Graduate · Smoothing a weak derivative · 481 of 650
Smoothing a weak derivative · Smoothing width ε=0.3; Corner location b=1
Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.3; Corner location b=1.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Smooth |x−(1)| by fε(x)=√((x−(1))²+(0.3)²), where ε=0.3>0. Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error.
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A starting point
Set d=x−b. The smooth derivative is d/√(d²+ε²); the function error is √(d²+ε²)−|d|.
Work through the reasoning
Step 1
Identify the model and target
Smooth |x−(1)| by fε(x)=√((x−(1))²+(0.3)²), where ε=0.3>0. Evaluate at x=0.6. The governing relation is fε(x)=√((x−b)²+ε²); fε′=(x−b)/fε; ||fε−|x−b|||∞≤ε. Set d=x−b. The smooth derivative is d/√(d²+ε²); the function error is √(d²+ε²)−|d|.
Step 2
Substitute and calculate
At d=x−b=-0.4, the smooth derivative is d/√(d²+ε²)=-0.4/√(-0.4²+0.3²)=-0.8. Subtracting |d| from the smooth function gives error 0.1, at most ε=0.3.
Step 3
Check the mathematical meaning
The error 0.1 lies between 0 and ε=0.3. The smooth derivative has magnitude at most one. This function-error bound does not imply uniform convergence of derivatives across the corner. Input x: 0.6; Smooth derivative: -0.8; Sign representative: -1; Smoothing error: 0.1. Decimal values are rounded, so use unrounded intermediate values.
The answer
At d=x−b=-0.4, the smooth derivative is d/√(d²+ε²)=-0.4/√(-0.4²+0.3²)=-0.8. Subtracting |d| from the smooth function gives error 0.1, at most ε=0.3. The error 0.1 lies between 0 and ε=0.3. The smooth derivative has magnitude at most one. This function-error bound does not imply uniform convergence of derivatives across the corner. Animation check: Input x: 0.6; Smooth derivative: -0.8; Sign representative: -1; Smoothing error: 0.1. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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The mathematical idea
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. Smooth |x−(1)| by fε(x)=√((x−(1))²+(0.3)²), where ε=0.3>0. Evaluate at x=0.6. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
fε(x)=√((x−b)²+ε²); fε′=(x−b)/fε; ||fε−|x−b|||∞≤ε
03 · Reflect and transfer
Explain what changes and why.
Why does changing the sign representative at the one corner leave the weak derivative unchanged as an almost-everywhere class?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.