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# Step-by-Step Math Answer

Find the first derivative of
*
f
*
(
*
x
*
).

Taking the derivative of
*
f
*
(
*
x
*
) using trigonometric rules:

Evaluate the
*
f
*
(
*
x
*
) and its first derivative at
*
x
*
=
*
π
*
/4.

The value and the slope of
*
f
*
at
*
x
*
=
*
π
*
/4 are given by
*
f
*
(
*
π
*
/4) and
*
f
*
'(
*
π
*
/4).

Replace
*
x
*
with
*
π
*
/4 in the
*
f
*
(
*
x
*
) and its first derivative.

Find the first degree polynomial
*
P
*
_{
1
}
(
*
x
*
) approximation of
*
f
*
(
*
x
*
).

The first degree polynomial approximation of
*
f
*
(
*
x
*
) such that the graph passes through
*
x
*
=
*
π
*
/4 is:

We need to solve for
*
a
*
_{
0
}
and
*
a
*
_{
1
}
.

*
a
*
_{
0
}
=
*
P
*
_{
1
}
(
*
π
*
/4) =
*
f
*
(
*
π
*
/4) = √ 2

*
a
*
_{
1
}
=
*
P
*
_{
1
}
'(
*
π
*
/4) =
*
f
*
'(
*
π
*
/4) = √ 2

Substitute the values of
*
a
*
_{
0
}
and
*
a
*
_{
1
}
.

The first degree polynomial approximation of
*
f
*
(
*
x
*
) is :

This is also called as first degree Taylor polynomial of
*
f
*
at
*
c
*
.

Use a graphing utility to graph the function and
*
P
*
_{
1
}
(
*
x
*
).

Use a graphing utility to graph the function and
*
P
*
_{
1
}
(
*
x
*
).