riaology
Unit 1 · Lesson 8 of 8

Choose an input tolerance that guarantees an output tolerance.

A calibration engineer specifies an allowed output error. The slope determines how tightly the input must be controlled.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its input tolerance δ.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase allowed output error ε? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Read the modelf(x)=mx+1,L=ma+1,∣f(x)−L∣=∣m∣∣x−a∣<ε if ∣x−a∣<δ=ε∣m∣f(x)=mx+1,\quad L=ma+1,\quad |f(x)-L|=|m||x-a|<\varepsilon\ \text{if }|x-a|<\delta=\frac\varepsilon{|m|}

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

Distance from input target versus Distance from output limit. Output error, Allowed error. Selected input tolerance δ: 0.1 . Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.001.50.531Distance from output limitDistance from input target
Output errorAllowed error · dashed

input tolerance δδ=0.33=0.1\delta=\frac{0.3}{3}=0.1

Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.

Assumptions, units, and limits

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

The formal limit definition starts with every positive output tolerance. You must produce a positive input tolerance that makes all sufficiently nearby inputs satisfy the output requirement. A linear function turns this proof into an explicit inequality.

f(x)=mx+1,L=ma+1,∣f(x)−L∣=∣m∣∣x−a∣<ε if ∣x−a∣<δ=ε∣m∣f(x)=mx+1,\quad L=ma+1,\quad |f(x)-L|=|m||x-a|<\varepsilon\ \text{if }|x-a|<\delta=\frac\varepsilon{|m|}
02

Keep the reasoning visible

0<∣x−a∣<δ⇒∣f(x)−L∣<ε0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon

Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.

03

What makes this model useful?

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

The definition is a guarantee for all inputs in a punctured interval, not a check at just one selected input.

A worked example

Calculate the starting scenario

  1. Use allowed output error ε = 0.3 ; slope magnitude = 3 .
  2. δ=0.33=0.1\delta=\frac{0.3}{3}=0.1
  3. Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band. The checked quantity is input tolerance δ.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the input tolerance δ using allowed output error ε 0.3 , slope magnitude 3 . State the meaning of its sign and units.

Checked quantity: input tolerance δ · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

The formal limit definition starts with every positive output tolerance. You must produce a positive input tolerance that makes all sufficiently nearby inputs satisfy the output requirement. A linear function turns this proof into an explicit inequality.

Explore the visual solution
Read the modelf(x)=mx+1,L=ma+1,∣f(x)−L∣=∣m∣∣x−a∣<ε if ∣x−a∣<δ=ε∣m∣f(x)=mx+1,\quad L=ma+1,\quad |f(x)-L|=|m||x-a|<\varepsilon\ \text{if }|x-a|<\delta=\frac\varepsilon{|m|}

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

Distance from input target versus Distance from output limit. Output error, Allowed error. Selected input tolerance δ: 0.1 . Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.001.50.531Distance from output limitDistance from input target
Output errorAllowed error · dashed

input tolerance δδ=0.33=0.1\delta=\frac{0.3}{3}=0.1

Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.

Assumptions, units, and limits

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

The formal limit definition starts with every positive output tolerance. You must produce a positive input tolerance that makes all sufficiently nearby inputs satisfy the output requirement. A linear function turns this proof into an explicit inequality.

f(x)=mx+1,L=ma+1,∣f(x)−L∣=∣m∣∣x−a∣<ε if ∣x−a∣<δ=ε∣m∣f(x)=mx+1,\quad L=ma+1,\quad |f(x)-L|=|m||x-a|<\varepsilon\ \text{if }|x-a|<\delta=\frac\varepsilon{|m|}
Calculate and explain

Apply the idea to a new scenario

For ε=0.2 and slope magnitude 4, find a suitable δ.

Checked quantity: input tolerance δ · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

The formal limit definition starts with every positive output tolerance. You must produce a positive input tolerance that makes all sufficiently nearby inputs satisfy the output requirement. A linear function turns this proof into an explicit inequality.

Explore the visual solution
Read the modelf(x)=mx+1,L=ma+1,∣f(x)−L∣=∣m∣∣x−a∣<ε if ∣x−a∣<δ=ε∣m∣f(x)=mx+1,\quad L=ma+1,\quad |f(x)-L|=|m||x-a|<\varepsilon\ \text{if }|x-a|<\delta=\frac\varepsilon{|m|}

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

Distance from input target versus Distance from output limit. Output error, Allowed error. Selected input tolerance δ: 0.05 . Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.0020.541Distance from output limitDistance from input target
Output errorAllowed error · dashed

input tolerance δδ=0.24=0.05\delta=\frac{0.2}{4}=0.05

Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.

Assumptions, units, and limits

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

The formal limit definition starts with every positive output tolerance. You must produce a positive input tolerance that makes all sufficiently nearby inputs satisfy the output requirement. A linear function turns this proof into an explicit inequality.

f(x)=mx+1,L=ma+1,∣f(x)−L∣=∣m∣∣x−a∣<ε if ∣x−a∣<δ=ε∣m∣f(x)=mx+1,\quad L=ma+1,\quad |f(x)-L|=|m||x-a|<\varepsilon\ \text{if }|x-a|<\delta=\frac\varepsilon{|m|}
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Read the modelf(x)=mx+1,L=ma+1,∣f(x)−L∣=∣m∣∣x−a∣<ε if ∣x−a∣<δ=ε∣m∣f(x)=mx+1,\quad L=ma+1,\quad |f(x)-L|=|m||x-a|<\varepsilon\ \text{if }|x-a|<\delta=\frac\varepsilon{|m|}

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

Distance from input target versus Distance from output limit. Output error, Allowed error. Selected input tolerance δ: 0.1 . Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.001.50.531Distance from output limitDistance from input target
Output errorAllowed error · dashed

input tolerance δδ=0.33=0.1\delta=\frac{0.3}{3}=0.1

Choose the input tolerance to guarantee the requested output tolerance. A steeper function requires a narrower input band.

Assumptions, units, and limits

Linear function with positive slope m. The strict inequalities apply inside the δ interval, not at its boundary.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. The formal limit definition starts with every positive output tolerance. You must produce a positive input tolerance that makes all sufficiently nearby inputs satisfy the output requirement. A linear function turns this proof into an explicit inequality.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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