5 Steps to a 5 AP Calculus BC 2019

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