5 Steps to a 5 AP Calculus BC 2019

(Marvins-Underground-K-12) #1
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
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