Linear approximation
approximation of a function by its tangent line at a point
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.
Definition
Given a twice continuously differentiable function
f
{\displaystyle f}
of one real variable, Taylor's theorem for the case
n
=
1
{\displaystyle n=1}
states that
f
(
x
)
=
f
(
a
)
+
f
′
(
a
)
(
x
−
a
)
+
R
2
{\displaystyle f(x)=f(a)+f'(a)(x-a)+R_{2}}
where
R
2
{\displaystyle R_{2}}
is the remainder term. The linear approximation is obtained by dropping the remainder:
f
(
x
)
≈
f
(
a
)
+
f
′
(
a
)
(
x
−
a
)
.
{\displaystyle f(x)\approx f(a)+f'(a)(x-a).}
This is a good approximation when
x
{\displaystyle x}
is close enough to
a
{\displaystyle a}
; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of
f
{\displaystyle f}
at
(
a
,
f
(
a
)
)
{\displaystyle (a,f(a))}
. For this reason, this process is also called the tangent line approximation. Linear approximations in this case are further improved when the second derivative of a,
f
″
(
a
)
{\displaystyle f''(a)}
, is sufficiently small (close to zero) (i.e., at or near an inflection point).
If
f
{\displaystyle f}
is concave down in the interval between
x
{\displaystyle x}
and
a
{\displaystyle a}
, the approximation will be an overestimate (since the derivative is decreasing in that interval). If
f
{\displaystyle f}
is concave up, the approximation will be an underestimate.
Begin with the source’s own compact description: “Linear approximation” is approximation of a function by its tangent line at a point. The dossier treats that line as a proposition to test through Linear, approximation and function, not as a finished interpretation.
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