J. Kim MS 125-03 Worksheet 3.9 (Oct. 9, 2006)
Section 3.9 Tangent Line Approximation
The Tangent Line Approximation
Suppose
)(xf
is differentiable at
ax
. Then, for values of x near a, the tangent line approximation to
)(xf
is
))(()()( axafafxf
.
The expression
))(()( axafaf
is called the local linearization of
f
near
ax
.
The error,
)(xE
, in the approximation is defined by
))(()()()( axafafxfxE
.
Example 1. What is the tangent line approximation for
xxf sin)(
near
0x
?
Example 2 Let
x
exf
3
)(
a) What is the local linearization of
)(xf
near
0x
?
b) Approximate
)1.0(f
using the local linearization.
c) Estimate the error in b).