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.

 xfx,y tells us the slope of tangents in the x direction, and  yfx,y 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 f__xx,y represent the partial derivative function with respect to x, and let f__yx,y represent the partial derivative function with respect to y.

The gradient of a function fx,y, symbolized fx,y, is defined

fx,y = f__xx,y,f__yx,y 

You can also evaluate the gradient at any particular values x,y = a,b.

The tangent plane of fx,y at a point a,b is defined

z  = fa,b + f__xa,bxa + f__ya, byb

Vector Definition

The gradient is often interpreted as a vector.

Let α  = ab, γ =xy, and  fγ = f__xγf__yγ

Then, we can write the formula for the tangent plane as

fα + fαγα

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: fx,y=

xf   =
 yf   =

 
x value of point =   

y value of point =  

Point on surface = 
Tangent plane =  z=

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