Tangent Planes and Linear Approximation
Tangent plane of a graph
Given and a point in the domain of . Suppose is differentiable at . How can we find the equation of the tangent plane to the graph of at ?
We can use the following trick: Define the function of three variables . Then the graph of can be regarded as the level surface . Let . In the previous section, we know that the equation of the tangent plane to the level surface at is as follows:
By computation, we have
, ,
i.e. (normal vector to the tangent plane)
Therefore, we have
Example: Suppose . Find the equation of the tangent plane to the graph of when . Also, find the parametric equation of the normal line when .
Answer:
, . Then , and . Therefore, the equation of the tangent plane is
As for the normal line at , the direction vector is . Hence, its parametric equations are
where is any real number.
Exercise: Find the equation of the tangent plane to the graph of when . Also, find the parametric equations of the normal line when .
Exercise: Find the parametric equation of the tangent line to the curve of intersection of and at . (Hint: Use the normals of the surfaces. You can use GeoGebra applet below to plot the two surfaces to see how their intersection looks like.)
Linear approximation
Let be a function of two variable such that it is differentiable at . Then we have following linear approximation:
Let , we can rewrite the above as follows:
is the linear approximation of when is near (or equivalently, is near ).
Example: Let . Find its linear approximation at and then use it to approximate .
Answer:
, and . Hence and .
We can use this to approximate the value of :
(Note: The approximation is justified only when and are small. In this example, and .)
Remark: For a function of three variables such that it is differentiable at , we can define its linear approximation at in a similar way:
where is near .
Exercise: Find the approximation of at . Then use it to approximate .