Calculator Inputs
Enter a base point, the function value there, and the three partial derivatives. Then enter the nearby target point.
Plotly Graph
The chart tracks the linear estimate from the base point to the target point along one straight path.
Formula Used
For a function f(x, y, z), the linear approximation near the base point (a, b, c) is:
L(x, y, z) = f(a, b, c)
+ fx(a, b, c) · (x - a)
+ fy(a, b, c) · (y - b)
+ fz(a, b, c) · (z - c)
In this calculator:
- f(a,b,c) is the known function value at the base point.
- fx, fy, fz are the partial derivatives at the base point.
- (x,y,z) is the nearby point where you want an estimate.
- (x-a, y-b, z-c) measures the movement from the base point.
This method estimates local behavior using the tangent plane in three variables.
How to Use This Calculator
- Enter the base coordinates a, b, and c.
- Enter the nearby target coordinates x, y, and z.
- Provide the known value f(a,b,c).
- Provide fx(a,b,c), fy(a,b,c), and fz(a,b,c).
- Optionally enter the actual function value for error analysis.
- Click the calculation button to show the estimate.
- Review the summary table, graph, and change metrics.
- Use the export buttons to save the result.
Example Data Table
These sample rows show typical situations where a local linear model gives a fast estimate.
| Function | Base Point | Target Point | Approximation |
|---|---|---|---|
| sqrt(x+y+z) | (4,1,0) | (4.1,0.9,0.1) | 2.2450 |
| ln(x+y+z) | (1,1,1) | (1.05,0.98,1.02) | 1.1153 |
| x^2+y^2+z^2 | (1,2,3) | (1.01,1.98,3.03) | 13.1600 |
| e^(x-y+z) | (0,0,0) | (0.02,-0.01,0.03) | 1.0600 |
| xyz | (2,1,3) | (2.1,1.02,2.95) | 6.0700 |
Frequently Asked Questions
1. What does this calculator estimate?
It estimates the value of a three-variable function near a known base point. The estimate uses the function value and three partial derivatives at that base point.
2. When is linear approximation most accurate?
It is usually most accurate when the target point is very close to the base point. Accuracy often drops as the distance from the base point increases.
3. Why do I need partial derivatives?
The partial derivatives describe how the function changes in the x, y, and z directions. Together, they define the local tangent plane used for the estimate.
4. Can this calculator find derivatives for me?
No. This tool uses derivative values you already know. You can obtain them from symbolic work, software, or manual differentiation before using the calculator.
5. What is the directional change shown in the results?
It is the derivative-based adjustment added to the base value. It equals fx·dx + fy·dy + fz·dz, which measures the local change along the chosen move.
6. Why is an actual value field included?
It lets you compare the estimate with the true function value. That comparison produces signed error, absolute error, and percent error for quick validation.
7. What does the graph represent?
The graph shows how the linear model changes from the base point to the target point along a straight path. It helps visualize local behavior and trend direction.
8. Can I use this for any three-variable function?
Yes, as long as you know the base value and partial derivatives at the reference point. The method works for many smooth functions in mathematics, physics, and engineering.