Linear Approximation 3 Variables Calculator

Calculator

f(x₀, y₀, z₀)

∂f/∂x at base point

∂f/∂y at base point

∂f/∂z at base point

x₀

y₀

z₀

Target x

Target y

Target z

Calculate Approximation Reset

Plotly Graph

This graph shows the linear model contribution from x, y, and z around the selected base point.

Example Data Table

Formula Used

For a function of three variables, the linear approximation near a base point is:

L(x,y,z) = f(x₀,y₀,z₀) + f_x(x₀,y₀,z₀)(x-x₀) + f_y(x₀,y₀,z₀)(y-y₀) + f_z(x₀,y₀,z₀)(z-z₀)

This model estimates the function near the chosen point by using tangent plane behavior. The method works best when the target point stays close to the base point.

How to Use This Calculator

1. Enter the known function value at the base point.
2. Enter the three partial derivatives at that same point.
3. Provide the base coordinates x₀, y₀, and z₀.
4. Enter the target coordinates x, y, and z.
5. Click the calculate button to estimate the nearby function value.
6. Review the step terms, total change, and graph.
7. Export the output to CSV or PDF when needed.

About Linear Approximation in Three Variables

Purpose

Linear approximation helps estimate complicated multivariable functions using local derivative information. It replaces a hard function with an easier tangent plane model near one selected point.

Why It Works

If the function is smooth, tiny input changes create nearly linear output changes. Partial derivatives measure how sensitive the function is along each coordinate direction.

Best Use Case

This method is strongest for nearby points. Accuracy often drops when the target moves farther away from the base point because curvature becomes more important.

Interpretation

The result combines the base value and three directional change terms. Each term shows how much one coordinate contributes to the final estimated change.

FAQs

1. What does this calculator estimate?

It estimates a nearby function value for a three-variable function. It uses a base value and partial derivatives to build a tangent plane approximation.

2. When is the estimate most accurate?

The estimate is usually best when the target point is close to the base point. Small coordinate changes keep the tangent plane closer to the true surface.

3. Why are partial derivatives required?

Each partial derivative measures change along one variable while holding the others fixed. Together, they describe the local slope pattern of the function.

4. What is the meaning of Δx, Δy, and Δz?

They represent the movement from the base point to the target point in each coordinate direction. These differences scale the derivative contributions.

5. Can this replace the exact function value?

No. It is an approximation, not the exact value. It is useful when exact evaluation is hard or when quick local estimates are sufficient.

6. What does the gradient magnitude show?

It summarizes the overall strength of local change at the base point. Larger values often indicate stronger sensitivity to small input changes.

7. Why include a graph?

The graph helps visualize how the linear model responds to variable changes. It makes the estimated behavior easier to inspect and explain.

8. What if my target point is far away?

The linear estimate may become less reliable. For distant points, higher-order methods or the exact function should be preferred.