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

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

More Applications of Derivatives 211

Step 3: Write equation of normal line.

mnormal=4; (2, 1/2)

Equation of normal line: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.

10.2 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 10.2-1.)

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 10.2-1