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Tangent Planes and Linear Approximation Calculator

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Example
Created on 2024-06-20Asked by Scarlett Carter (Solvelet student)
Find the equation of the tangent plane to the surface z=x2+y2z = x^2 + y^2 at the point (1,1,2)(1, 1, 2).

Solution

To find the equation of the tangent plane to the surface z=x2+y2z = x^2 + y^2 at the point (1,1,2)(1, 1, 2): 1. **Compute the partial derivatives:** ∂z∂x=2x,∂z∂y=2y. \frac{\partial z}{\partial x} = 2x, \quad \frac{\partial z}{\partial y} = 2y. 2. **Evaluate the partial derivatives at (1,1,2)(1, 1, 2):** ∂z∂x(1,1)=2,∂z∂y(1,1)=2. \frac{\partial z}{\partial x}(1, 1) = 2, \quad \frac{\partial z}{\partial y}(1, 1) = 2. 3. **Equation of the tangent plane:** z−z0=∂z∂x(x0,y0)(x−x0)+∂z∂y(x0,y0)(y−y0). z - z_0 = \frac{\partial z}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial z}{\partial y}(x_0, y_0)(y - y_0). 4. **Substitute the values:** z−2=2(x−1)+2(y−1). z - 2 = 2(x - 1) + 2(y - 1). 5. **Simplify:** z=2x+2y−2. z = 2x + 2y - 2. 6. **Result:** The equation of the tangent plane is z=2x+2y−2z = 2x + 2y - 2. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Harper Wright on Solvelet
1. Find the equation of the tangent plane to the surface z=x2+y2z=x^2+y^2 at the point (1,1,2)(1,1,2).2. Use linear approximation to estimate the value of 4.1 4.1 using the function f(x)=x f(x) = x and the linearization at x=4 x = 4 .,
DefinitionTangent planes are those planes which touch the surface at a point and has the same slope as the surface on that point. We use the tangent plane to approximate the value of a function locally, which is called linear approximation. Recall from examples tangent plane to z=x2+y2 is z=2x+2y — 2 at (1,1,­2)
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