Calculated Result
Calculator Input
Use variables x and y. Example functions include
x^2 + y^2, sin(x) + cos(y), and x*y + y^2.
Interactive Graph
Generated Data Table
| x | y | f(x,y) | g(x,y) | Constraint Status |
|---|---|---|---|---|
| 2 | 2 | 8 | 0 | On constraint |
Example Data Table
This example uses f(x,y)=x²+y² with the constraint x+y-4=0.
| x | y | f(x,y) | g(x,y) | Meaning |
|---|---|---|---|---|
| 0 | 4 | 16 | 0 | Point lies on the constraint. |
| 2 | 2 | 8 | 0 | Lower value on the line. |
| 4 | 0 | 16 | 0 | Point lies on the constraint. |
| 1 | 1 | 2 | -2 | Point is below the equality line. |
Formula Used
A multivariable function is evaluated as z = f(x,y). The calculator samples values of x and y across the selected domain. It then computes the output value for every grid point.
A constraint is tested as g(x,y)=0, g(x,y)≤0, or g(x,y)≥0. For equality constraints, the tolerance value decides whether a point is close enough to the constraint curve.
The numerical gradient near a test point is estimated with central difference: fx ≈ [f(x+h,y)-f(x-h,y)] / 2h and fy ≈ [f(x,y+h)-f(x,y-h)] / 2h.
How to Use This Calculator
Enter your multivariable function in the first field. Use x and y as variables.
Add a constraint if needed. Choose equality, less than, or greater than form.
Set the x and y domain limits. A wider domain shows more of the graph.
Choose a grid resolution. Higher resolution gives smoother graphs but needs more processing.
Enter a test point. The calculator will evaluate the function, constraint, and gradient there.
Press the calculate button. Results appear below the header and above the form.
Multivariable Function and Constraint Graphing Guide
What This Calculator Does
A multivariable function uses more than one input. Most classroom examples use x and y. The output is often called z. This calculator turns that relationship into a surface. It also creates a contour view. The contour view shows level curves from above. This helps you compare height, slope, and shape.
Why Constraints Matter
A constraint limits the points that can be used. It can be an equation or inequality. For example, x plus y equals four is a line. A function may have many values. The constraint selects the allowed values. This is useful in optimization problems. It is also helpful in economics, physics, engineering, and statistics.
Understanding the Graph
The surface graph shows the full function over the chosen domain. Higher areas mean larger function values. Lower areas mean smaller values. The contour graph shows the same function as flat curves. Close contour lines suggest fast change. Wide spacing suggests slow change. Constraint points are marked when they satisfy the selected rule.
Using the Test Point
The test point gives a focused calculation. The tool evaluates f at that point. It also evaluates the constraint expression. A zero value means the point lies on an equality constraint. For inequalities, the sign decides whether the point is feasible. The gradient estimate shows the local direction of fastest increase.
Practical Study Value
This tool is useful before solving by hand. It gives a visual check. It can reveal likely maxima, minima, and boundary behavior. It also helps students understand Lagrange multiplier problems. You can export the sampled data as a CSV file. You can also save a PDF summary for notes, reports, or assignments.
FAQs
1. What is a multivariable function?
A multivariable function uses two or more inputs. In this calculator, the common form is f(x,y). The output becomes z, which can be graphed as a surface or contour plot.
2. What does the constraint field do?
The constraint field limits the valid points. You can test equality, less than, or greater than conditions. This helps show feasible regions and boundary points.
3. Can I graph x squared plus y squared?
Yes. Enter x^2 + y^2 in the function box. You can add a constraint like x + y - 4 to study restricted values.
4. What is constraint tolerance?
Tolerance controls how close a point must be to equality. Smaller values are stricter. Larger values show more nearby points on the graph and table.
5. What does the gradient show?
The gradient estimates the direction where the function increases fastest near the test point. It is calculated with a small numerical step.
6. Why use contour graphs?
Contour graphs show level curves from above. They make it easier to compare height changes, slopes, and possible optimization points.
7. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report with function values and summary details.
8. Is this exact symbolic solving?
No. This calculator uses numerical sampling and graphing. It is best for visualization, checking work, and exploring possible solutions before exact solving.