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Find the (linear) equation of the tangent plane to the surface z = 6 y^2 ? 4 x^2 + x at the...

Question:

Find the (linear) equation of the tangent plane to the surface {eq}z=6y^2-4x^2+x {/eq} at the point (3,-1,-27) . Your answer should be in the form of an equation.

Tangent Plane

The equation of the plane tangent to a function {eq}z(x,y) {/eq} at a point {eq}(x_0,y_0) {/eq} is found linearizing the function at that point. This corresponds to arrest the Taylor series of the function at the first order terms

$$\displaystyle T(x,y) = z(x_0,y_0) + z_x(x_0,y_0) (x-x_0) + z_y(x_0,y_0) (y-y_0) $$

where

$$z_x, \; z_y $$

are the partial derivatives of the function.

Answer and Explanation:

The equation of the tangent plane to the surface

{eq}z(x,y)=6y^2-4x^2+x {/eq}

at the point (3,-1,-27) is found linearizing the function at that point

$$\displaystyle T(x,y) = z(3,-1) + z_x(3,-1) (x-3) + z_y(3,-1) (y+1) \\ \displaystyle z(3,-1) = -27 \\ \displaystyle z_x(x,y) = -8x+1 \rightarrow z_x(3,-1) = -23 \\ \displaystyle z_y(x,y) = 12y \rightarrow z_y(3,-1) = -12 \\ \displaystyle T(x,y) = z(3,-1) + z_x(3,-1) (x-3) + z_y(3,-1) (y+1) = \rightarrow \\ -27 -23(x-3) -12(y+1) =30-23x-12y $$


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Linearization of Functions

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Chapter 10 / Lesson 1
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Over the river and through the woods to Grandmother's house we go ... Are we there yet? In this lesson, apply linearization to estimate when we will finally get to Grandma's house!


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