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17 sie 2024 · Determine the equation of a plane tangent to a given surface at a point. Use the tangent plane to approximate a function of two variables at a point. Explain when a function of two variables is differentiable. Use the total differential to approximate the change in a function of two variables.
16 lis 2022 · In this section formally define just what a tangent plane to a surface is and how we use partial derivatives to find the equations of tangent planes to surfaces that can be written as z=f(x,y). We will also see how tangent planes can be thought of as a linear approximation to the surface at a given point.
2 mar 2022 · Find the equation of the tangent plane to the surface at \(x = a\text{,}\) \(y = 2a\text{.}\) For what value of \(a\) is the tangent plane parallel to the plane \(x - y + z = 1\text{?}\)
25 lip 2021 · Find the equation of the tangent plane to \[ z = 3x^2 - xy \nonumber \] at the point \((1,2,1)\). Solution. We let \[F(x,y,z) = 3x^2 - xy - z\nonumber \] then \[\nabla F = \langle 6x - y, -x, -1\rangle . \nonumber \] At the point \((1,2,1)\), the normal vector is \[\nabla F(1,2,1) = \langle 4, -1, -1\rangle . \nonumber \]
Tangent Plane to a Surface. Let $(x_0 , y_0 , z_0 )$ be any point on the surface $z = f (x, y)$. If $f (x, y)$ is differentiable at $(x_0 , y_0 )$, then the surface has a tangent plane at $(x_0 , y_0 , z_0 )$ given by $$ f_x (x_0 , y_0 )(x – x_0 ) + f_y (x_0 , y_0 )(y – y_0 ) – (z – z_0 ) = 0. $$
4.4.1 Determine the equation of a plane tangent to a given surface at a point. 4.4.2 Use the tangent plane to approximate a function of two variables at a point. 4.4.3 Explain when a function of two variables is differentiable. 4.4.4 Use the total differential to approximate the change in a function of two variables.
Find the tangent plane and normal line to the surface \(z=f(x,y)=\frac{2y}{x^2+y^2}\) at \((x,y)=(-1,2)\text{.}\)
