3.10
Linear Approximation and Differentials
We
have seen that a curve lies very close to its tangent line near the point of
tangency. In fact, by zooming in toward
a point on the graph of a differentiable function, we noticed that the graph
looks more and more like its tangent line.
This observation is the basis for a method of finding approximate values
of functions.
The
idea is that it might be easy to calculate a value f(a) of a function,
but difficult (or even impossible) to compute nearby values of f. So we settle for the easily computed values
of the linear function L whose graph is the tangent line of f at (a,
f(a)).
In
other words, we use the tangent line at (a, f(a)) as an approximation to
the curve y = f(x) when x is near a. An equation of this tangent line is
Y = f(a) + 
(a)(x-a)
and
the approximation
f(x) 
f(a) + 
(a)(x-a)
is
called the linear approximation or tangent line approximation of f at a. The linear function whose graph is the
tangent line, that is,
L(x) = f(a) + 
(a)(x-a)
is
called the linearization of f at a.
Example 1
Find
the linearization of the function 
at a=0 and use
it to approximate the numbers 
and 
.
The
derivative of 
is 
And
so we have 
and 
. Using these values, the linearization is
The
corresponding linear approximation (0) is
In particular, we have
and 
Example 2
Compute 
and 
for the given values of x and 
. 
and 
In general, 
Example 3
The radius of a sphere was measured and found to be
32 cm with a possible error in measurement of at most 0.15 cm. What is the maximum error in using this
value of the radius to compute the volume of the sphere?
. If the error in the
measured value of r is denied by 
, then the corresponding error in the calculated value of V
is 
, which can be approximated by the differential 
.
When r = 32 and dr = 0.15, this
becomes
The maximum error in the calculated volume is about 1930 cm3.