The line tangent to a curve 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.
Lesson 12: Linear Approximation and Differentials (Section 21 slides)
1. 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
. . . . . .
2. Announcements
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 October 14, 2010 2 / 36
3. Midterm FAQ
Question
What sections are covered on the midterm?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
4. Midterm FAQ
Question
What sections are covered on the midterm?
Answer
The midterm will cover Sections 1.1–2.5 (The Chain Rule).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
5. Midterm FAQ
Question
What sections are covered on the midterm?
Answer
The midterm will cover Sections 1.1–2.5 (The Chain Rule).
Question
Is Section 2.6 going to be on the midterm?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
6. Midterm FAQ
Question
What sections are covered on the midterm?
Answer
The midterm will cover Sections 1.1–2.5 (The Chain Rule).
Question
Is Section 2.6 going to be on the midterm?
Answer
The midterm will cover Sections 1.1–2.5 (The Chain Rule).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
7. Midterm FAQ
Question
What sections are covered on the midterm?
Answer
The midterm will cover Sections 1.1–2.5 (The Chain Rule).
Question
Is Section 2.6 going to be on the midterm?
Answer
The midterm will cover Sections 1.1–2.5 (The Chain Rule).
Question
Is Section 2.8 going to be on the midterm?...
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 3 / 36
8. Midterm FAQ, continued
Question
What format will the exam take?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
9. Midterm FAQ, continued
Question
What format will the exam take?
Answer
There will be both fixed-response (e.g., multiple choice) and
free-response questions.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
10. Midterm FAQ, continued
Question
What format will the exam take?
Answer
There will be both fixed-response (e.g., multiple choice) and
free-response questions.
Question
Will explanations be necessary?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
11. Midterm FAQ, continued
Question
What format will the exam take?
Answer
There will be both fixed-response (e.g., multiple choice) and
free-response questions.
Question
Will explanations be necessary?
Answer
Yes, on free-response problems we will expect you to explain yourself.
This is why it was required on written homework.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 4 / 36
12. Midterm FAQ, continued
Question
Is (topic X) going to be tested?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 5 / 36
13. Midterm FAQ, continued
Question
Is (topic X) going to be tested?
Answer
Everything covered in class or on homework is fair game for the exam.
No topic that was not covered in class nor on homework will be on the
exam. (This is not the same as saying all exam problems are similar to
class examples or homework problems.)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 5 / 36
14. Midterm FAQ, continued
Question
Will there be a review session?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 6 / 36
15. Midterm FAQ, continued
Question
Will there be a review session?
Answer
We’re working on it.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 6 / 36
16. Midterm FAQ, continued
Question
Will calculators be allowed?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 7 / 36
17. Midterm FAQ, continued
Question
Will calculators be allowed?
Answer
No. The exam is designed for pencil and brain.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 7 / 36
18. Midterm FAQ, continued
Question
How should I study?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 8 / 36
19. Midterm FAQ, continued
Question
How should I study?
Answer
The exam has problems; study by doing problems. If you get one
right, think about how you got it right. If you got it wrong or didn’t
get it at all, reread the textbook and do easier problems to build up
your understanding.
Break up the material into chunks. (related) Don’t put it all off until
the night before.
Ask questions.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 8 / 36
20. Midterm FAQ, continued
Question
How many questions are there?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 9 / 36
21. Midterm FAQ, continued
Question
How many questions are there?
Answer
Does this question contribute to your understanding of the material?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 9 / 36
22. Midterm FAQ, continued
Question
Will there be a curve on the exam?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 10 / 36
23. Midterm FAQ, continued
Question
Will there be a curve on the exam?
Answer
Does this question contribute to your understanding of the material?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 10 / 36
24. Midterm FAQ, continued
Question
When will you grade my get-to-know-you and photo extra credit?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 11 / 36
25. Midterm FAQ, continued
Question
When will you grade my get-to-know-you and photo extra credit?
Answer
Does this question contribute to your understanding of the material?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 11 / 36
26. Objectives
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. . . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 12 / 36
27. Outline
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 13 / 36
28. The Big Idea
Question
Let f be differentiable at a. What linear function best approximates f
near a?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
29. The Big Idea
Question
Let f be differentiable at a. What linear function best approximates f
near a?
Answer
The tangent line, of course!
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
30. The Big Idea
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))?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
31. The Big Idea
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)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 14 / 36
32. The tangent line is a linear approximation
y
.
L(x) = f(a) + f′ (a)(x − a)
is a decent approximation to f L
. (x) .
near a. f
.(x) .
f
.(a) .
.
x−a
. x
.
a
. x
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 15 / 36
33. The tangent line is a linear approximation
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
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 15 / 36
34. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
35. Example
.
Example
Estimate sin(61◦ ) = sin(61π/180) by using a linear approximation
(i) about a = 0 (ii) about a = 60◦ = π/3.
Solution (i)
If f(x) = sin x, then f(0) = 0
and f′ (0) = 1.
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
36. Example
.
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)
(π)
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3
and f′ (0) = 1. f′ π = .
3
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
37. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = .
3
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
38. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2
So the linear approximation
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
39. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2
So the linear approximation
So L(x) =
near 0 is L(x) = 0 + 1 · x = x.
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
40. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus
( )
61π 61π
sin ≈ ≈ 1.06465
180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
41. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈
180 180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
42. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈ 0.87475
180 180 180
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
43. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈ 0.87475
180 180 180
Calculator check: sin(61◦ ) ≈
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
44. Example
.
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)
(π) √
3
We have f = and
If f(x) = sin x, then f(0) = 0 ( ) 3 2
and f′ (0) = 1. f′ π = 1 .
3 2 √
So the linear approximation 3 1( π)
So L(x) = + x−
near 0 is L(x) = 0 + 1 · x = x. 2 2 3
Thus Thus
( ) ( )
61π 61π 61π
sin ≈ ≈ 1.06465 sin ≈ 0.87475
180 180 180
Calculator check: sin(61◦ ) ≈ 0.87462.
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
45. Illustration
y
.
y
. = sin x
. x
.
. 1◦
6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
46. Illustration
y
.
y
. = L1 (x) = x
y
. = sin x
. x
.
0
. . 1◦
6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
47. Illustration
y
.
y
. = L1 (x) = x
b
. ig difference! y
. = sin x
. x
.
0
. . 1◦
6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
48. Illustration
y
.
y
. = L1 (x) = x
√ ( )
y
. = L2 (x) = 2
3
+ 1
2 x− π
3
y
. = sin x
.
. . x
.
0
. .
π/3 . 1◦
6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
49. Illustration
y
.
y
. = L1 (x) = x
√ ( )
y
. = L2 (x) = 2
3
+ 1
2 x− π
3
y
. = sin x
. . ery little difference!
v
. . x
.
0
. .
π/3 . 1◦
6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
50. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
51. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
52. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
f(10) ≈ L(10) = f(9) + f′ (9)(10 − 9)
1 19
=3+ (1) = ≈ 3.167
2·3 6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
53. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
f(10) ≈ L(10) = f(9) + f′ (9)(10 − 9)
1 19
=3+ (1) = ≈ 3.167
2·3 6
( )2
19
Check: =
6
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
54. Another Example
Example
√
Estimate 10 using the fact that 10 = 9 + 1.
Solution
√
The key step is to use a linear approximation to f(x) =
√ x near a = 9
to estimate f(10) = 10.
f(10) ≈ L(10) = f(9) + f′ (9)(10 − 9)
1 19
=3+ (1) = ≈ 3.167
2·3 6
( )2
19 361
Check: = .
6 36
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
55. Dividing without dividing?
Example
A student has an irrational fear of long division and needs to estimate
577 ÷ 408. He writes
577 1 1 1
= 1 + 169 = 1 + 169 × × .
408 408 4 102
1
Help the student estimate .
102
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 19 / 36
56. Dividing without dividing?
Example
A student has an irrational fear of long division and needs to estimate
577 ÷ 408. He writes
577 1 1 1
= 1 + 169 = 1 + 169 × × .
408 408 4 102
1
Help the student estimate .
102
Solution
1
Let f(x) = . We know f(100) and we want to estimate f(102).
x
1 1
f(102) ≈ f(100) + f′ (100)(2) = − (2) = 0.0098
100 1002
577
=⇒ ≈ 1.41405
408 . . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 19 / 36
57. Questions
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?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 20 / 36
58. Answers
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?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 21 / 36
59. Answers
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?)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 21 / 36
60. Questions
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?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 22 / 36
61. Answers
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
$100
$150
$600 (?)
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 23 / 36
62. Questions
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?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 24 / 36
63. Answers
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?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 25 / 36
64. Answers
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
The slope of the line is
rise
m=
run
We are given a “run” of dx, so the corresponding “rise” is m dx.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 25 / 36
65. Outline
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 26 / 36
66. Differentials are another way to express derivatives
The fact that the the tangent
line is an approximation means
that
y
.
f(x + ∆x) − f(x) ≈ f′ (x) ∆x
∆y dy
Rename ∆x = dx, so we can
write this as .
.
dy
.
∆y
′
∆y ≈ dy = f (x)dx. .
.
dx = ∆x
Note this looks a lot like the
Leibniz-Newton identity
. x
.
dy x x
. . + ∆x
= f′ (x)
dx
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 27 / 36
67. Using differentials to estimate error
y
.
Estimating error with
differentials
If y = f(x), x0 and ∆x is known,
and an estimate of ∆y is
desired: .
.
Approximate: ∆y ≈ dy
dy
.
∆y
Differentiate: dy = f′ (x) dx .
.
dx = ∆x
Evaluate at x = x0 and
dx = ∆x.
. x
.
x x
. . + ∆x
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 28 / 36
68. Example
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?
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
69. Example
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(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
2
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
70. Example
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(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
2 ( )
97 9409 9409
(I) A(ℓ + ∆ℓ) = A = So ∆A = − 32 ≈ 0.6701.
12 288 288
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
71. Example
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(ℓ) = ℓ . We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
2 ( )
97 9409 9409
(I) A(ℓ + ∆ℓ) = A = So ∆A = − 32 ≈ 0.6701.
12 288 288
dA
(II) = ℓ, so dA = ℓ dℓ, which should be a good estimate for ∆ℓ.
dℓ
When ℓ = 8 and dℓ = 12 , we have dA = 12 = 2 ≈ 0.667. So we
1 8
3
get estimates close to the hundredth of a square foot.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 29 / 36
72. Why?
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).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 30 / 36
73. Outline
The linear approximation of a function near a point
Examples
Questions
Differentials
Using differentials to estimate error
Advanced Examples
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 31 / 36
74. Gravitation
Pencils down!
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.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 32 / 36
75. Gravitation
Pencils down!
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 ) = = −mg.
r2
e
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? . . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 32 / 36
76. Gravitation Solution
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
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 33 / 36
77. Systematic linear approximation
√ √
2 is irrational, but 9/4 is rational and 9/4 is close to 2.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
78. Systematic linear approximation
√ √
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
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
79. Systematic linear approximation
√ √
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
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
80. Systematic linear approximation
√ √
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
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
81. Illustration of the previous example
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
82. Illustration of the previous example
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
83. Illustration of the previous example
.
2
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
84. Illustration of the previous example
.
.
2
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
85. Illustration of the previous example
.
.
2
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
86. Illustration of the previous example
. 2, 17 )
( 12
. .
.
2
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
87. Illustration of the previous example
. 2, 17 )
( 12
. .
.
2
.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
88. Illustration of the previous example
.
. 2, 17/12)
(
. . 4, 3)
(9 2
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
89. Illustration of the previous example
.
. 2, 17/12)
(
.. ( . 9, 3)
(
)4 2
289 17
. 144 , 12
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
90. Illustration of the previous example
.
. 2, 17/12)
(
.. ( . 9, 3)
(
( 577 ) )4 2
. 2, 408 289 17
. 144 , 12
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
91. Summary
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.
. . . . . .
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 36 / 36