Tangent Planes and Linear Approximations: Examples

1. Find the Tangent plane of the equation below at the point (1,2,2). Also demonstrate that the tangent plane is a good linear approximation of the function using the point (1.1,2.1).

Answer:

First, find $f_{x}$ and $f_{y}$

Now plug everything into the equation to find the tangent plane:

Now, put the point (1.1,2.1) into both the original equation and the tangent plane:

Tangent Plane

2. Use differentials to find the change in the following function from (1,1) to (1.01, 0.97):

Answer:

First, find $\Delta x$ and $\Delta y:$

Now, use the definition of the differential to find the change:
MATH

MATH

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