5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 15:14

210 STEP 4. Review the Knowledge You Need to Score High

Example 2

Find the point on the graph ofy=lnxsuch that the normal line at this point is parallel to

the liney=−ex−1.

Step 1: Findmtangent.

y=lnx;

dy

dx

=

1

x

Step 2: Findmnormal.

mnormal=

− 1

mtangent

=

− 1

1 /x

=−x

Slope ofy=−ex−1is−e.

Since normal line is parallel to the liney=−ex−1, setmnormal=−e⇒−x=−eor

x=e.

Step 3: Find the point on the graph. Atx=e,y=lnx=lne=l. Thus, the point of

the graph ofy =lnx at which the normal is parallel toy=−ex−1is(e, 1).

(See Figure 10.1-14.)

[–6.8, 9.8] by [–5, 3]

Figure 10.1-14

Example 3

Given the curvey=

1

x

: (a) write an equation of the normal line to the curvey=

1

x

at the

point (2, 1/2), and (b) does this normal line intersect the curve at any other point? If yes,

find the point.

Step 1: Findmtangent.

y=

1

x

;

dy

dx

=(−1)(x−^2 )=−

1

x^2

Step 2: Findmnormal.

mnormal=

− 1

mtangent

=

− 1

− 1 /x^2

=x^2

At (2, 1/2),mnormal= 22 =4.