Example Data Table
| f(x, y) |
Center (a, b) |
Target (x, y) |
Use Case |
| x^2*y + sin(x*y) |
(1, 2) |
(1.05, 1.95) |
Mixed polynomial and trigonometric surface. |
| sqrt(x*y) + log(x+y) |
(4, 5) |
(4.1, 4.8) |
Domain controlled root and logarithm example. |
| exp(x-y) + x*y^2 |
(1, 0.5) |
(0.9, 0.7) |
Exponential model with interaction terms. |
Formula Used
The calculator uses the tangent plane linearization formula.
L(x, y) = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b)
Partial derivatives are estimated with central differences.
fx(a, b) ≈ [f(a + h, b) - f(a - h, b)] / 2h
fy(a, b) ≈ [f(a, b + h) - f(a, b - h)] / 2h
How to Use This Calculator
- Enter a valid function using x and y.
- Enter the center point where the plane touches the surface.
- Enter the target point for approximation.
- Choose a small derivative step, such as 0.0001.
- Enter a direction angle for directional derivative output.
- Press the calculate button.
- Review the tangent plane, errors, and diagnostics.
- Download the CSV or PDF when needed.
Understanding Linearization of f(x, y)
Linearization gives a nearby plane for a two variable surface. It replaces a curved function with a simple tangent plane. This is useful when exact evaluation is slow, messy, or unavailable. The method uses one point as a base. That point is called (a, b). The calculator estimates the height f(a, b). It also estimates the slope in the x direction. It estimates the slope in the y direction too.
Why This Calculator Helps
Many multivariable problems need fast local estimates. A small change in x and y often creates a predictable change in z. Linearization captures that first order change. It is common in calculus, optimization, economics, engineering, and error analysis. The tool also compares the linear estimate with the real function value. This helps you judge accuracy. A small error means the target point is close enough, or the surface is nearly flat.
What Inputs Matter
The expression must use x and y. You can enter powers, products, quotients, and common functions. Use radians for trigonometric inputs. Choose a center point close to the target point. Pick a small derivative step. Very large steps reduce precision. Extremely tiny steps may increase rounding noise. The default step usually works well for classroom examples.
Reading the Output
The main result is L(x, y). It is the tangent plane formula. The approximation value is L at the target point. The actual value is f at that same point. The absolute error shows direct distance between them. The relative error gives a percentage comparison when the actual value is not zero. The gradient vector shows the steepest local rise. The directional derivative shows slope in your chosen angle. Second partial values are diagnostic only. They warn when curvature may affect accuracy.
Best Practice
Use clean syntax and multiplication signs. Test simple functions first. Move the target point near the center for better estimates. Export the CSV when you need a spreadsheet record. Export the PDF when you need a printable report. Always treat linearization as a local approximation, not a global replacement.
When It Works Best
It works best near smooth points. Avoid sharp corners, discontinuities, and undefined values. Recalculate when the center changes and inputs shift.
FAQs
What is linearization of f(x, y)?
It is a tangent plane approximation for a two variable function near a chosen point. It uses the function value and two partial derivatives.
What variables can I use?
Use x and y only. The expression parser also accepts constants pi and e, plus common functions such as sin, cos, sqrt, exp, and log.
Should trigonometric inputs use degrees?
No. Trigonometric functions use radians. Convert degrees to radians before entering angle values inside sin, cos, tan, or inverse functions.
What does the derivative step mean?
The step h controls the central difference estimate. A moderate small value, such as 0.0001, often balances precision and rounding noise.
Why is my result undefined?
The function may leave its domain near the center or target point. Check roots, logarithms, divisions, and powers with restricted values.
What is relative error?
Relative error compares absolute error with the actual function value. It is shown as a percentage when the actual value is not zero.
What does the gradient magnitude show?
It shows the local steepness of the surface at the center point. Larger values usually mean the surface changes faster nearby.
Can I use this for exact derivatives?
This tool estimates derivatives numerically. It is excellent for quick work, but symbolic tools are better when exact derivative formulas are required.