Lesson 12: Linear Approximation (Section 41 handout)

V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010

Notes
Section 2.8
Linear Approximation and Differentials

V63.0121.041, Calculus I

New York University

October 13, 2010

Announcements

Quiz 2 in recitation this week on §§1.5, 1.6, 2.1, 2.2
Midterm on §§1.1–2.5

Announcements
Notes

Quiz 2 in recitation this
week on §§1.5, 1.6, 2.1, 2.2
Midterm on §§1.1–2.5

V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 2 / 27

Objectives
Notes
Use tangent lines to make
linear approximations to a
function.
Given a function and a
point in the domain,
compute the linearization
of the function at that
point.
Use linearization to
approximate values of
functions
Given a function, compute
the differential of that
function
Use the differential notation
to estimate error in linear
approximations.

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Outline
Notes

The linear approximation of a function near a point
Examples
Questions

Differentials
Using differentials to estimate error

Advanced Examples


The Big Idea
Notes
Question
Let f be differentiable at a. What linear function best approximates f near
a?

Answer
The tangent line, of course!

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)


The tangent line is a linear approximation
Notes

y

L(x) = f (a) + f (a)(x − a)

is a decent approximation to f L(x)
near a. f (x)
How decent? The closer x is to
a, the better the approxmation f (a) x −a
L(x) is to f (x)

x
a x


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

Solution (i) Solution (ii)
√
If f (x) = sin x, then f (0) = 0 We have f π = 23 and
3
and f (0) = 1. f π = 1.
3 2 √
So the linear approximation near 3 1 π
So L(x) = + x−
0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus
Thus
61π
61π 61π sin ≈ 0.87475
sin ≈ ≈ 1.06465 180
180 180

Calculator check: sin(61◦ ) ≈ 0.87462.


Illustration
Notes
y
y = L1 (x) = x

√
3 1 π
y = L2 (x) = 2 + 2 x− 3
big diﬀerence! y = sin x
very little diﬀerence!

x
0 π/3 61◦


Another Example
Notes
Example
√
Estimate 10 using the fact that 10 = 9 + 1.

Solution


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Dividing without dividing?
Notes
Example
Suppose I have an irrational fear of division and need to estimate
577 ÷ 408. I write
577 1 1 1
= 1 + 169 = 1 + 169 × × .
408 408 4 102
1
But still I have to ﬁnd .
102

Solution


Questions
Notes

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?

Example
Suppose our factory makes MP3 players and the marginal cost is currently
$50/lot. How much will it cost to make 2 more lots? 3 more lots? 12
more lots?

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


Answers
Notes

Example

Answer


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Questions
Notes

Example

Example
more lots?

Example


Answers
Notes

Example
more lots?

Answer


Questions
Notes

Example

Example
more lots?

Example


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Answers
Notes

Example

Answer


Outline
Notes

Examples
Questions

Diﬀerentials

Advanced Examples


Diﬀerentials are another way to express derivatives
Notes

f (x + ∆x) − f (x) ≈ f (x) ∆x y
∆y dy

Rename ∆x = dx, so we can
write this as

∆y ≈ dy = f (x)dx. dy
∆y

And this looks a lot like the dx = ∆x
Leibniz-Newton identity
dy
= f (x) x
dx x x + ∆x
Linear approximation means ∆y ≈ dy = f (x0 ) dx near x0 .


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Notes

y
If y = f (x), x0 and ∆x is known,
and an estimate of ∆y is desired:

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

Evaluate at x = x0 and
dx = ∆x
dx = ∆x.

x
x x + ∆x


Example Notes
A sheet of plywood measures 8 ft × 4 ft. Suppose our plywood-cutting
machine 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 the area
of the sheet be off by?

Solution
1 2
Write A( ) = . We want to know ∆A when = 8 ft and ∆ = 1 in.
2
97 9409 9409
(I) A( + ∆ ) = A = So ∆A = − 32 ≈ 0.6701.
12 288 288
dA
(II) = , so dA = d , which should be a good estimate for ∆ .
d
When = 8 and d = 12 , we have dA = 12 = 2 ≈ 0.667. So we get
1 8
3
estimates close to the hundredth of a square foot.


Why?
Notes

Why use linear approximations dy when the actual difference ∆y is
known?
Linear approximation is quick and reliable. Finding ∆y exactly
depends on the function.
These examples are overly simple. See the “Advanced Examples”
later.
In real life, sometimes only f (a) and f (a) are known, and not the
general f (x).


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Outline
Notes

Examples
Questions

Diﬀerentials

Advanced Examples


Gravitation
Pencils down! Notes

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 = −mg from gravity.
In fact, the force felt is
GMm
F (r ) = − ,
r2
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.
GMm
At r = re the force really is F (re ) = 2
= −mg .
re
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 level F (re )?
The relative error? The percentage error?

Gravitation Solution
Notes
Solution
We wonder if ∆F = F (re + ∆r ) − F (re ) is small.
Using a linear approximation,

dF GMm
∆F ≈ dF = dr = 2 3
dr
dr re re
GMm dr ∆r
= 2
= 2mg
re re re

∆F ∆r
The relative error is ≈ −2
F re
re = 6378.1 km. If ∆r = 50 m,
∆F ∆r 50
≈ −2 = −2 = −1.56 × 10−5 = −0.00156%
F re 6378100


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Systematic linear approximation
Notes
√
2 is irrational, but 9/4 is rational and 9/4 is close to 2. So
√ 1 17
2= 9/4 − 1/4 ≈ 9/4 + (−1/4) =
2(3/2) 12

This is a better approximation since (17/12)2 = 289/144
Do it again!
√ 1
2= 289/144 − 1/144 ≈ 289/144 + (−1/144) = 577/408
2(17/12)
2
577 332, 929 1
Now = which is away from 2.
408 166, 464 166, 464


Illustration of the previous example
Notes

(2, 17 )
12

(9, 2)
4
3
2

V63.0121.041, Calculus I (NYU) Section 2.8 17 12Approximation and Differentials
(2, / )
Linear 9 3 October 13, 2010 26 / 27
(4, 2)
577 289 17
2, 408 144 , 12

Summary
Notes

Linear approximation: If f is differentiable at a, the best linear
approximation to f near a is given by

Lf ,a (x) = f (a) + f (a)(x − a)

Differentials: If f is differentiable at x, a good approximation to
∆y = f (x + ∆x) − f (x) is

dy dy
∆y ≈ dy = · dx = · ∆x
2
dx dx
Don’t buy plywood from me.


9

Lesson 12: Linear Approximation (Section 41 handout)

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Lesson 12: Linear Approximation (Section 41 handout)