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Implicit Curves Explorer

About This MicroSim

Many important curves in mathematics cannot be written in the form \(y = f(x)\). Instead, they are defined implicitly by equations where \(x\) and \(y\) are mixed together, like:

  • Circle: \(x^2 + y^2 = r^2\)
  • Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
  • Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
  • Folium of Descartes: \(x^3 + y^3 = 3axy\)
  • Lemniscate: \((x^2 + y^2)^2 = a^2(x^2 - y^2)\)

Delta Moment

"Wait, you're telling me I can find the slope at any point on this curve, even though I can't solve for \(y\)? That's like knowing the steepness of a hill without having a map that shows the elevation!"

How to Use

  1. Select a Curve: Use the dropdown menu to choose from five different implicit curves
  2. Move the Point: Click anywhere on the curve or drag the blue point to explore different locations
  3. Toggle Tangent: Use the checkbox to show or hide the tangent line
  4. Adjust Parameters: The slider changes the size/shape parameter of each curve
  5. Random Point: Click this button to jump to a random location on the curve

Key Observations

As you explore, notice that:

  1. Tangent lines exist everywhere (except at special points like cusps or self-intersections)
  2. The derivative \(\frac{dy}{dx}\) changes continuously as you move along the curve
  3. Vertical tangents occur when \(\frac{\partial F}{\partial y} = 0\) (undefined slope)
  4. At every smooth point, we can find the slope using implicit differentiation

The Math Behind It

For a curve defined implicitly by \(F(x, y) = 0\), we find the slope using:

\[\frac{dy}{dx} = -\frac{\partial F / \partial x}{\partial F / \partial y} = -\frac{F_x}{F_y}\]

This works because we're applying the chain rule to both sides of \(F(x, y) = 0\) and solving for \(\frac{dy}{dx}\).

Why This Matters

Implicit differentiation is a powerful technique because:

  • Many real-world curves (orbits, contour lines, level curves) are naturally implicit
  • It works even when solving for \(y\) explicitly is impossible
  • The tangent line captures local behavior regardless of the curve's global complexity

Lesson Plan

Learning Objective: Students will recognize that implicit equations define curves where tangent lines exist even when \(y\) cannot be explicitly solved for \(x\) (Bloom Level 2: Understand)

Suggested Activities

  1. Exploration (5 min): Let students freely explore different curves, noting where tangent lines are horizontal, vertical, or at various angles

  2. Pattern Recognition (5 min): For the circle, have students identify:

    • Where is \(\frac{dy}{dx} = 0\)? (horizontal tangent)
    • Where is \(\frac{dy}{dx}\) undefined? (vertical tangent)
    • How does the sign of \(\frac{dy}{dx}\) relate to whether the point is "going up" or "going down"?

  3. Compare Curves (5 min): Switch between different curves at the same point (approximately) and discuss how the tangent lines differ

  4. Challenge (5 min): For the Folium of Descartes, find the point where the curve crosses itself. What happens to the tangent line there?

Technical Details

This MicroSim uses the marching squares algorithm to render implicit curves. The algorithm:

  1. Divides the viewing area into a grid of small squares
  2. Evaluates \(F(x, y)\) at each corner
  3. Draws line segments where the sign of \(F\) changes (curve crosses the cell)

The tangent line is computed using the gradient \(\nabla F = (F_x, F_y)\), which is perpendicular to the curve at every point.

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