A.
APPROXIMATIONS
In science and engineering, you often approaimate complicated functions
by
simpler ones
which are easier to calculate with, and which show the relations between the variables more
clearly.
Of
course, the approximation must be close enough
to give you reasonable accuracy.
For this reason, approximation is a skill, one your other teachers will expect you to have.
This is a good place to start acquiring it.
Throughout, we will use the symbol a to mean "approximately equal to"; this is a bit
vague, but making approximations in engineering is more
art than science.
1.
The
linear
approximation;
linearizations.
The simplest way to approximate
a function
f(z)
for values of
z
near
a
jr
is to use a linear
funcion.
The linear function we shall use is the one whose
graph is the tangent line
to
f(z)
at
x
=
a.
This
makes
sense because
the
tangent line at (a,
f(a))
gives
a good approximation to the graph of
f(z),
_.
i
l.
T
.
.
a
s
at
a
to
cose
s
a
(1)
height
of
the
graph
of
f(z)
s height
of
the
tangent
at (a, f(a))
To turn
(1)
into calculus, we need the equation for the tangent line. Since the line goes
through
(a,
f(a)) and has slope
f'(a),
its equation is
S
f(a)
+
f'(a)(x

),
and therefore
(1)
can be expressed as
(2)
f(z)
f()
+
f'(a)(

a),
for
Tua.
This says
that for x near
a,
the function
f(z)
can be approximated
by
the linear function
on the right of (2). This function

the one whose graph is the tangent line

is called
the
linerization
of f(z)
at
z
=
a.
The appraoimation (2) is often written in an equivalent form
that you should become
familiar with;
it makes use of a dependent variable. Writing
(3)
v
= f(z),
A
=
z
a,
Ay
=
f()

f(a),
the approximation (2) takes the
form
(2')
AL
f'(a)
Az,
for
Az
.
f(a)Ax
In this form, the quantity on the
right
represents the change in height of
the
taent line, while the left measures the change in height of the eraph.
Here are some examples of linear approximations.
In all
of them,
we
are taking
a
= 0,
this being the most important
case All
can be found
by
using
(2)above and calculating
the
derivative. You should verify each of them, and memorize the approximation.