lesson 12: linear approximation and differentials (section 21 slides)
DESCRIPTION
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.TRANSCRIPT
Section 2.8Linear Approximation and Differentials
V63.0121.021, Calculus I
New York University
October 14, 2010
Announcements
I Quiz 2 in recitation this week on §§1.5, 1.6, 2.1, 2.2I Midterm on §§1.1–2.5
. . . . . .
. . . . . .
Announcements
I Quiz 2 in recitation thisweek on §§1.5, 1.6, 2.1,2.2
I Midterm on §§1.1–2.5
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 2 / 36
. . . . . .
Midterm FAQ
QuestionWhat sections are covered on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs Section 2.6 going to be on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs 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
. . . . . .
Midterm FAQ
QuestionWhat sections are covered on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs Section 2.6 going to be on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs 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
. . . . . .
Midterm FAQ
QuestionWhat sections are covered on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs Section 2.6 going to be on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs 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
. . . . . .
Midterm FAQ
QuestionWhat sections are covered on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs Section 2.6 going to be on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs 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
. . . . . .
Midterm FAQ
QuestionWhat sections are covered on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs Section 2.6 going to be on the midterm?
AnswerThe midterm will cover Sections 1.1–2.5 (The Chain Rule).
QuestionIs 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
. . . . . .
Midterm FAQ, continued
QuestionWhat format will the exam take?
AnswerThere will be both fixed-response (e.g., multiple choice) andfree-response questions.
QuestionWill explanations be necessary?
AnswerYes, 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
. . . . . .
Midterm FAQ, continued
QuestionWhat format will the exam take?
AnswerThere will be both fixed-response (e.g., multiple choice) andfree-response questions.
QuestionWill explanations be necessary?
AnswerYes, 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
. . . . . .
Midterm FAQ, continued
QuestionWhat format will the exam take?
AnswerThere will be both fixed-response (e.g., multiple choice) andfree-response questions.
QuestionWill explanations be necessary?
AnswerYes, 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
. . . . . .
Midterm FAQ, continued
QuestionWhat format will the exam take?
AnswerThere will be both fixed-response (e.g., multiple choice) andfree-response questions.
QuestionWill explanations be necessary?
AnswerYes, 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
. . . . . .
Midterm FAQ, continued
QuestionIs (topic X) going to be tested?
AnswerEverything 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 theexam. (This is not the same as saying all exam problems are similar toclass examples or homework problems.)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 5 / 36
. . . . . .
Midterm FAQ, continued
QuestionIs (topic X) going to be tested?
AnswerEverything 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 theexam. (This is not the same as saying all exam problems are similar toclass examples or homework problems.)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 5 / 36
. . . . . .
Midterm FAQ, continued
QuestionWill there be a review session?
AnswerWe’re working on it.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 6 / 36
. . . . . .
Midterm FAQ, continued
QuestionWill there be a review session?
AnswerWe’re working on it.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 6 / 36
. . . . . .
Midterm FAQ, continued
QuestionWill calculators be allowed?
AnswerNo. The exam is designed for pencil and brain.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 7 / 36
. . . . . .
Midterm FAQ, continued
QuestionWill calculators be allowed?
AnswerNo. The exam is designed for pencil and brain.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 7 / 36
. . . . . .
Midterm FAQ, continued
QuestionHow should I study?
Answer
I The exam has problems; study by doing problems. If you get oneright, think about how you got it right. If you got it wrong or didn’tget it at all, reread the textbook and do easier problems to build upyour understanding.
I Break up the material into chunks. (related) Don’t put it all off untilthe night before.
I Ask questions.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 8 / 36
. . . . . .
Midterm FAQ, continued
QuestionHow should I study?
Answer
I The exam has problems; study by doing problems. If you get oneright, think about how you got it right. If you got it wrong or didn’tget it at all, reread the textbook and do easier problems to build upyour understanding.
I Break up the material into chunks. (related) Don’t put it all off untilthe night before.
I Ask questions.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 8 / 36
. . . . . .
Midterm FAQ, continued
QuestionHow many questions are there?
AnswerDoes 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
. . . . . .
Midterm FAQ, continued
QuestionHow many questions are there?
AnswerDoes 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
. . . . . .
Midterm FAQ, continued
QuestionWill there be a curve on the exam?
AnswerDoes 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
. . . . . .
Midterm FAQ, continued
QuestionWill there be a curve on the exam?
AnswerDoes 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
. . . . . .
Midterm FAQ, continued
QuestionWhen will you grade my get-to-know-you and photo extra credit?
AnswerDoes 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
. . . . . .
Midterm FAQ, continued
QuestionWhen will you grade my get-to-know-you and photo extra credit?
AnswerDoes 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
. . . . . .
Objectives
I Use tangent lines to makelinear approximations to afunction.
I Given a function and apoint in the domain,compute thelinearization of thefunction at that point.
I Use linearization toapproximate values offunctions
I Given a function, computethe differential of thatfunction
I Use the differentialnotation to estimate errorin linear approximations.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 12 / 36
. . . . . .
Outline
The linear approximation of a function near a pointExamplesQuestions
DifferentialsUsing differentials to estimate error
Advanced Examples
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 13 / 36
. . . . . .
The Big Idea
QuestionLet f be differentiable at a. What linear function best approximates fnear a?
AnswerThe tangent line, of course!
QuestionWhat 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
. . . . . .
The Big Idea
QuestionLet f be differentiable at a. What linear function best approximates fnear a?
AnswerThe tangent line, of course!
QuestionWhat 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
. . . . . .
The Big Idea
QuestionLet f be differentiable at a. What linear function best approximates fnear a?
AnswerThe tangent line, of course!
QuestionWhat 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
. . . . . .
The Big Idea
QuestionLet f be differentiable at a. What linear function best approximates fnear a?
AnswerThe tangent line, of course!
QuestionWhat 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
. . . . . .
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 closer x is toa, the better the approxmationL(x) is to f(x)
. .x
.y
.
.
.
.f(a)
.f(x).L(x)
.a .x
.x − a
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 15 / 36
. . . . . .
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 closer x is toa, the better the approxmationL(x) is to f(x)
. .x
.y
.
.
.
.f(a)
.f(x).L(x)
.a .x
.x − a
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 15 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32
andf′(π3)=
12
.
I So L(x) =
√32
+12
(x− π
3
)
I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32
andf′(π3)=
12
.
I So L(x) =
√32
+12
(x− π
3
)
I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32
andf′(π3)=
12
.
I So L(x) =
√32
+12
(x− π
3
)
I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)=
12
.
I So L(x) =
√32
+12
(x− π
3
)
I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)= 1
2 .
I So L(x) =
√32
+12
(x− π
3
)
I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)= 1
2 .
I So L(x) =
√32
+12
(x− π
3
)I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)= 1
2 .
I So L(x) =√32
+12
(x− π
3
)
I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)= 1
2 .
I So L(x) =√32
+12
(x− π
3
)I Thus
sin(61π180
)≈
0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)= 1
2 .
I So L(x) =√32
+12
(x− π
3
)I Thus
sin(61π180
)≈ 0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)= 1
2 .
I So L(x) =√32
+12
(x− π
3
)I Thus
sin(61π180
)≈ 0.87475
Calculator check: sin(61◦) ≈
0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 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)
I If f(x) = sin x, then f(0) = 0and f′(0) = 1.
I So the linear approximationnear 0 is L(x) = 0+ 1 · x = x.
I Thus
sin(61π180
)≈ 61π
180≈ 1.06465
Solution (ii)
I We have f(π3)=
√32 and
f′(π3)= 1
2 .
I So L(x) =√32
+12
(x− π
3
)I Thus
sin(61π180
)≈ 0.87475
Calculator check: sin(61◦) ≈ 0.87462.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 16 / 36
. . . . . .
Illustration
. .x
.y
.y = sin x
.61◦
.y = L1(x) = x
..0
.
.big difference!
.y = L2(x) =√32 + 1
2(x− π
3)
..π/3
.
.very little difference!
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
. . . . . .
Illustration
. .x
.y
.y = sin x
.61◦
.y = L1(x) = x
..0
.
.big difference!.y = L2(x) =
√32 + 1
2(x− π
3)
..π/3
.
.very little difference!
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
. . . . . .
Illustration
. .x
.y
.y = sin x
.61◦
.y = L1(x) = x
..0
.
.big difference!
.y = L2(x) =√32 + 1
2(x− π
3)
..π/3
.
.very little difference!
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
. . . . . .
Illustration
. .x
.y
.y = sin x
.61◦
.y = L1(x) = x
..0
.
.big difference!
.y = L2(x) =√32 + 1
2(x− π
3)
..π/3
.
.very little difference!
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
. . . . . .
Illustration
. .x
.y
.y = sin x
.61◦
.y = L1(x) = x
..0
.
.big difference!
.y = L2(x) =√32 + 1
2(x− π
3)
..π/3
. .very little difference!
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 17 / 36
. . . . . .
Another Example
Example
Estimate√10 using the fact that 10 = 9+ 1.
SolutionThe 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)
= 3+1
2 · 3(1) =
196
≈ 3.167
Check:(196
)2=
36136
.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
. . . . . .
Another Example
Example
Estimate√10 using the fact that 10 = 9+ 1.
SolutionThe 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)
= 3+1
2 · 3(1) =
196
≈ 3.167
Check:(196
)2=
36136
.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
. . . . . .
Another Example
Example
Estimate√10 using the fact that 10 = 9+ 1.
SolutionThe 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)
= 3+1
2 · 3(1) =
196
≈ 3.167
Check:(196
)2=
36136
.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
. . . . . .
Another Example
Example
Estimate√10 using the fact that 10 = 9+ 1.
SolutionThe 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)
= 3+1
2 · 3(1) =
196
≈ 3.167
Check:(196
)2=
36136
.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
. . . . . .
Another Example
Example
Estimate√10 using the fact that 10 = 9+ 1.
SolutionThe 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)
= 3+1
2 · 3(1) =
196
≈ 3.167
Check:(196
)2=
36136
.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 18 / 36
. . . . . .
Dividing without dividing?
Example
A student has an irrational fear of long division and needs to estimate577÷ 408. He writes
577408
= 1+ 1691
408= 1+ 169× 1
4× 1
102.
Help the student estimate1
102.
Solution
Let f(x) =1x. We know f(100) and we want to estimate f(102).
f(102) ≈ f(100) + f′(100)(2) =1
100− 1
1002(2) = 0.0098
=⇒ 577408
≈ 1.41405
Calculator check:577408
≈ 1.41422.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 19 / 36
. . . . . .
Dividing without dividing?
Example
A student has an irrational fear of long division and needs to estimate577÷ 408. He writes
577408
= 1+ 1691
408= 1+ 169× 1
4× 1
102.
Help the student estimate1
102.
Solution
Let f(x) =1x. We know f(100) and we want to estimate f(102).
f(102) ≈ f(100) + f′(100)(2) =1
100− 1
1002(2) = 0.0098
=⇒ 577408
≈ 1.41405
Calculator check:577408
≈ 1.41422.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 19 / 36
. . . . . .
Questions
Example
Suppose we are traveling in a car and at noon our speed is 50mi/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 iscurrently $50/lot. How much will it cost to make 2 more lots? 3 morelots? 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?
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 20 / 36
. . . . . .
Answers
Example
Suppose we are traveling in a car and at noon our speed is 50mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight?
Answer
I 100miI 150miI 600mi (?) (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
. . . . . .
Answers
Example
Suppose we are traveling in a car and at noon our speed is 50mi/hr.How far will we have traveled by 2:00pm? by 3:00pm? By midnight?
Answer
I 100miI 150miI 600mi (?) (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
. . . . . .
Questions
Example
Suppose we are traveling in a car and at noon our speed is 50mi/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 iscurrently $50/lot. How much will it cost to make 2 more lots? 3 morelots? 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?
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 22 / 36
. . . . . .
Answers
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 morelots? 12 more lots?
Answer
I $100I $150I $600 (?)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 23 / 36
. . . . . .
Questions
Example
Suppose we are traveling in a car and at noon our speed is 50mi/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 iscurrently $50/lot. How much will it cost to make 2 more lots? 3 morelots? 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?
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 24 / 36
. . . . . .
Answers
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?
AnswerThe slope of the line is
m =riserun
We are given a “run” of dx, so the corresponding “rise” is mdx.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 25 / 36
. . . . . .
Answers
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?
AnswerThe slope of the line is
m =riserun
We are given a “run” of dx, so the corresponding “rise” is mdx.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 25 / 36
. . . . . .
Outline
The linear approximation of a function near a pointExamplesQuestions
DifferentialsUsing differentials to estimate error
Advanced Examples
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 26 / 36
. . . . . .
Differentials are another way to express derivatives
The fact that the the tangentline is an approximation meansthat
f(x+∆x)− f(x)︸ ︷︷ ︸∆y
≈ f′(x)∆x︸ ︷︷ ︸dy
Rename ∆x = dx, so we canwrite this as
∆y ≈ dy = f′(x)dx.
Note this looks a lot like theLeibniz-Newton identity
dydx
= f′(x)
. .x
.y
.
.
.x .x+∆x
.dx = ∆x
.∆y.dy
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 27 / 36
. . . . . .
Using differentials to estimate error
Estimating error withdifferentialsIf y = f(x), x0 and ∆x is known,and an estimate of ∆y isdesired:
I Approximate: ∆y ≈ dyI Differentiate: dy = f′(x)dxI Evaluate at x = x0 and
dx = ∆x.. .x
.y
.
.
.x .x+∆x
.dx = ∆x
.∆y.dy
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 28 / 36
. . . . . .
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, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?
Solution
Write A(ℓ) =12ℓ2. We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
(I) A(ℓ+∆ℓ) = A(9712
)=
9409288
So ∆A =9409288
− 32 ≈ 0.6701.
(II) dAdℓ
= ℓ, so dA = ℓdℓ, which should be a good estimate for ∆ℓ.
When ℓ = 8 and dℓ = 112 , we have dA = 8
12 = 23 ≈ 0.667. So we
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
. . . . . .
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, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?
Solution
Write A(ℓ) =12ℓ2. We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
(I) A(ℓ+∆ℓ) = A(9712
)=
9409288
So ∆A =9409288
− 32 ≈ 0.6701.
(II) dAdℓ
= ℓ, so dA = ℓdℓ, which should be a good estimate for ∆ℓ.
When ℓ = 8 and dℓ = 112 , we have dA = 8
12 = 23 ≈ 0.667. So we
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
. . . . . .
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, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?
Solution
Write A(ℓ) =12ℓ2. We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
(I) A(ℓ+∆ℓ) = A(9712
)=
9409288
So ∆A =9409288
− 32 ≈ 0.6701.
(II) dAdℓ
= ℓ, so dA = ℓdℓ, which should be a good estimate for ∆ℓ.
When ℓ = 8 and dℓ = 112 , we have dA = 8
12 = 23 ≈ 0.667. So we
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
. . . . . .
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, butthe length is prone to errors. If the length is off by 1 in, how bad can thearea of the sheet be off by?
Solution
Write A(ℓ) =12ℓ2. We want to know ∆A when ℓ = 8 ft and ∆ℓ = 1 in.
(I) A(ℓ+∆ℓ) = A(9712
)=
9409288
So ∆A =9409288
− 32 ≈ 0.6701.
(II) dAdℓ
= ℓ, so dA = ℓdℓ, which should be a good estimate for ∆ℓ.
When ℓ = 8 and dℓ = 112 , we have dA = 8
12 = 23 ≈ 0.667. So we
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
. . . . . .
Why?
Why use linear approximations dy when the actual difference ∆y isknown?
I Linear approximation is quick and reliable. Finding ∆y exactlydepends on the function.
I These examples are overly simple. See the “Advanced Examples”later.
I In real life, sometimes only f(a) and f′(a) are known, and not thegeneral f(x).
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 30 / 36
. . . . . .
Outline
The linear approximation of a function near a pointExamplesQuestions
DifferentialsUsing differentials to estimate error
Advanced Examples
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 31 / 36
. . . . . .
GravitationPencils down!
Example
I Drop a 1 kg ball off the roof of the Silver Center (50m high). Weusually say that a falling object feels a force F = −mg from gravity.
I In fact, the force felt is
F(r) = −GMmr2
,
where M is the mass of the earth and r is the distance from thecenter of the earth to the object. G is a constant.
I At r = re the force really is F(re) =GMmr2e
= −mg.
I What is the maximum error in replacing the actual force felt at thetop of the building F(re +∆r) by the force felt at ground levelF(re)? The relative error? The percentage error?
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 32 / 36
. . . . . .
GravitationPencils down!
Example
I Drop a 1 kg ball off the roof of the Silver Center (50m high). Weusually say that a falling object feels a force F = −mg from gravity.
I In fact, the force felt is
F(r) = −GMmr2
,
where M is the mass of the earth and r is the distance from thecenter of the earth to the object. G is a constant.
I At r = re the force really is F(re) =GMmr2e
= −mg.
I What is the maximum error in replacing the actual force felt at thetop of the building F(re +∆r) by the force felt at ground levelF(re)? The relative error? The percentage error?
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 32 / 36
. . . . . .
Gravitation Solution
SolutionWe wonder if ∆F = F(re +∆r)− F(re) is small.
I Using a linear approximation,
∆F ≈ dF =dFdr
∣∣∣∣redr = 2
GMmr3e
dr
=
(GMmr2e
)drre
= 2mg∆rre
I The relative error is∆FF
≈ −2∆rre
I re = 6378.1 km. If ∆r = 50m,
∆FF
≈ −2∆rre
= −250
6378100= −1.56× 10−5 = −0.00156%
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 33 / 36
. . . . . .
Systematic linear approximation
I√2 is irrational, but
√9/4 is rational and 9/4 is close to 2.
So
√2 =
√9/4 − 1/4 ≈
√9/4 +
12(3/2)
(−1/4) =1712
I This is a better approximation since (17/12)2 = 289/144
I Do it again!
√2 =
√289/144 − 1/144 ≈
√289/144 +
12(17/12)
(−1/144) = 577/408
Now(577408
)2=
332,929166,464
which is1
166,464away from 2.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
. . . . . .
Systematic linear approximation
I√2 is irrational, but
√9/4 is rational and 9/4 is close to 2. So
√2 =
√9/4 − 1/4 ≈
√9/4 +
12(3/2)
(−1/4) =1712
I This is a better approximation since (17/12)2 = 289/144
I Do it again!
√2 =
√289/144 − 1/144 ≈
√289/144 +
12(17/12)
(−1/144) = 577/408
Now(577408
)2=
332,929166,464
which is1
166,464away from 2.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
. . . . . .
Systematic linear approximation
I√2 is irrational, but
√9/4 is rational and 9/4 is close to 2. So
√2 =
√9/4 − 1/4 ≈
√9/4 +
12(3/2)
(−1/4) =1712
I This is a better approximation since (17/12)2 = 289/144
I Do it again!
√2 =
√289/144 − 1/144 ≈
√289/144 +
12(17/12)
(−1/144) = 577/408
Now(577408
)2=
332,929166,464
which is1
166,464away from 2.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
. . . . . .
Systematic linear approximation
I√2 is irrational, but
√9/4 is rational and 9/4 is close to 2. So
√2 =
√9/4 − 1/4 ≈
√9/4 +
12(3/2)
(−1/4) =1712
I This is a better approximation since (17/12)2 = 289/144
I Do it again!
√2 =
√289/144 − 1/144 ≈
√289/144 +
12(17/12)
(−1/144) = 577/408
Now(577408
)2=
332,929166,464
which is1
166,464away from 2.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 34 / 36
. . . . . .
Illustration of the previous example
.
.2
.
(94 ,32)
..(2, 1712)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
.
.2
.
(94 ,32)
..(2, 1712)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
.
(94 ,32)
..(2, 1712)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
.
(94 ,32)
..(2, 1712)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
.
(94 ,32)
..(2, 1712)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
.
(94 ,32)
..(2, 1712)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
.
(94 ,32)
..(2, 1712)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
..(94 ,
32)..(2, 17/12)
..(289144 ,
1712
)..(2, 577408
)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
..(94 ,
32)..(2, 17/12) .
.(289144 ,
1712
)
..(2, 577408
)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Illustration of the previous example
..2
..(94 ,
32)..(2, 17/12) .
.(289144 ,
1712
)..(2, 577408
)
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 35 / 36
. . . . . .
Summary
I Linear approximation: If f is differentiable at a, the best linearapproximation to f near a is given by
Lf,a(x) = f(a) + f′(a)(x− a)
I Differentials: If f is differentiable at x, a good approximation to∆y = f(x+∆x)− f(x) is
∆y ≈ dy =dydx
· dx =dydx
·∆x
I Don’t buy plywood from me.
V63.0121.021, Calculus I (NYU) Section 2.8 Linear Approximation October 14, 2010 36 / 36