Second Derivative Def - Limit of [2(Average of y values)Around - initial y value)]/dx^2
Inspired by this Video
GeoGebra Findings
With the function e^x (exponential function):
When involving the average of y-values around the initial y-value, the definition of the second derivative can be described by the following formula/equation:
f"(x)=limit (1/h²) * [f(a + h) + f(a - h) - 2f(a)] as delta x approaches 0
f''(x) =limit (1/dx^2) * [2 * (average of f(x) around f(x_initial)) - f(x)] as delta x approaches 0
With the function x^3:
When involving the average of y-values around the initial y-value, the definition of the second derivative can be described by the following formula/equation:
f''(x) = (1/dx^2) * [2 * (average of f(x) around f(x_initial)) - f(x)]
f''(a) ≈ (1/h²) * [f(a + h) + f(a - h) - 2f(a)]
How to Use This GeoGebra Applet?
Open the GeoGebra Applet in app:
Step 1: Change the function from x^3 to e^x.
Step 2: Simulating the Limit Operation
Move the slider k (representing the small change in x) to values close to zero, like 0.0001, or similar small values. Observe that f′′(a)=d=n.
What does this mean?
Step 3: At k=0.0001 (or for other small values of k close to zero):
- Move the slider a (representing the initial x-value) and observe at which parts of the graph you get a negative d and a negative f′′(a). What does this mean?
- Move the slider a (representing the initial x-value) and observe at which parts of the graph you get a positive d and a positive f′′(a). What does this mean?
- Move the slider a (representing the initial x-value) and observe at which parts of the graph you get d and f′′(a)<3. What does this mean?
