riaology
Unit 3 · Lesson 3 of 6

Approximate a trajectory by following short tangents.

An Euler solver predicts the next temperature using the present cooling rate. Compare its line segments with the exact curve.

Change the input. Follow the mathematics.

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

What will change if you increase step size δt? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Coffee cup at 59.4 degrees CelsiusSurroundings: 22°C
Time (min) versus Temperature (°C). Euler segments, Exact cooling. Selected Euler temperature estimate: 59.366 °C. Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.00402804Temperature (°C)Time (min)
Euler segmentsExact cooling · dashed

Euler temperature estimateTEuler(4)=59.366,Texact(4)=60.2192T_{Euler}(4)=59.366,\quad T_{exact}(4)=60.2192

Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.

Assumptions, units, and limits

Euler approximates the same cooling IVP. A final shorter step lands exactly on the chosen endpoint.

Slope field for euler, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

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

01

From the situation to an equation

Evaluate the slope at the current state, multiply by step duration, and add the increment. Repeat using the updated state. Accumulated numerical error depends on step size; stability can also fail when steps are too large.

Tn+1=Tn+Δt[−0.1(Tn−20)],T0=80T_{n+1}=T_n+\Delta t[-0.1(T_n-20)],\quad T_0=80
02

Keep the reasoning visible

yn+1=yn+hf(tn,yn)y_{n+1}=y_n+h f(t_n,y_n)

Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.

03

What makes this model useful?

Euler approximates the same cooling IVP. A final shorter step lands exactly on the chosen endpoint.

Euler is a numerical approximation. The exact curve is shown because this example has an analytic solution, not because every IVP does.

A worked example

Calculate the starting scenario

  1. Use step size δt = 1 min; end time = 4 min.
  2. TEuler(4)=59.366,Texact(4)=60.2192T_{Euler}(4)=59.366,\quad T_{exact}(4)=60.2192
  3. Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example. The checked quantity is Euler temperature estimate.
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 Euler temperature estimate using step size δt 1 min, end time 4 min. State the meaning of its sign and units.

Checked quantity: Euler temperature estimate · °C. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Evaluate the slope at the current state, multiply by step duration, and add the increment. Repeat using the updated state. Accumulated numerical error depends on step size; stability can also fail when steps are too large.

Explore the visual solution
Coffee cup at 59.4 degrees CelsiusSurroundings: 22°C
Time (min) versus Temperature (°C). Euler segments, Exact cooling. Selected Euler temperature estimate: 59.366 °C. Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.00402804Temperature (°C)Time (min)
Euler segmentsExact cooling · dashed

Euler temperature estimateTEuler(4)=59.366,Texact(4)=60.2192T_{Euler}(4)=59.366,\quad T_{exact}(4)=60.2192

Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.

Assumptions, units, and limits

Euler approximates the same cooling IVP. A final shorter step lands exactly on the chosen endpoint.

Slope field for euler, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

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.

Evaluate the slope at the current state, multiply by step duration, and add the increment. Repeat using the updated state. Accumulated numerical error depends on step size; stability can also fail when steps are too large.

Tn+1=Tn+Δt[−0.1(Tn−20)],T0=80T_{n+1}=T_n+\Delta t[-0.1(T_n-20)],\quad T_0=80
Calculate and explain

Apply the idea to a new scenario

Use step size 0.5 min to compute cooling temperature at 6 min from 80°C.

Checked quantity: Euler temperature estimate · °C. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Evaluate the slope at the current state, multiply by step duration, and add the increment. Repeat using the updated state. Accumulated numerical error depends on step size; stability can also fail when steps are too large.

Explore the visual solution
Coffee cup at 52.4 degrees CelsiusSurroundings: 22°C
Time (min) versus Temperature (°C). Euler segments, Exact cooling. Selected Euler temperature estimate: 52.42161 °C. Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.00403806Temperature (°C)Time (min)
Euler segmentsExact cooling · dashed

Euler temperature estimateTEuler(6)=52.4216,Texact(6)=52.9287T_{Euler}(6)=52.4216,\quad T_{exact}(6)=52.9287

Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.

Assumptions, units, and limits

Euler approximates the same cooling IVP. A final shorter step lands exactly on the chosen endpoint.

Slope field for euler, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

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.

Evaluate the slope at the current state, multiply by step duration, and add the increment. Repeat using the updated state. Accumulated numerical error depends on step size; stability can also fail when steps are too large.

Tn+1=Tn+Δt[−0.1(Tn−20)],T0=80T_{n+1}=T_n+\Delta t[-0.1(T_n-20)],\quad T_0=80
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
Coffee cup at 59.4 degrees CelsiusSurroundings: 22°C
Time (min) versus Temperature (°C). Euler segments, Exact cooling. Selected Euler temperature estimate: 59.366 °C. Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.00402804Temperature (°C)Time (min)
Euler segmentsExact cooling · dashed

Euler temperature estimateTEuler(4)=59.366,Texact(4)=60.2192T_{Euler}(4)=59.366,\quad T_{exact}(4)=60.2192

Euler follows the current tangent for one step. The finite step introduces error; reducing step size improves this stable example.

Assumptions, units, and limits

Euler approximates the same cooling IVP. A final shorter step lands exactly on the chosen endpoint.

Slope field for euler, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

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

The reasoning. Evaluate the slope at the current state, multiply by step duration, and add the increment. Repeat using the updated state. Accumulated numerical error depends on step size; stability can also fail when steps are too large.

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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