Tangent Plane & Linear Approximation Calculator

This Tangent Plane & Linear Approximation Calculator helps you find the tangent plane, linearization, and differential of a function z = f(x,y) at a given point. The calculator finds the required partial derivatives, builds the tangent plane equation, and uses it to form the linear approximation near the point. If a nearby point is given, it can also estimate the function value there. It is a simple way to check answers, understand the method clearly, and practise multivariable calculus step by step.

Step-by-step method

1. Identify the function and the base point.
2. Compute f_x and f_y with substeps.
3. Evaluate f(a,b), f_x(a,b), and f_y(a,b).
4. Use the tangent plane formula.
5. Write the linearization L(x,y).
6. Write the differential dz.
7. If a nearby point is given, use L(x,y) to approximate the function value there.

Formulas:

Example 1: f(x,y)=x^2+y^2; (1,2)

Step 1A - Identify the function.

In this problem: Read the given function.

Step 1B - Identify the base point.

In this problem: Read the point where the tangent plane and linearization are required.

Step 2A - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

Step 2B - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

Step 2C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 3A - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

Step 3B - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

Step 3C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 4A - Evaluate f at the base point.

In this problem: Substitute the base point values.

Step 4B - Evaluate fx at the base point.

In this problem: Substitute the base point values.

Step 4C - Evaluate fy at the base point.

In this problem: Substitute the base point values.

Step 5A - Use the tangent plane formula.

In this problem: Start with the general formula.

Step 5B - Substitute the values.

In this problem: Substitute f(a,b), f_x(a,b), and f_y(a,b).

Step 6 - Write the linearization.

In this problem: The linearization uses the same expression as the tangent plane right-hand side.

Step 7 - Write the differential.

In this problem: Use dz = f_x(a,b)dx + f_y(a,b)dy.

Final answer: Tangent plane, linearization, and differential computed

Example 2: f(x,y)=x^2+y^2; (1,2); (1.1,2.05)

Step 1A - Identify the function.

In this problem: Read the given function.

Step 1B - Identify the base point.

In this problem: Read the point where the tangent plane and linearization are required.

Step 1C - Identify the nearby point.

In this problem: This point will be used for approximation.

Step 2A - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

Step 2B - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

Step 2C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 3A - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

Step 3B - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

Step 3C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 4A - Evaluate f at the base point.

In this problem: Substitute the base point values.

Step 4B - Evaluate fx at the base point.

In this problem: Substitute the base point values.

Step 4C - Evaluate fy at the base point.

In this problem: Substitute the base point values.

Step 5A - Use the tangent plane formula.

In this problem: Start with the general formula.

Step 5B - Substitute the values.

In this problem: Substitute f(a,b), f_x(a,b), and f_y(a,b).

Step 6 - Write the linearization.

In this problem: The linearization uses the same expression as the tangent plane right-hand side.

Step 7 - Write the differential.

In this problem: Use dz = f_x(a,b)dx + f_y(a,b)dy.

Step 8A - Substitute the nearby point into the linearization.

In this problem: Use the linearization to estimate the function value.

Step 8B - Simplify the approximation.

In this problem: Write the estimated value.

Final answer: Tangent plane, linearization, and differential computed