More Applications of Derivatives 181
Normal
f
Tangent Tangent f
Normal
Figure 9.1-12
Example 1
Write an equation for each normal to the graph ofy=2 sinxfor 0≤x≤ 2 πthat has a
slope of
1
2
.
Step 1: Findmtangent.
y=2 sinx;
dy
dx
=2 cosx
Step 2: Findmnormal.
mnormal=−
1
mtangent
=−
1
2 cosx
Setmnormal=
1
2
⇒−
1
2 cosx
=
1
2
⇒cosx=− 1
⇒x=cos−^1 (−1) orx=π.(See Figure 9.1-13.)
[−1.5π, 2.5π] by [−3, 3]
Figure 9.1-13
Step 3: Write equation of normal line.
Atx=π,y=2 sinx=2(0)=0; (π,0).
Sincem=
1
2
, equation of normal is:
y− 0 =
1
2
(x−π)ory=
1
2
x−
π
2