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# Step-by-Step Math Answer

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

Taking the derivative of
*
f
*
(
*
x
*
) using power rule:

Evaluate the
*
f
*
(
*
x
*
) and its first derivative at
*
x
*
= 1.

The value and the slope of
*
f
*
at
*
x
*
= 1 are given by
*
f
*
(1) and
*
f
*
'(1).

Replace
*
x
*
with 1 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
*
= 1 is
*
P
*
_{
1
}
(
*
x
*
) =
*
a
*
_{
0
}
+
*
a
*
_{
1
}
(
_{
}
*
x
*
– 1).

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

*
a
*
_{
0
}
=
*
P
*
_{
1
}
(1) =
*
f
*
(1) = 4

*
a
*
_{
1
}
=
*
P
*
_{
1
}
'(1) =
*
f
*
'(1) = –2

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

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

*
P
*
_{
1
}
(
*
x
*
) = 4 + –2(
_{
}
*
x
*
– 1)

*
P
*
_{
1
}
(
*
x
*
) = 4 –2
_{
}
*
x
*
+ 2

*
P
*
_{
1
}
(
*
x
*
) = –2
_{
}
*
x
*
+ 6

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
*
).