Graduate · Smoothing a weak derivative
Smoothing a weak derivative: A broad rounded corner
Smoothing a weak derivative: investigate a broad rounded corner with smoothing width ε = 1; corner location b = 0.
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The absolute-value function has a corner but admits a weak first derivative represented by a sign function. A smooth approximation makes the distinction visible: its derivative transitions continuously, while its function error is uniformly bounded by ε. Values assigned to the sign function at the single corner do not affect the weak derivative class. This investigation starts with Smoothing width ε = 1; Corner location b = 0. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.
A relationship to keep
fε(x)=√((x−b)²+ε²); fε′=(x−b)/fε; ||fε−|x−b|||∞≤ε
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Set up the mathematical model
Locate the corner b and compare the exact absolute-value curve with its smooth approximation. The starting case is “A broad rounded corner.”
- STEP 2
Follow the changing quantity
Trace the derivative of the smooth curve. Away from the corner it approaches −1 or +1 as ε shrinks.
- STEP 3
Explain and test the result
Use √(d²+ε²)−|d|≤ε for the function-error bound. This does not give uniform convergence of the derivatives across the corner.
Your turn to explain
Make a prediction. Test your reasoning.
Keep Smoothing width ε = 1; Corner location b = 0. Pause the timeline at 20%. Given input x = -1.8, calculate smooth derivative, sign representative, smoothing error. Show the substitution into the displayed formula.
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
At d=x−b=-1.8, the smooth derivative is d/√(d²+ε²)=-1.8/√(-1.8²+1²)=-0.874157. Subtracting |d| from the smooth function gives error 0.259126, at most ε=1. Results: Smooth derivative: -0.874; Sign representative: -1; Smoothing error: 0.259. Decimal values are rounded; retain the original parameters when checking.
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