Part II: The Answers
276
Answers
401–500
420.
y = –x + 1
You know a point on the tangent line, so you just need to find the slope. Begin by
taking the derivative of x
2
+ 2xy + y
2
:
2
2
2
2
0
x
y
x
dy
dx
y
dy
dx
+
+
+
=






Then use the values x = 0 and y = 1 and solve for the slope
dy
dx
:
2 0
2 1 0 2 1
0
2 2
0
1
( ) ( ( ) ) ( )
+
+ +
=
+
=
= −
dy
dx
dy
dx
dy
dx
The tangent line has a slope of –1 and passes through (0, 1), so its equation is
y
x
y
x
− = −
−
= − +
1
1
0
1
(
)
421.
y
x
= −
+ ( − )
2 2
2
π
π
( )
You know a point on the tangent line, so you just need to find the slope. Begin by
taking the derivative of cos(xy) + x
2
= sin y:
−
+
+
=
sin( )
( cos )
xy
y x
dy
dx
x
y
dy
dx
1
2

 

 
Then use the values x = 1 and y = π
2
and solve for the slope
dy
dx
:
−
+
+ =
= −
sin π π
π
2 2
1
2 0
2 2
( )

 

 
dy
dx
dy
dx
The tangent line has a slope of 2 2
− π and passes through 1 2
, π
( ) , so its equation is
y
x
y
x
− = −
= −
+
π
π
π
π
2
2 2
1
2 2
2
( ) −
( ) −
(
)
(
)