Example Data Table
Try these sample entries to test local approximations and compare error behavior.
| Function | Variables | Base point | Target point | Use case |
|---|---|---|---|---|
sin(x*y) + x^2 - y/3 |
2 | (1, 2) | (1.15, 2.1) | Tangent plane practice |
sqrt(x^2 + y^2 + z^2) |
3 | (3, 4, 2) | (3.1, 3.9, 2.2) | Distance sensitivity |
exp(x-y) + log(z+5) |
3 | (0.5, 0.2, 1) | (0.55, 0.15, 1.1) | Growth model estimate |
Formula Used
For a two-variable function, the first-order linearization at (a,b) is:
L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)For a three-variable function, the model 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)The calculator estimates each partial derivative with central differences:
fx(a,b,c) ≈ [f(a+h,b,c) - f(a-h,b,c)] / (2h)How to Use This Calculator
- Enter a smooth function using variables x, y, and optionally z.
- Select whether the function uses two or three variables.
- Enter the base point where the tangent model should be built.
- Enter the nearby target point you want to estimate.
- Adjust the derivative step size only when precision needs tuning.
- Press submit, then review the gradient, estimate, actual value, and error.
- Use the graph to compare the true function with the linear model.
- Download CSV or PDF reports for records, study notes, or project files.
Linearization Guide
What linearization means
Linearization gives a simple local model for a hard function. It replaces a curved surface with a tangent plane near a chosen point. This is useful when exact evaluation is slow, noisy, or hard to explain.
Why gradients matter
For a function of two variables, the base point is usually written as (a, b). The calculator finds the function value at that point. Then it estimates the partial derivative in each direction. These derivatives describe how fast the function changes when only one input moves.
Accuracy and error
The linear model is best near the base point. Small input changes usually give small error. Large jumps can bend away from the tangent plane. That is why the tool also compares the estimate with the actual target value. The error section helps you decide whether the approximation is acceptable.
Three variable models
For three variables, the same idea extends to a tangent hyperplane. The gradient holds the slopes for x, y, and z. Each slope is multiplied by the matching input change. The sum gives the first order correction to the base value.
Numerical derivative method
This calculator uses numerical central differences. It evaluates the function slightly forward and backward in each variable. The difference ratio gives a stable slope estimate for many smooth functions. A smaller step can improve precision, but it may also increase rounding noise. The default step is a practical starting point.
Reading the graph
The graph samples points along the straight path from the base point to the target point. It plots the actual function and the linear estimate. When the lines stay close, the approximation is strong along that path. When they separate quickly, the function has strong curvature.
Study and project use
Students can use this page to check tangent plane homework. Engineers can estimate quick changes in models. Data analysts can study local sensitivity. The downloadable CSV and PDF reports make it easier to save assumptions, inputs, derivatives, and errors.
Best practice
Always enter a smooth function when possible. Corners, jumps, or undefined regions can break a local model. If the actual value returns an error, adjust the target point or expression. Keep units consistent, and label outputs clearly when your function represents cost, distance, temperature, pressure, or probability. Review results before sharing them.
FAQs
1. What is multivariable linearization?
It is a first-order approximation of a function near a base point. It uses the function value and partial derivatives to build a tangent plane or tangent hyperplane.
2. When is the estimate most accurate?
It is most accurate near the base point. Accuracy usually decreases as the target point moves farther away or the function has stronger curvature.
3. Does this calculator find symbolic derivatives?
No. It uses numerical central differences. This makes it practical for many entered functions without needing a symbolic algebra engine.
4. Which functions are supported?
You can use common functions such as sin, cos, tan, sqrt, abs, exp, log, ln, log10, pow, min, and max.
5. What step size should I use?
The default value works for many smooth functions. Try a smaller or larger value if the gradient looks unstable or the function changes sharply.
6. Why does my expression return an error?
The expression may divide by zero, use an invalid logarithm, use an invalid square root, or contain unsupported syntax. Check the input values.
7. What does the graph show?
It compares actual function values with linearized estimates along the straight path from the base point to the target point.
8. Can I use this for three variables?
Yes. Select three variables and use x, y, and z in the expression. The calculator will build a local tangent hyperplane.