riaology
Unit 2 · Lesson 3 of 11

Changing one dimension can change another.

A fixed-perimeter rectangle grows one side while shrinking the other. Its area depends on both changes.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its area derivative with respect to x.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase rectangle side x? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Rectangle sides 2 and 3 meters; geometry changes at constant perimeter2 m3 mBoth dimensions respond to x
Side x (m) versus Area (m²). Area. Selected area derivative with respect to x: 1 m. Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.003.1252.56.255Area (m²)Side x (m)
Area

area derivative with respect to xA′(2)=(5−2)−2=1A'(2)=(5-2)-2=1

Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.

Assumptions, units, and limits

Rectangle sides x and 5−x5-x; total perimeter is fixed at 10 m. Both factors vary.

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

01

From the situation to an equation

Expand the product or apply the product rule. The two first-order area changes come from a thin strip on each side; the tiny corner product is higher order. These two contributions explain why a product derivative needs two terms.

A(x)=x(5−x),A′=1(5−x)+x(−1)=5−2xA(x)=x(5-x),\quad A'=1(5-x)+x(-1)=5-2x
02

Keep the reasoning visible

(fg)′=f′g+fg′(fg)'=f'g+fg'

Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.

03

What makes this model useful?

Rectangle sides x and 5−x5-x; total perimeter is fixed at 10 m. Both factors vary.

Differentiating each factor and multiplying the derivatives omits the main first-order contributions.

A worked example

Calculate the starting scenario

  1. Use rectangle side x = 2 m.
  2. A′(2)=(5−2)−2=1A'(2)=(5-2)-2=1
  3. Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one. The checked quantity is area derivative with respect to x.
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 area derivative with respect to x using rectangle side x 2 m. State the meaning of its sign and units.

Checked quantity: area derivative with respect to x · m. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Expand the product or apply the product rule. The two first-order area changes come from a thin strip on each side; the tiny corner product is higher order. These two contributions explain why a product derivative needs two terms.

Explore the visual solution
Rectangle sides 2 and 3 meters; geometry changes at constant perimeter2 m3 mBoth dimensions respond to x
Side x (m) versus Area (m²). Area. Selected area derivative with respect to x: 1 m. Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.003.1252.56.255Area (m²)Side x (m)
Area

area derivative with respect to xA′(2)=(5−2)−2=1A'(2)=(5-2)-2=1

Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.

Assumptions, units, and limits

Rectangle sides x and 5−x5-x; total perimeter is fixed at 10 m. Both factors vary.

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.

Expand the product or apply the product rule. The two first-order area changes come from a thin strip on each side; the tiny corner product is higher order. These two contributions explain why a product derivative needs two terms.

A(x)=x(5−x),A′=1(5−x)+x(−1)=5−2xA(x)=x(5-x),\quad A'=1(5-x)+x(-1)=5-2x
Calculate and explain

Apply the idea to a new scenario

At side x=3x=3 m with the other side 5−x5-x, find the area-change rate.

Checked quantity: area derivative with respect to x · m. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Expand the product or apply the product rule. The two first-order area changes come from a thin strip on each side; the tiny corner product is higher order. These two contributions explain why a product derivative needs two terms.

Explore the visual solution
Rectangle sides 3 and 2 meters; geometry changes at constant perimeter3 m2 mBoth dimensions respond to x
Side x (m) versus Area (m²). Area. Selected area derivative with respect to x: -1 m. Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.003.1252.56.255Area (m²)Side x (m)
Area

area derivative with respect to xA′(3)=(5−3)−3=−1A'(3)=(5-3)-3=-1

Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.

Assumptions, units, and limits

Rectangle sides x and 5−x5-x; total perimeter is fixed at 10 m. Both factors vary.

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.

Expand the product or apply the product rule. The two first-order area changes come from a thin strip on each side; the tiny corner product is higher order. These two contributions explain why a product derivative needs two terms.

A(x)=x(5−x),A′=1(5−x)+x(−1)=5−2xA(x)=x(5-x),\quad A'=1(5-x)+x(-1)=5-2x
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
Rectangle sides 2 and 3 meters; geometry changes at constant perimeter2 m3 mBoth dimensions respond to x
Side x (m) versus Area (m²). Area. Selected area derivative with respect to x: 1 m. Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.003.1252.56.255Area (m²)Side x (m)
Area

area derivative with respect to xA′(2)=(5−2)−2=1A'(2)=(5-2)-2=1

Differentiating a product uses two terms. Changing x changes both rectangle sides, not just one.

Assumptions, units, and limits

Rectangle sides x and 5−x5-x; total perimeter is fixed at 10 m. Both factors vary.

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

The reasoning. Expand the product or apply the product rule. The two first-order area changes come from a thin strip on each side; the tiny corner product is higher order. These two contributions explain why a product derivative needs two terms.

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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