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Chapter 12: Q. 4 (page 988)

Fill in the blanks to complete each of the following theorem statements:

Let f(x,y)be a function of two variables that is differentiable at the point (x0,y0)The equation of the plane tangent to the surface defined by f(x,y)at (x0,y0)is _____ .

Short Answer

Expert verified

Let f(x,y)be a function of two variables that is differentiable at the point (x0,y0). The equation of the plane tangent to the surface defined by f(x,y) at (x0,y0) is fx(x0,y0)x-x0+fy(x0,y0)y-y0=z-f(x0,y0).

Step by step solution

01

Step 1. Given information

Let f(x,y)be a function of two variables that is differentiable at the point (x0,y0). The equation of the plane tangent to the surface defined by f(x,y) at (x0,y0) is _____ .

02

Step 2. Filling in the blanks

Let f(x,y)be a function of two variables that is differentiable at the point (x0,y0). The equation of the plane tangent to the surface defined by f(x,y)at (x0,y0)is fx(x0,y0)x-x0+fy(x0,y0)y-y0=z-f(x0,y0).

Finding the tangent plane to a surface, which is analogous to finding the equation of a tangent line to a curve. A tangent plane at a point contains all of the lines tangent to that point. For a function of two variables a tangent plane is given by the partial derivatives at a given point on the surface as above.

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