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n · ( ( x x 0 ) i + ( y y 0 ) j + ( z f ( x 0 , y 0 ) ) k ) = 0 ( f x ( x 0 , y 0 ) i + f y ( x 0 , y 0 ) j - k ) · ( ( x x 0 ) i + ( y y 0 ) j + ( z f ( x 0 , y 0 ) ) k ) = 0 f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 ) ( z f ( x 0 , y 0 ) ) = 0 .

Solving this equation for z gives [link] .

Finding a tangent plane

Find the equation of the tangent plane to the surface defined by the function f ( x , y ) = 2 x 2 3 x y + 8 y 2 + 2 x 4 y + 4 at point ( 2 , −1 ) .

First, we must calculate f x ( x , y ) and f y ( x , y ) , then use [link] with x 0 = 2 and y 0 = −1 :

f x ( x , y ) = 4 x 3 y + 2 f y ( x , y ) = −3 x + 16 y 4 f ( 2 , −1 ) = 2 ( 2 ) 2 3 ( 2 ) ( −1 ) + 8 ( −1 ) 2 + 2 ( 2 ) 4 ( −1 ) + 4 = 34. f x ( 2 , −1 ) = 4 ( 2 ) 3 ( −1 ) + 2 = 13 f y ( 2 , −1 ) = −3 ( 2 ) + 16 ( −1 ) 4 = −26.

Then [link] becomes

z = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 ) z = 34 + 13 ( x 2 ) 26 ( y ( −1 ) ) z = 34 + 13 x 26 26 y 26 z = 13 x 26 y 18.

(See the following figure).

A curved surface f(x, y) = 2x2 – 3xy + 8y2 + 2x + 4y + 4 with tangent plane z = 13x – 26y – 18 at point (2, –1, 34).
Calculating the equation of a tangent plane to a given surface at a given point.
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Find the equation of the tangent plane to the surface defined by the function f ( x , y ) = x 3 x 2 y + y 2 2 x + 3 y 2 at point ( −1 , 3 ) .

z = 7 x + 8 y 3

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Finding another tangent plane

Find the equation of the tangent plane to the surface defined by the function f ( x , y ) = sin ( 2 x ) cos ( 3 y ) at the point ( π / 3 , π / 4 ) .

First, calculate f x ( x , y ) and f y ( x , y ) , then use [link] with x 0 = π / 3 and y 0 = π / 4 :

f x ( x , y ) = 2 cos ( 2 x ) cos ( 3 y ) f y ( x , y ) = −3 sin ( 2 x ) sin ( 3 y ) f ( π 3 , π 4 ) = sin ( 2 ( π 3 ) ) cos ( 3 ( π 4 ) ) = ( 3 2 ) ( 2 2 ) = 6 4 f x ( π 3 , π 4 ) = 2 cos ( 2 ( π 3 ) ) cos ( 3 ( π 4 ) ) = 2 ( 1 2 ) ( 2 2 ) = 2 2 f y ( π 3 , π 4 ) = −3 sin ( 2 ( π 3 ) ) sin ( 3 ( π 4 ) ) = −3 ( 3 2 ) ( 2 2 ) = 3 6 4 .

Then [link] becomes

z = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 ) z = 6 4 + 2 2 ( x π 3 ) 3 6 4 ( y π 4 ) z = 2 2 x 3 6 4 y 6 4 π 2 6 + 3 π 6 16 .
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A tangent plane to a surface does not always exist at every point on the surface. Consider the function

f ( x , y ) = { x y x 2 + y 2 ( x , y ) ( 0 , 0 ) 0 ( x , y ) = ( 0 , 0 ) .

The graph of this function follows.

A curved surface that passes through (0, 0, 0) and that folds up on either side of the z axis.
Graph of a function that does not have a tangent plane at the origin.

If either x = 0 or y = 0 , then f ( x , y ) = 0 , so the value of the function does not change on either the x - or y -axis. Therefore, f x ( x , 0 ) = f y ( 0 , y ) = 0 , so as either x o r y approach zero, these partial derivatives stay equal to zero. Substituting them into [link] gives z = 0 as the equation of the tangent line. However, if we approach the origin from a different direction, we get a different story. For example, suppose we approach the origin along the line y = x . If we put y = x into the original function, it becomes

f ( x , x ) = x ( x ) x 2 + ( x ) 2 = x 2 2 x 2 = | x | 2 .

When x > 0 , the slope of this curve is equal to 2 / 2 ; when x < 0 , the slope of this curve is equal to ( 2 / 2 ) . This presents a problem. In the definition of tangent plane , we presumed that all tangent lines through point P (in this case, the origin) lay in the same plane. This is clearly not the case here. When we study differentiable functions, we will see that this function is not differentiable at the origin.

Linear approximations

Recall from Linear Approximations and Differentials that the formula for the linear approximation of a function f ( x ) at the point x = a is given by

y f ( a ) + f ( a ) ( x a ) .

The diagram for the linear approximation of a function of one variable appears in the following graph.

A curve in the xy plane with a point and a tangent to that point. The figure is marked tangent line approximation.
Linear approximation of a function in one variable.

The tangent line can be used as an approximation to the function f ( x ) for values of x reasonably close to x = a . When working with a function of two variables, the tangent line is replaced by a tangent plane, but the approximation idea is much the same.

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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