The line tangent to a curve, which is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.

1. 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.
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 3 / 27
1

2. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
Outline
Notes
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 4 / 27
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)
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 5 / 27
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 6 / 27
2

3. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
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.
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 7 / 27
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◦
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 8 / 27
Another Example
Notes
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 9 / 27
3

4. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 10 / 27
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?
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 11 / 27
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 12 / 27
4

5. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
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?
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 13 / 27
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 14 / 27
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?
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 15 / 27
5

6. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 16 / 27
Outline
Notes
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 17 / 27
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 .
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 18 / 27
6

7. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
Using differentials to estimate error
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 19 / 27
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.
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 20 / 27
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).
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 21 / 27
7

8. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
Outline
Notes
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 22 / 27
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?
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 23 / 27
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 24 / 27
8

9. V63.0121.041, CalculusSection 2.8 : Linear Approximation and Differentials
I October 13, 2010
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
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 25 / 27
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.
V63.0121.041, Calculus I (NYU) Section 2.8 Linear Approximation and Differentials October 13, 2010 27 / 27
9