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# Step-by-Step Math Answer

Find the first five derivatives of
*
f
*
(
*
x
*
).

To complete the given table, first find the first five derivatives of
*
f
*
(
*
x
*
).

Since
*
f
*
(
*
x
*
) = sin
*
x
*
, the repeated differentiation of
*
f
*
(
*
x
*
) yields

*
f
*
′ (
*
x
*
) = cos
*
x
*

*
f
*
″ (
*
x
*
) = –sin
*
x
*

*
f
*
^{
(3)
}
(
*
x
*
) = –cos
*
x
*

*
f
*
^{
(4)
}
(
*
x
*
) = sin
*
x
*

and

*
f
*
^{
(5)
}
(
*
x
*
) = cos
*
x
*
.

Replace
*
x
*
with 0 in
*
f
*
(
*
x
*
) and its derivatives and simplify.

Next, evaluate
*
f
*
(
*
x
*
) and its derivatives when
*
x
*
is 0.

So, replace
*
x
*
with 0 in
*
f
*
(
*
x
*
) and its derivatives and simplify.

*
f
*
(0) = sin 0 or 0

*
f
*
′ (0) = cos 0 or 1

*
f
*
″ (0) = –sin 0 or 0

*
f
*
^{
(3)
}
(
*
x
*
) = –cos 0 or –1

*
f
*
^{
(4)
}
(
*
x
*
) = sin 0 or 0

and

*
f
*
^{
(5)
}
(
*
x
*
) = cos 0 or 1.

Use the definition of
*
n
*
^{
th
}
Maclaurin polynomial.

If
*
c
*
= 0, then the polynomial

is called the
*
n
*
^{
th
}
Maclaurin polynomial for
*
f
*
. So,

Substitute the known values in
*
P
*
_{
1
}
(
*
x
*
),
*
P
*
_{
3
}
(
*
x
*
) and
*
P
*
_{
5
}
(
*
x
*
) and simplify.

Now, substitute the known values in
*
P
*
_{
1
}
(
*
x
*
),
*
P
*
_{
3
}
(
*
x
*
) and
*
P
*
_{
5
}
(
*
x
*
) and simplify.

Replace
*
x
*
with 0 in
*
P
*
_{
1
}
(
*
x
*
),
*
P
*
_{
3
}
(
*
x
*
), and
*
P
*
_{
5
}
(
*
x
*
) and simplify.

Next, replace
*
x
*
with 0 in the Maclaurin polynomials
*
P
*
_{
1
}
(
*
x
*
),
*
P
*
_{
3
}
(
*
x
*
), and
*
P
*
_{
5
}
(
*
x
*
) and simplify. So,

Repeat the above procedure to complete the table.

Repeat the above procedure for the remaining values of
*
x
*
.

The completed table is as shown.