4.4 Tangent planes and linear approximations  (Page 5/11)

Recall that earlier we showed that the function

was not differentiable at the origin. Let’s calculate the partial derivatives $f_x$ and $f_y\text{:}$

The contrapositive of the preceding theorem states that if a function is not differentiable, then at least one of the hypotheses must be false. Let’s explore the condition that $f_x\left(0,0\right)$ must be continuous. For this to be true, it must be true that $\underset{\left(x,y\right)\to \left(0,0\right)}{\text{lim}}{f}_x\left(0,0\right)=f_x\left(0,0\right)\text{:}$

Let $x=ky.$ Then

If $y>0,$ then this expression equals $1\text{/}{\left(k^2+1\right)}^{3\text{/}2};$ if $y<0,$ then it equals $\text{−}\left(1\text{/}{\left(k^2+1\right)}^{3\text{/}2}\right).$ In either case, the value depends on $k,$ so the limit fails to exist.

Differentials

In Linear Approximations and Differentials we first studied the concept of differentials. The differential of $y,$ written $dy,$ is defined as $f^{\prime }\left(x\right)dx.$ The differential is used to approximate $\text{Δ}y=f\left(x+\text{Δ}x\right)-f\left(x\right),$ where $\text{Δ}x=dx.$ Extending this idea to the linear approximation of a function of two variables at the point $\left(x_0,y_0\right)$ yields the formula for the total differential for a function of two variables.

Notice that the symbol $\partial$ is not used to denote the total differential; rather, $d$ appears in front of $z.$ Now, let’s define $\text{Δ}z=f\left(x+\text{Δ}x,y+\text{Δ}y\right)-f\left(x,y\right).$ We use $dz$ to approximate $\text{Δ}z,$ so

Therefore, the differential is used to approximate the change in the function $z=f\left(x_0,y_0\right)$ at the point $\left(x_0,y_0\right)$ for given values of $\text{Δ}x$ and $\text{Δ}y.$ Since $\text{Δ}z=f\left(x+\text{Δ}x,y+\text{Δ}y\right)-f\left(x,y\right),$ this can be used further to approximate $f\left(x+\text{Δ}x,y+\text{Δ}y\right)\text{:}$

See the following figure.

One such application of this idea is to determine error propagation. For example, if we are manufacturing a gadget and are off by a certain amount in measuring a given quantity, the differential can be used to estimate the error in the total volume of the gadget.