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.