Calculator Inputs
Model the implicit equation F(x, y) = 0 using quadratic, linear, trigonometric, exponential, and logarithmic terms.
Example Data Table
| Scenario | Equation | Point | dy/dx | d²y/dx² | Tangent Type |
|---|---|---|---|---|---|
| Circle | x² + y² - 25 = 0 | (3, 4) | -0.75 | -0.390625 | Oblique tangent |
| Ellipse | 4x² + 9y² - 36 = 0 | (0, 2) | 0 | -0.222222 | Horizontal tangent |
| Mixed conic | x² + xy + y² - 7 = 0 | (2, 1) | -1.25 | -0.65625 | Oblique tangent |
Formula Used
This page models the implicit relation:
F(x, y) = Ax² + Bxy + Cy² + Dx + Ey + Gsin(x) + Hcos(y) + Ieˣ + Jln(y + shift) + K = 0
First partial derivatives
Fx = 2Ax + By + D + Gcos(x) + Ieˣ
Fy = Bx + 2Cy + E - Hsin(y) + J / (y + shift)
Second partial derivatives
Fxx = 2A - Gsin(x) + Ieˣ
Fxy = B
Fyy = 2C - Hcos(y) - J / (y + shift)²
Implicit differentiation formulas
dy/dx = -Fx / Fy
d²y/dx² = -(Fxx + 2Fxy(dy/dx) + Fyy(dy/dx)²) / Fy
Curvature = |y''| / (1 + (y')²)3/2
When Fy = 0, the tangent may become vertical. When both Fx and Fy are near zero, the point is singular.
How to Use This Calculator
- Enter the coefficients for the implicit equation terms.
- Choose a point (x₀, y₀) for evaluation.
- Set graph limits and resolution for the contour view.
- Click Calculate Implicit Results.
- Review the residual, derivatives, tangent line, normal line, curvature, and plotted curve.
- Export the result summary with the CSV or PDF button.
FAQs
1. What does this calculator solve?
It evaluates an implicit equation at a chosen point, then computes the residual, partial derivatives, dy/dx, d²y/dx², tangent line, normal line, and curvature.
2. Does the point need to satisfy the equation?
Yes for a true tangent to F(x, y) = 0. If the residual is not near zero, the slope describes the local level set through that point instead.
3. Why can the slope be undefined?
When Fy equals zero, the tangent may be vertical. In that case dy/dx is not a finite number, so the tool labels the tangent accordingly.
4. What does curvature mean here?
Curvature measures how sharply the local branch bends near the chosen point. Larger values mean tighter turning, while smaller values indicate a flatter local shape.
5. Can I include trigonometric or exponential terms?
Yes. The form supports sin(x), cos(y), ex, and ln(y + shift) along with quadratic, mixed, linear, and constant terms.
6. Why is the logarithm sometimes rejected?
The logarithm term requires y + shift to stay positive. If that condition fails at the selected point, derivative evaluation stops and a domain warning appears.
7. What exactly is shown in the graph?
The chart plots contour lines of F(x, y). The highlighted zero contour represents the implicit curve, while the point, tangent, and normal traces provide context.
8. Can I export my results?
Yes. Use CSV for spreadsheet-ready values and PDF for a clean summary of the equation, selected point, derivatives, slopes, and line equations.