5 Steps to a 5 AP Calculus BC 2019

182 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 is parallel to the liney=−ex−1, setmnormal=−e⇒−x=−eorx=e.

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

graph of y=lnxat which the normal is parallel toy =−ex−1is(e, 1). (See

Figure 9.1-14.)

[−6.8, 9.8] by [−5, 3]

Figure 9.1-14

Example 3

Given the curvey=

1

x

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

1

x

at the point (2,

1/2), and (b) does this normal 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.