**total differential**is an estimate for the change in the output of a function

*f*(

*x*,

*y*) in response to (small) changes

*dx*and

*dy*in the input variables. This differential is

*dz*=

*f*

_{x}*dx*+

*f*

_{y}*dy*.

*z*=

*f*(

*x*+

*dx*,

*y*+

*dy*) -

*f*(

*x*,

*y*)

A quick note about differentials! We said in class that

The **total differential** is an estimate for the change
in the output of a function *f*(*x*,*y*) in response to
(small) changes *dx* and *dy* in the input variables. This
differential is
*dz* = *f*_{x} *dx* +
*f*_{y} *dy*.

And we noted that this is in some sense equivalent to approximating
the actual change in the function value,

delta *z* = *f*(*x*+*dx*, *y*+*dy*) -
*f*(*x*,*y*)

with the difference in the linear (tangent plane) approximation.
Now, this is a little confusing, because normally we think about the
linear approximation being centered around a specific point,
(*x*_{0},
*y*_{0}), not (as we've done
above) about some arbitrary point (*x*, *y*). But we can
get through this without any trouble, as is (hopefully!) explained
below.

So, let's start with some function *f*(*x*,*y*). The
linear approximation to *f* at the point (*a*,*b*) is
just

=

Check that the second line of that makes sense! So, if we started
asking what the change in the function value was between the point
(*a*,*b*) and another point near it, we could approximate
this with the change in the linear approximation, and say that

delta *z* = *f*(*x*,*y*) - *f*(*a*,*b*)

~*L*(*x*,*y*) - *L*(*a*,*b*)

~

(where "~" means "is approximately equal to"). Let's work this out:

Now, to quantify the distance we go from the point (*a*,*b*),
we could take *x* = *a* + *dx* and
*y* = *b* + *dy*. Let's also call the
difference in *L* values delta *L*, so that

delta *L* =
*L*(*a*+*dx*,*b*+*dy*) -
*L*(*a*,*b*) =
*f*_{x}(*a*,*b*) (*dx*) +
*f*_{y}(*a*,*b*) (*dy*).

Notice that the expression on the right-hand side is the same as the
differential that we defined above, with the only exception being that
we're considering it at (*a*,*b*) instead of
(*x*,*y*). But we never said what *a* and *b*
are, so we could happily say that they give any point
(*x*,*y*), at which point it's clear that the change
delta *L* is in fact what we called the differential:

delta *L* =
*f*_{x}(*x*,*y*) (*dx*) +
*f*_{y}(*x*,*y*) (*dy*)

*dz* =
*f*_{x}(*x*,*y*) (*dx*) +
*f*_{y}(*x*,*y*) (*dy*).

**An Example**

The volume of a paraboloid (graph to the right) with base radius
*r* and height *h* is

Suppose we measure that the radius is *r* = 2 cm and
height is *h* = 4 cm, with a possible measurement error
of up to 0.2 cm. How far off could our volume estimate be?
We'll estimate this with the differential.

The differential for this is

So, with our measured dimensions of 2 and 4 cm, we have

And an estimate for the maximum possible error is if *r* and
*h* are off by *dr* = 0.2 cm and *dh* =
0.2 cm, respectively:

(Note that the volume itself is *V*(2,4) =
8 pi cm^{3}, so this is quite
a big maximum error! Time to go back and measure more carefully!)

differentials

Last modified: Mon Feb 2 16:35:48 EST 2004

Comments to:glarose@umich.edu

©2004 Gavin LaRose, UM Math Dept.