13-linearapproximationanddifferentials

8/3/2019 13-linearapproximationanddifferentials

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Section 2.8

Linear Approximation and Differentials

V63.0121.002.2010Su, Calculus I

New York University

May 26, 2010

Announcements

Quiz 2 Thursday on Sections 1.52.5

No class Monday, May 31

Assignment 2 due Tuesday, June 1

. . . . . .


2/69

. . . . . .

Announcements

Quiz 2 Thursday onSections 1.52.5

No class Monday, May 31

Assignment 2 due

Tuesday, June 1

V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 2 / 27


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. . . . . .

Objectives

Use tangent lines to makelinear approximations to a

function. Given a function and a

point in the domain,compute thelinearization of the

function at that point. Use linearization to

approximate values offunctions

Given a function, compute

the differential of thatfunction

Use the differential

notation to estimate error

in linear approximations.V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 3 / 27


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. . . . . .

Outline

The linear approximation of a function near a point

Examples

Questions

Differentials

Using differentials to estimate error

Advanced Examples



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. . . . . .

The Big Idea

Question

Let fbe differentiable at a. What linear function best approximates fnear a?



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. . . . . .

The Big Idea

Question


Answer

The tangent line, of course!



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. . . . . .

The Big Idea

Question


Answer


Question

What is the equation for the line tangent to y= f(x) at (a, f(a))?



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. . . . . .

The Big Idea

Question


Answer


Question

What is the equation for the line tangent to y= f(x) at (a, f(a))?

Answer

L(x) = f(a) + f(a)(x a)



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. . . . . .

The tangent line is a linear approximation

L(x) = f(a) + f(a)(x a)

is a decent approximation to fnear a.

. .x

.y

...

.f(a)

.f(x).L(x)

.a .x

.x a



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. . . . . .

The tangent line is a linear approximation

L(x) = f(a) + f(a)(x a)

is a decent approximation to fnear a.

How decent? The closerx is to

a, the better the approxmation

L(x) is to f(x)

. .x

.y

...

.f(a)

.f(x).L(x)

.a .x

.x a



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. . . . . .

Example.

.

Example

Estimate sin(61) = sin(61/180) by using a linear approximation(i) about a = 0 (ii) about a = 60 = /3.



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. . . . . .

Example.

.

Example


Solution (i)

If f(x) = sinx, then f(0) = 0and f(0) = 1.

So the linear approximation

near0 is L(x) = 0 + 1 x= x. Thus

sin61

180

61

180 1.06465



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. . . . . .

Example.

.

Example


Solution (i)




sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) = andf (3 ) = .



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. . . . . .

Example.

.

Example

Estimate sin(61

) =sin(61

/180

)by using a linear approximation

(i) about a = 0 (ii) about a = 60 = /3.

Solution (i)




sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f(

3

)= .



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. . . . . .

Example.

.

Example

Estimate sin(61

) =sin(61

/180

)by using a linear approximation


Solution (i)




sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f(

3

)= 12 .


l


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. . . . . .

Example.

.

Example

Estimate sin(61) = sin(61/180) by using a linear approximation


Solution (i)

If f(x) = sinx, then f(0) = 0and f(0) = 1. So the linear approximation


sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f(

3

)= 12 .

So L(x) =


E l


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. . . . . .

Example.

.

Example


Solution (i)



sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f(

3

)= 12 .

So L(x) =

3

2+

1

2

x

3


E l


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. . . . . .

Example.

.

Example


Solution (i)



sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f(

3

)= 12 .

So L(x) =

3

2+

1

2

x

3

Thus

sin61180


E l


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. . . . . .

Example.

.

Example


Solution (i)



sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f(

3

)= 12 .

So L(x) =

3

2+

1

2

x

3

Thus

sin61180 0.87475


Example


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. . . . . .

Example.

.

Example


Solution (i)



sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f(

3)= 12 .

So L(x) =

3

2+

1

2

x

3

Thus

sin61180 0.87475

Calculator check: sin(61)


Example


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. . . . . .

Example.

.

Example


Solution (i)



sin61

180

61180

1.06465

Solution (ii)

We have f(3 ) =

3

2

and

f (3 ) = 12 . So L(x) =

3

2+

1

2

x

3

Thus

sin61180 0.87475

Calculator check: sin(61) 0.87462.


Illustration


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. . . . . .

Illustration

. .x

.y

.y= sinx

.61V63.0121.002.2010Su, Calculus I (NYU) Section 2.8 Linear Approximation May 26, 2010 8 / 27

Illustration


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. . . . . .

Illustration

. .x

.y

.y= sinx

.61

.y= L1(x) = x

..0

.


Illustration


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. . . . . .

Illustration

. .x

.y

.y= sinx

.61

.y= L1(x) = x

..0

.

.big difference!


Illustration


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. . . . . .

Illustration

. .x

.y

.y= sinx

.61

.y= L1(x) = x

..0

.

.y= L2(x) = 32 + 12 (x 3)

../3

.


Illustration


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. . . . . .

Illustration

. .x

.y

.y= sinx

.61

.y= L1(x) = x

..0

.

.y= L2(x) = 32 + 12 (x 3)

../3

. .very little difference!


Another Example


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. . . . . .

Another Example

Example

Estimate 10 using the fact that 10 = 9 + 1.


Another Example


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. . . . . .

Another Example

Example


Solution

The key step is to use a linear approximation to f(x) =

x near a = 9

to estimate f(10) = 10.


Another Example


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. . . . . .

Another Example

Example


Solution


x near a = 9


10

9 +d

dx

x

x=9

(1)

= 3 +

1

2 3(1) =19

6 3.167


Another Example


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. . . . . .

Another Example

Example


Solution


x near a = 9


10

9 +d

dx

x

x=9

(1)

= 3 +

1

2 3(1) =19

6 3.167

Check:

19

6

2=


Another Example


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. . . . . .

p

Example


Solution


x near a = 9


10

9 +d

dx

x

x=9

(1)

= 3 +

1

2 3(1) =19

6 3.167

Check:

19

6

2=

361

36.


Dividing without dividing?


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. . . . . .

g g

Example

Suppose I have an irrational fear of division and need to estimate

577 408. I write

577

408= 1 + 169

1

408= 1 + 169 1

4 1

102.

But still I have to find 1102

.


Dividing without dividing?


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. . . . . .

g g

Example

Suppose I have an irrational fear of division and need to estimate

577 408. I write

577

408= 1 + 169

1

408= 1 + 169 1

4 1

102.

But still I have to find 1102

.

Solution

Let f(x) =1

x. We know f(100) and we want to estimate f(102).

f(102) f(100) + f(100)(2) = 1100

11002

(2) = 0.0098

=

577

408 1.41405


Questions


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. . . . . .

Example

Suppose we are traveling in a car and at noon our speed is 50 mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight?


Answers


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. . . . . .

Example

Suppose we are traveling in a car and at noon our speed is 50 mi/hr.

How far will we have traveled by 2:00pm? by 3:00pm? By midnight?


Answers


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. . . . . .

Example

Suppose we are traveling in a car and at noon our speed is 50 mi/hr.

How far will we have traveled by 2:00pm? by 3:00pm? By midnight?

Answer

100 mi

150 mi

600 mi (?) (Is it reasonable to assume 12 hours at the same

speed?)


Questions


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. . . . . .

Example


Example

Suppose our factory makes MP3 players and the marginal cost iscurrently $50/lot. How much will it cost to make 2 more lots? 3 more

lots? 12 more lots?


Answers


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. . . . . .

Example


lots? 12 more lots?

Answer

$100

$150

$600 (?)


Questions


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. . . . . .

Example


Example


lots? 12 more lots?

Example

Suppose a line goes through the point (x0, y0) and has slope m. If thepoint is moved horizontally by dx, while staying on the line, what is thecorresponding vertical movement?


Answers


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. . . . . .

Example



Answers


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. . . . . .

Example


Answer

The slope of the line is

m =rise

run

We are given a run ofdx, so the corresponding rise is m dx.


Outline


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. . . . . .


Examples

Questions

Differentials


Advanced Examples


Differentials are another way to express derivatives


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. . . . . .

f(x+x) f(x) y

f(x)x dy

Rename x= dx, so we canwrite this as

y dy= f(x)dx.

And this looks a lot like theLeibniz-Newton identity

dy

dx= f(x) . .x

.y

..

.x .x+x

.dx = x.y .dy


Differentials are another way to express derivatives


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. . . . . .

f(x+x) f(x) y

f(x)x dy

Rename x= dx, so we canwrite this as

y dy= f(x)dx.

And this looks a lot like theLeibniz-Newton identity

dy

dx= f(x) . .x

.y

..

.x .x+x

.dx = x.y .dy

Linear approximation means y dy= f(x0)dxnear x0.




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. . . . . .

Ify= f(x), x0 and x is known,and an estimate ofy isdesired:

Approximate: y dy Differentiate: dy= f(x)dx

Evaluate at x= x0 anddx= x.

. .x

.y

..

.x .x+x

.dx = x.y .dy


Example


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. . . . . .

Example

A sheet of plywood measures 8 ft 4 ft. Suppose our plywood-cuttingmachine will cut a rectangle whose width is exactly half its length, but

the length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?


Example


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. . . . . .

p



Solution

Write A() =1

22. We want to knowA when = 8 ft and = 1 in.


Example


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. . . . . .

p



Solution

Write A() =1


(I) A(+) = A

97

12

=

9409

288So A =

9409

288 32 0.6701.


Example


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. . . . . .

p



Solution

Write A() =1


(I) A(+) = A

97

12

=

9409

288So A =

9409

288 32 0.6701.

(II)dA

d = , so dA

= d

, which should be a good estimate for

.

When = 8 and d = 112 , we have dA =8

12 =23 0.667. So we

get estimates close to the hundredth of a square foot.


Why?


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. . . . . .

Why use linear approximations dywhen the actual difference y isknown?

Linear approximation is quick and reliable. Finding yexactlydepends on the function.

These examples are overly simple. See the Advanced Exampleslater.

In real life, sometimes only f(a) and f(a) are known, and not thegeneral f(x).


Outline


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. . . . . .


Examples

Questions

Differentials


Advanced Examples


GravitationPencils down!


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. . . . . .

Example

Drop a 1 kg ball off the roof of the Silver Center (50m high). We

usually say that a falling object feels a force F= mgfrom gravity.


GravitationPencils down!


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. . . . . .

Example

Drop a 1 kg ball off the roof of the Silver Center (50m high). We

usually say that a falling object feels a force F= mgfrom gravity. In fact, the force felt is

F(r) = GMmr2

,

where M is the mass of the earth and r is the distance from the

center of the earth to the object. G is a constant.

At r= re the force really is F(re) = GMmr2e

= mg. What is the maximum error in replacing the actual force felt at the

top of the building F(re +r) by the force felt at ground levelF(re)? The relative error? The percentage error?


Gravitation Solution


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. . . . . .

Solution

We wonder ifF= F(re +r)

F(re) is small.

Using a linear approximation,

F dF= dFdr

re

dr= 2GMm

r3edr

= GMmr2e

drre= 2mgr

re

The relative error isF

F 2r

re

re = 6378.1 km. Ifr= 50 m,

F

F 2r

re= 2 50

6378100= 1.56 105 = 0.00156%


Systematic linear approximation


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2 is irrational, but

9/4 is rational and 9/4 is close to 2.




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. . . . . .


9/4 is rational and 9/4 is close to 2. So

2 =

9/4 1/4

9/4 +

1

2(3/2)(1/4) = 17

12




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. . . . . .



2 =

9/4 1/4

9/4 +

1

2(3/2)(1/4) = 17

12

This is a better approximation since(

17

/12

)

2

=289

/144




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. . . . . .



2 =

9/4 1/4

9/4 +

1

2(3/2)(1/4) = 17

12

This is a better approximation since (17/12)2 = 289/144

Do it again!

2 =

289/144 1/144

289/144 +

1

2(17/12)(1/144) = 577/408

Now

577

408

2=

332, 929

166, 464which is

1

166, 464away from 2.


Illustration of the previous example


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.




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.




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. .2




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. . . . . .

. .2

.




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. . . . . .

. .2

.




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. . . . . .

. .2

.(2, 1712)




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. . . . . .

. .2

.(2, 1712)




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. . . . . .

..(94 , 32)(2, 17/12)




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. . . . . .

..(94 , 32)(2, 17/12) . .289144 , 1712




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. . . . . .

..(94 , 32)(2, 17/12) . .289144 , 1712..(2, 577408)


Summary


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. . . . . .

Linear approximation: Iff is differentiable at a, the best linearapproximation to fnear a is given by

Lf,a(x) = f(a) + f(a)(x a)

Differentials: Iff is differentiable at x, a good approximation toy= f(x+x) f(x) is

y dy= dydx dx= dy

dxx

Dont buy plywood from me.


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13-linearapproximationanddifferentials