Q23:

The first condition for a linear function
πΏ
(
π₯
)
to be an approximation to
π
(
π₯
)
near
π₯
=
π
is that
πΏ
(
π
)
=
π
(
π
)
. So
πΏ
(
π₯
)
=
π
(
π
)
+
π
(
π₯
β
π
)
for some constant
π
. The error in using
πΏ
instead of
π
at a point
π₯
is the function
πΈ
(
π₯
)
=
π
(
π₯
)
β
πΏ
(
π₯
)
.

The tangent line approximation to
π
near
π₯
=
π
is the linear approximation
πΏ
(
π₯
)
=
π
(
π
)
+
π
β²
(
π
)
(
π₯
β
π
)
. What is the tangent line approximation to
π
(
π₯
)
=
π
π
π₯
near
π₯
=
0
?

What is the error in using the tangent line approximation to
π
(
π₯
)
=
π
4
.
5
π₯
near
π₯
=
0
at the point 0.1? Give your answer to 5 decimal places.

What is the error in using the tangent line approximation to
π
(
π₯
)
=
π
4
.
5
π₯
near
π₯
=
0
at the point 0.01? Give your answer to 5 decimal places.

What is the error in using the tangent line approximation to
π
(
π₯
)
=
π
4
.
5
π₯
near
π₯
=
0
at the point 0.001? Give your answer to 5 decimal places.