
Use a slider for \( x = a \) to track the tangent slope vs. y-value.
The Challenge: Identify exactly which base causes the function and its derivative to overlap perfectly.
Prep Checklist
Laptops/Tablets for Desmos/GeoGebra
"Golden Function" Guided Worksheets
Discussion Cards (Pre-cut)
Closure (5 min)
"If the area from \( -\infty \) to \( x \) is finite (it equals \( e^x \)), does that mean if we rotate this shape around the x-axis, the volume is also finite?"
Teacher Note:
This seeds the idea for Gabriel's Horn and improper integrals.
Key Differentiation
Scaffold: Provide the Desmos link with functions pre-loaded but derivative display toggled off.
Extend: Ask students to find the derivative of \( 2^x \) and determine what constant \( k \) makes \( \frac{d}{dx}[2^x] = k \cdot 2^x \). (Hint: \( k = \ln 2 \)).
Golden Function Presentation Slides \( \int \)
\( e^x \)
The Golden Function
Uncovering the unique calculus properties of Euler's Number
Slope = Height = Area
Warm-Up: The Tangent Test
On your worksheet, sketch the function \( y = 2^x \).
Challenge:
Draw a tangent line at \( x = 0 \). Look at the slope. Is it equal to the y-value at that point?
Sketch area on worksheet
\( (0, 1) \)
Visual Evidence
Embedded media
9:29 - 11:58
Calculus Properties of \( e \)
Watch for Slope and Area!
Main Activity: Discovery
01
Graph Bases
Plot \( y = 2^x \), \( y = 3^x \), and \( y = e^x \). Color code them!
02
Find Derivatives
Type d/dx for each function.
03
The Overlap
Observe which derivative function lies exactly on top of its parent.
Goal: Why is \( e \) called "the fixed point of differentiation"?
Desmos GeoGebra
The Integral Property
Definition
\[ \int_{-\infty}^{x} e^t dt = e^x \]
The accumulated area from "forever ago" up to any point \( x \) is exactly equal to the height of the function at that point.
Height Area
Final Thought
"If the area under \( e^x \) from \( -\infty \) to \( 0 \) is finite (it equals 1), does the shape have a finite volume if we rotate it?"
Discussion Question
Golden Function Student Worksheet The Golden Function
Name:
Date:
Learning Objective
Visualize and verify the unique calculus property of Euler's Number: \( f(x) = f'(x) = \int_{-\infty}^x f(t) dt \).
1
Warm-Up: The Tangent Test
Sketch the graph of \( y = 2^x \) on the grid to the right. Then, draw a tangent line at \( x = 0 \).
Observation
At \( x = 0 \), the y-value is 1. Looking at your tangent line, would you estimate the slope (\( m \)) is:
Greater than 1 (Steeper)
Exactly 1
Less than 1 (Shallower)
0 x y
2
Video Evidence: The Calculus of \( e \)
Note: Focus on the properties discussed from 9:29 onwards.
The Slope Relationship
Explain the significance of the tangent slope equaling the y-value.
The Area Relationship
How does the accumulated area from \( -\infty \) relate to the height?
3
Main Activity: The Discovery
Using graphing software, explore different exponential bases. Record the exact values at \( x = 0 \).
Function Height at \( x=0 \) Slope at \( x=0 \) Perfect Overlap? \( y = 2^x \) \( y = 3^x \) \( y = e^x \)
Analysis Questions
1. For which base was the derivative steeper than the original function? For which base was it shallower?
2. As the base increases, what happens to the slope of the tangent at the y-intercept?
3. Based on your data, why is \( e \) called the "fixed point" of differentiation?
4
Closure Discussion
"If the area from \( -\infty \) to \( 0 \) is finite (it equals 1), does that mean the shape created by rotating this curve around the x-axis has finite volume?"
Prepare your argument for the class discussion.
Golden Function Discussion Cards Golden Function Discussion Cards
Cut along the dotted lines. Use these prompts to drive deep inquiry during the Desmos activity.
Growth Rates
The "Too Slow" Base
Why does base \( 2 \) grow "too slowly" for its slope to match its height? If the y-value is 1, why is the slope less than 1?
Card #01 • Geometric Intuition
The Overlap
Perfect Symmetry
When the function and its derivative overlap exactly, what does that tell you about the speed of growth relative to the total value?
Card #02 • Analytical Discovery
Accumulation
The Infinite Past
The video mentions area from \( -\infty \). How is it possible to have a finite area when the curve goes back forever?
Card #03 • Improper Integrals
Logarithms
The Base Transition
If \( \frac{d}{dx}[2^x] \neq 2^x \), how could we use a constant to fix it? What is that constant called?
Card #04 • Derivatives of \( a^x \)
Discussion Tip: Ask groups to rotate cards every 4 minutes during the activity.
Golden Function Answer Key Answer Key & Teacher Guide
The Golden Function: Euler's Number in Calculus
1
Warm-Up: The Tangent Test
Observation for \( y = 2^x \) at \( x=0 \):
Expected Answer: Less than 1 (Shallower).
The actual slope is \( \ln(2) \approx 0.693 \). Students should notice visually that the line is not at a 45-degree angle (which would be slope 1).
2
Video Evidence
Derivative Viewpoint
"The rate of change is equal to the value of the function. At any point, the slope of the tangent line equals the y-coordinate."
Integral Viewpoint
"The accumulated area under the curve from negative infinity up to point x is equal to the value of the function at that point."
3
Main Activity: Discovery
Function Height (\( x=0 \)) Slope (\( x=0 \)) Overlap? \( y = 2^x \) 1 0.693 No \( y = 3^x \) 1 1.099 No \( y = e^x \) 1 1.000 YES
Analysis 1: Larger/Smaller Derivatives
Base 3 had a derivative larger than the function. Base 2 had a derivative smaller than the function.
Analysis 2: Effect of Base
Increasing the base increases the slope of the tangent at the y-intercept. This suggests there must be a base between 2 and 3 where the slope is exactly 1.
4
Closure: Finite vs. Infinite
Conceptual Guidance
Yes, the volume is finite! This leads to the "Gabriel's Horn" paradox (though usually associated with \( 1/x \)). For \( e^x \) rotated around the x-axis, the integral of \( \pi (e^x)^2 \) from \( -\infty \) to \( 0 \) is \( \int \pi e^{2x} dx = [\frac{\pi}{2} e^{2x}] = \frac{\pi}{2} \).
