4.4 Tangent planes and linear approximations  (Page 3/11)

Notice that this equation also represents the tangent plane to the surface defined by $z=f\left(x,y\right)$ at the point $\left(x_0,y_0\right).$ The idea behind using a linear approximation is that, if there is a point $\left(x_0,y_0\right)$ at which the precise value of $f\left(x,y\right)$ is known, then for values of $\left(x,y\right)$ reasonably close to $\left(x_0,y_0\right),$ the linear approximation (i.e., tangent plane) yields a value that is also reasonably close to the exact value of $f\left(x,y\right)$ ( [link] ). Furthermore the plane that is used to find the linear approximation is also the tangent plane to the surface at the point $\left(x_0,y_0\right).$

Differentiability

When working with a function $y=f\left(x\right)$ of one variable, the function is said to be differentiable at a point $x=a$ if $f^{\prime }\left(a\right)$ exists. Furthermore, if a function of one variable is differentiable at a point, the graph is “smooth” at that point (i.e., no corners exist) and a tangent line is well-defined at that point.

The idea behind differentiability of a function of two variables is connected to the idea of smoothness at that point. In this case, a surface is considered to be smooth at point $P$ if a tangent plane to the surface exists at that point. If a function is differentiable at a point, then a tangent plane to the surface exists at that point. Recall the formula for a tangent plane at a point $\left(x_0,y_0\right)$ is given by

For a tangent plane to exist at the point $\left(x_0,y_0\right),$ the partial derivatives must therefore exist at that point. However, this is not a sufficient condition for smoothness, as was illustrated in [link] . In that case, the partial derivatives existed at the origin, but the function also had a corner on the graph at the origin.