Part II: The Answers
280
Answers
401–500
Next, find the derivative of the function to get the slope of the line:
′
′ ( )
f x
x
x
f
( )
sin
cos
sin
cos
= −
+
= −
+
=−
π
π
π
2
2
2
1
Using the point-slope formula with the slope m = –1 and the point π
2
1
,
( ) gives you
y y m x x
y
x
y
x
y
x
− =
−
− = −
−
= − + +
= − + +
1
1
1
1
2
2
1
2
2
(
)
( )
π
π
π
Replace y with L(x) to get the solution: L x
x
( ) = − + +
π 2
2
.
428.
L x
x
( )
( )
( )
=
+
5
3 6
8
3 6
2 3
2 3
Finding the linearization L(x) is the same as finding the equation of the tangent line.
First find a point on the line by entering the value of a in the function. For the given
value of a, the corresponding y value is
f ( )
2
2 2
6 6
2
3
3
1 3
=
+ =
=
Next, find the derivative of the function to get the slope of the line:
′
(
)
(
)
′
( )
f x
x
x
x
x
x
x
f
( )
( )
( )
=
+
( + ) =
+
+
=
+
+
−
1
3
2 1
2 1
3
2
2 2 1
3 2 2
2
2 3
2
2 3
2
2 3
= =
5
3 6
2 3
( )
Using the point-slope formula with the slope m =
5
3 6
2 3
( )
and the point (2, 6
1/3
)
gives you
y y m x x
y
x
y
x
y
− =
−
−
=
−
=
−
+
=
1
1
1 3
2 3
2 3
2 3
1 3
6
5
3 6
2
5
3 6
10
3 6
6
5
3
(
)
( )
(
)
( )
( )
(6 6
10
3 6
18
3 6
5
3 6
8
3 6
2 3
2 3
2 3
2 3
2 3
)
( )
( )
( )
( )
x
y
x
−
+
=
+
Replace y with L(x) to get the solution: L x
x
( ) ( )
( )
=
+
5
3 6
8
3 6
2 3
2 3 .