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2 Derivatives

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Copyright © Cengage Learning. All rights reserved.

2.9Linear Approximations and

Differentials

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3

Linear Approximations and Differentials

We have seen that a curve lies very close to its tangent linenear the point of tangency. In fact, y !ooming in to"ard a

point on the graph of a differentiale function, "e noticed

that the graph loo#s more and more li#e its tangent line.

$his oservation is the asis for a method of finding

approximate values of functions.

$he idea is that it might e easy to calculate a value f %a& of

a function, ut difficult %or even impossile& to compute

neary values of f .

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(o "e settle for the easily computed values of the linearfunction L "hose graph is the tangent line of f at %a, f %a&&.

%(ee )igure *.&

Figure 1

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In other "ords, "e use the tangent line at %a, f %a&& as anapproximation to the curve y f % x & "hen x is near a. An

e-uation of this tangent line is

y f %a& f ′ %a&% x / a&

and the approximation

f % x & f %a& f ′ %a&% x / a&

is called the linear approximation or tangent line

approximation of f at a.

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$he linear function "hose graph is this tangent line, that is,

L % x & f %a& f ′%a&% x / a&

is called the linearization of f at a.

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1

2xample *

)ind the lineari!ation of the function f % x & at a *and use it to approximate the numers and . Are

these approximations overestimates or underestimates

(olution4

$he derivative of f % x & % x 3&*56 is

f ′% x & % x + 3& /*56

and so "e have f %*& 6 and f ′%*& .

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2xample * / Solution

8utting these values into 2-uation 6, "e see that thelineari!ation is

L % x & f %*& f ′%*& % x / *&

6 % x / *&

$he corresponding linear approximation is

%"hen x is near *&

cont9d

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In particular, "e have

and

cont9d

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$he linear approximation is illustrated in )igure 6.

cont9d

Figure 2

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We see that, indeed, the tangent line approximation is agood approximation to the given function "hen x is near *.

We also see that our approximations are overestimates

ecause the tangent line lies aove the curve.

<f course, a calculator could give us approximations for

and , ut the linear approximation gives an

approximation over an entire interval.

cont9d

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*6


In the follo"ing tale "e compare the estimates from thelinear approximation in 2xample * "ith the true values.

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*3


=otice from this tale, and also from )igure 6, that thetangent line approximation gives good estimates "hen x is

close to * ut the accuracy of the approximation

deteriorates "hen x is farther a"ay from *.

Figure 2

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*'


$he next example sho"s that y using a graphingcalculator or computer "e can determine an interval

throughout "hich a linear approximation provides a

specified accuracy.

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*+

2xample 6

)or "hat values of x is the linear approximation

accurate to "ithin ;.+ What aout accuracy to "ithin ;.*

(olution4

Accuracy to "ithin ;.+ means that the functions shoulddiffer y less than ;.+4

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2xample 6 / Solution

2-uivalently, "e could "rite

$his says that the linear approximation should lie et"een

the curves otained y shifting the curve

up"ard and do"n"ard y an amount ;.+.

cont9d

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*1

2xample 6 / Solutioncont9d

)igure 3 sho"s the tangent line y %1 x &5' intersectingthe upper curve y ;.+ at P and Q.

Figure 3

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*7


>ooming in and using the cursor, "e estimate that the x ?coordinate of P is aout /6.00 and the x ?coordinate of Q

is aout 7.00.

$hus "e see from the graph that the approximation

is accurate to "ithin ;.+ "hen /6.0 @ x @ 7.0. %We have

rounded to e safe.&

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*:


(imilarly, from )igure ' "e see that the approximation isaccurate to "ithin ;.* "hen /*.* @ x @ 3.:.

Figure 4

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6;

Applications to 8hysics

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6*


Linear approximations are often used in physics. Inanaly!ing the conse-uences of an e-uation, a physicist

sometimes needs to simplify a function y replacing it "ith

its linear approximation.

)or instance, in deriving a formula for the period of a

pendulum, physics textoo#s otain the expression

aT /g sin θ for tangential acceleration and then replace

θ y θ "ith the remar# that sin θ is very close to θ if θ is nottoo large.

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ou can verify that the lineari!ation of the functionf % x & sin x at a ; is L % x & x and so the linear

approximation at ; is

sin x ≈ x

(o, in effect, the derivation of the formula for the period of a

pendulum uses the tangent line approximation for the sine

function.

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Another example occurs in the theory of optics, "here lightrays that arrive at shallo" angles relative to the optical axis

are called paraxial rays.

In paraxial %or Baussian& optics, oth sin θ and cos θ arereplaced y their lineari!ations. In other "ords, the linear

approximations

sin θ θ and cos θ *

are used ecause θ is close to ;.

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6'

Differentials

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6+

Differentials

$he ideas ehind linear approximations are sometimesformulated in the terminology and notation of differentials.

If y f % x &, "here f is a differentiale function, then the

differential dx is an independent variale that is, dx cane given the value of any real numer.

$he differential dy is then defined in terms of dx y the

e-uation

dy f ′% x & dx

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Differentials

$he geometric meaning of differentials is sho"n in)igure +.

Figure 5

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Differentials

Let P % x , f % x && and Q% x ∆ x , f % x ∆ x && e points on thegraph of f and let dx ∆ x . $he corresponding change in y is

∆y f % x + ∆ x & / f % x &

$he slope of the tangent line PR is the derivative f ′% x &. $hus

the directed distance from S to R is f ′ % x & dx dy .

$herefore dy represents the amount that the tangent line

rises or falls %the change in the lineari!ation&, "hereas ∆y

represents the amount that the curve y f % x & rises or falls

"hen x changes y an amount dx .

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2xample 3

Compare the values of ∆y and dy ify f % x & x 3 x 6 / 6 x + * and x changes %a& from 6 to 6.;+

and %& from 6 to 6.;*.

(olution4(a) We have

f %6& 63 66 / 6%6& *

:

f %6.;+& %6.;+&3 %6.;+&6 / 6%6.;+& *

:.1*106+

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∆y f %6.;+& / f %6&

;.1*106+

In general,

dy f ′% x & dx

%3 x 6 6 x – 6& dx

When x 6 and dx ∆ x ;.;+, this ecomes

dy 3%6&6 6%6& / 6E;.;+

;.1

cont9d

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3*


(b) f %6.;*& %6.;*&3 %6.;*&6 / 6%6.;*& *

:.*';1;*

∆y f

%6.;*& / f

%6&

;.*';1;*

When dx ∆ x ;.;*,

dy 3%6&6 6%6& / 6E;.;*

;.*'

cont9d

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Differentials

<ur final example illustrates the use of differentials inestimating the errors that occur ecause of approximate

measurements.

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2xample '

$he radius of a sphere "as measured and found to e6* cm "ith a possile error in measurement of at most

;.;+ cm. What is the maximum error in using this value of

the radius to compute the volume of the sphere

(olution4

If the radius of the sphere is r , then its volume is V π r 3. If

the error in the measured value of r is denoted y dr ∆r ,

then the corresponding error in the calculated value of V is∆V , "hich can e approximated y the differential

dV 'π r 6 dr

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2xample ' / Solution

When r 6* and dr ;.;+, this ecomes

dV 'π %6*&6;.;+

611

$he maximum error in the calculated volume is aout

611 cm3.

cont9d

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3+

Differentials

ote!

Although the possile error in 2xample ' may appear to e

rather large, a etter picture of the error is given y the

relative error , "hich is computed y dividing the error y

the total volume4

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Differentials

$hus the relative error in the volume is aout three timesthe relative error in the radius.

In 2xample ' the relative error in the radius is

approximately dr 5r ;.;+56* ;.;;6' and it produces arelative error of aout ;.;;1 in the volume.

$he errors could also e expressed as per"entage errors

of ;.6'F in the radius and ;.1F in the volume.

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