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6. Linear Approximations
An equation of the tangent line to the curve y = f(x) at (a, f(a)) is
y = f(a)+f
(a)(x − a)
Definition 20
The approximation f(x) ≈ f(a)+f
(a)(x − a) is called the
linear approximation or tangent line approximation of f at a, and
the function
L(x)=f(a)+f
(a)(x − a)
(whose graph is the tangent line) is called the linearization of f at
a. The linear approximation is a good approximation when x is near
a.
Example 25
Find the linearization of the function f(x)=
√
x +3 at x =1
and use it to approximate the numbers
√
3.98 and
√
4.05
The derivative of f(x)=
√
x +3is
f
(x)=
1
2
(x +3)
−1/2
=
1
2
√
x +3
and so we have f(1) = 2 and f
(1) = 1/4. Putting these values into
the expression for L(x), we see that the linearization is
L(x)=f(1) + f
(1)(x − 1) = 2 +
1
4
(x − 1) =
7
4
+
x
4
.
The corresponding linear approximation is
√
x +3 ≈
7
4
+
x
4
.In
particular, we have
√
3.98 ≈
7
4
+
0.98
4
=1.995 and
√
4.05 ≈
7
4
+
1.05
4
=2.0125.
35
