Lesson 12: Linear Approximation and Differentials (Section 21 handout)

V63.0121.021, CalculusSection 2.8 : Linear Approximation and Differentials
I October 14, 2010

Notes
Section 2.8
Linear Approximation and Differentials

V63.0121.021, Calculus I

New York University

October 14, 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

V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 14, 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
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


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Answers
Notes

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?

Answer


Answers
Notes

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?

Answer


Outline
Notes

Examples
Questions

Diﬀerentials

Advanced Examples


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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 .


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.


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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).


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?

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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


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

V63.0121.021, Calculus I (NYU) Section 2.8 17 12Approximation and Diﬀerentials
(2, / )
Linear 9 3 October 14, 2010 26 / 27
(4, 2)
577 289 17
2, 408 144 , 12

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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
dx dx
Don’t buy plywood from me.


Notes

Notes

9

Lesson 12: Linear Approximation and Differentials (Section 21 handout)

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