Calculator Inputs
Choose a model, enter the base point and small changes, then compare the linear estimate with the exact function value.
Interactive Plot
Single-variable models show the exact curve and tangent-line approximation. The multivariable model shows the exact surface and tangent plane.
Example Data Table
| Model | Function | Inputs | Base Value | Exact Value | Approximation | Absolute Error |
|---|---|---|---|---|---|---|
| Power | f(x)=3x^2 | x0=2, Δx=0.1 | 12.000000 | 13.230000 | 13.200000 | 0.030000 |
| Exponential | f(x)=2e^(0.5x) | x0=1, Δx=0.05 | 3.297443 | 3.380905 | 3.379879 | 0.001026 |
| Logarithmic | f(x)=4ln(x+2) | x0=1, Δx=0.08 | 4.394449 | 4.499783 | 4.501116 | 0.001333 |
| Multivariable | z=2x^1y^2+1 | x0=2,y0=3,Δx=0.1,Δy=0.2 | 37.000000 | 42.120000 | 42.280000 | 0.160000 |
Formula Used
Core single-variable approximation
Linearization: L(x) = f(x0) + f′(x0)(x − x0)
Differential estimate: Δy ≈ dy = f′(x0)Δx
Approximate function value: f(x0 + Δx) ≈ f(x0) + f′(x0)Δx
Core multivariable approximation
Total differential: dz ≈ fx(x0, y0)Δx + fy(x0, y0)Δy
Tangent plane: z ≈ z0 + fx(x − x0) + fy(y − y0)
Approximate nearby value: z(x0+Δx, y0+Δy) ≈ z0 + fxΔx + fyΔy
Error metrics
Absolute error: |exact − approximation|
Percentage error: (absolute error / |exact|) × 100
Relative sensitivity index: |x0·f′(x0)/f(x0)| for single-variable models
How to Use This Calculator
- Choose a model matching the function form you want to study.
- Enter coefficients and exponents for the selected equation.
- Set the base point where the derivative or partial derivatives are evaluated.
- Enter small changes Δx, and Δy for the multivariable option.
- Click Calculate Approximation to show results above the form.
- Compare the exact value, differential estimate, and error metrics.
- Review the graph to see how well the tangent model matches nearby behavior.
- Use the export buttons to save the result summary as CSV or PDF.
FAQs
1. What does differential approximation mean?
It estimates how a function changes near a known point by using its derivative or partial derivatives. The method replaces the original function with a nearby tangent line or tangent plane.
2. When is the approximation most accurate?
Accuracy is usually highest when the change values are small and the function is smooth near the base point. Large jumps move farther from the tangent model and often increase error.
3. Why compare the exact value with the approximation?
The comparison shows whether the local linear model is good enough for your use case. It also helps you understand curvature and how quickly error grows away from the base point.
4. What is the total differential in two variables?
The total differential combines the x and y effects into one estimate: dz ≈ fxΔx + fyΔy. It measures the approximate output change caused by small simultaneous changes in both inputs.
5. Why can percentage error become large?
Percentage error can grow when the exact value is very small, when the function is strongly curved, or when the chosen input changes are too large for a local approximation.
6. Can the calculator handle negative inputs?
Yes, many models allow them. However, logarithms require a positive argument, denominators cannot be zero, and fractional powers of negative values may become undefined in real numbers.
7. What does the graph show?
For one-variable models, the graph shows the exact curve and its tangent approximation. For the two-variable model, it shows the exact surface and tangent plane near the chosen point.
8. When should I use linearization instead of exact calculation?
Use linearization when you need a fast nearby estimate, sensitivity insight, or a hand-check for small changes. Use the exact calculation when precision across a wider range matters more.