5 Steps to a 5 AP Calculus BC 2019

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

.