More Applications of Derivatives 183
Step 3: Write equation of normal.
mnormal=4; (2, 1/2)
Equation of normal:y−
1
2
=4(x−2), ory= 4 x−
15
2
.
Step 4: Find other points of intersection.
y=
1
x
; y= 4 x−
15
2
Using the [Intersection] function of your calculator, entery 1 =
1
x
andy 2 = 4 x−
15
2
and obtainx=− 0 .125 andy=−8. Thus, the normal line intersects the graph of
y=
1
x
at the point (−0.125,−8) as well.
TIP • Remember that
∫
1 dx=x+Cand
d
dx
(1)=0.
9.2 Linear Approximations
Main Concepts:Tangent Line Approximation, Estimating thenth Root of a Number,
Estimating the Value of a Trigonometric Function of an Angle
Tangent Line Approximation (or Linear Approximation)
An equation of the tangent line to a curve at the point (a,f(a)) is:
y=f(a)+f′(a)(x−a), providing thatf is differentiable ata. (See Figure 9.2-1.)
Since the curve off(x) and the tangent line are close to each other for points nearx=a,
f(x)≈f(a)+f′(a)(x−a).
(a, f (a))
y = f(a) + f'(a)(x – a)
y
x
f (x)
0
Figure 9.2-1