Tangent Planes - Maple Help
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Tangent Planes

Main Concept

Tangent planes are the three-dimensional equivalent of tangent lines.

We can evaluate the derivative of a two-variable function f(x,y) with respect to either variable.

tells us the slope of tangents in the x direction, and  tells us the slope of tangents in the y direction.  If we combine these, we can determine a three-dimensional tangential direction at a given point.  This leads to the creation of a tangent plane.

 Definition Let $\mathrm{f__x}\left(x,y\right)$ represent the partial derivative function with respect to x, and let $\mathrm{f__y}\left(x,y\right)$ represent the partial derivative function with respect to y. The gradient of a function $f\left(x,y\right)$, symbolized $\nabla f\left(x,y\right)$, is defined You can also evaluate the gradient at any particular values . The tangent plane of $f\left(x,y\right)$ at a point $\left(a,b\right)$ is defined
 Vector Definition The gradient is often interpreted as a vector. Let Then, we can write the formula for the tangent plane as using the dot product.

Select a function below, then use the sliders below to select a point.  The plot will display the function and a portion of the tangent plane at the selected point.
Function: $\mathbit{f}\left(\mathbit{x}\mathbf{,}\mathbit{y}\right)\mathbf{=}$

x value of point =    

y value of point =   

Point on surface =
Tangent plane =  $z=$

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