m53 lec2.3.2 local linear approx and differentials.pdf
TRANSCRIPT
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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For today
1 Local Linear Approximation and Differentials
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 2 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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For today
1 Local Linear Approximation and Differentials
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 3 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 4 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
f (x0)
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
f (x0) = limx→x0
f (x) − f (x0)
x − x0
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 4 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
f (x0) = limx→x0
f (x) − f (x0)
x − x0
= lim∆x→0
∆y
∆x.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 4 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
f (x0) = limx→x0
f (x) − f (x0)
x − x0
= lim∆x→0
∆y
∆x.
If ∆x is small enough,
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 4 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
f (x0) = limx→x0
f (x) − f (x0)
x − x0
= lim∆x→0
∆y
∆x.
If ∆x is small enough,
=
⇒
∆y
∆x ≈f (x0)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 4 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
f (x0) = limx→x0
f (x) − f (x0)
x − x0
= lim∆x→0
∆y
∆x.
If ∆x is small enough,
=
⇒
∆y
∆x ≈f (x0)
=⇒ ∆y ≈ f (x0)∆x
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 4 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Recall:
f (x0) = limx→x0
f (x) − f (x0)
x − x0
= lim∆x→0
∆y
∆x.
If ∆x is small enough,
=
⇒
∆y
∆x ≈f (x0)
=⇒ ∆y ≈ f (x0)∆xy = f (x)
P
x0
f (x0)
Q
x
f (x)
S
dx=∆x
= x0 + dx
∆y
R
dy
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 4 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
Definitions
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
DefinitionsLet the function y = f (x) be differentiable at x.
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
DefinitionsLet the function y = f (x) be differentiable at x.
1 The differential dx of the independent variable x denotes an
arbitrary increment of x.
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
DefinitionsLet the function y = f (x) be differentiable at x.
1 The differential dx of the independent variable x denotes an
arbitrary increment of x.
2 The differential dy of the dependent variable y associated with
x is given by dy = f (x)dx.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 5 / 21
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
DefinitionsLet the function y = f (x) be differentiable at x.
1 The differential dx of the independent variable x denotes an
arbitrary increment of x.
2 The differential dy of the dependent variable y associated with
x is given by dy = f (x)dx.
If dx
= 0,
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 5 / 21
l d ff l
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
DefinitionsLet the function y = f (x) be differentiable at x.
1 The differential dx of the independent variable x denotes an
arbitrary increment of x.
2 The differential dy of the dependent variable y associated with
x is given by dy = f (x)dx.
If dx
= 0, then
dy = f (x)dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 5 / 21
L l Li A i i d Diff i l
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Local Linear Approximation and Differentials
DefinitionsLet the function y = f (x) be differentiable at x.
1 The differential dx of the independent variable x denotes an
arbitrary increment of x.
2 The differential dy of the dependent variable y associated with
x is given by dy = f (x)dx.
If dx
= 0, then
dy = f (x)dx =⇒ dy
dx= f (x).
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 5 / 21
L l Li A i i d Diff i l
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Local Linear Approximation and Differentials
Remark
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L l Li A i ti d Diff ti l
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Local Linear Approximation and Differentials
Remark
The symboldy
dxmay be interpreted as:
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L l Li A i ti d Diff ti l
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Local Linear Approximation and Differentials
Remark
The symboldy
dxmay be interpreted as:
the derivative of y = f (x) with respect to x
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 6 / 21
L l Li A i ti d Diff ti ls
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Local Linear Approximation and Differentials
Remark
The symboldy
dxmay be interpreted as:
the derivative of y = f (x) with respect to x
the quotient of the differential of y by the differential of x
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 6 / 21
Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Theorem
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Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Theorem
Let u and v be differentiable functions of x and c a constant.
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Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Theorem
Let u and v be differentiable functions of x and c a constant.
1 d(c) = 0
2 d(xn) = nxn−1dx
3 d(cu) = cdu
4 d(uv) = udv − vdu
5 d(uv
) = vdu−udvv2
6 d(un
) = nun−
1du
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Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Example
Find dy if
y = x5
− x3
+ 2x.
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Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Example
Find dy if
y = x5
− x3
+ 2x.
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 8 / 21
Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Example
Find dy if
y = x5
− x3
+ 2x.
Solution.
dy
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 8 / 21
Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Example
Find dy if
y = x5
− x3
+ 2x.
Solution.
dy = (5x4 − 3x2 + 2)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 8 / 21
Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Example
Find dy if
y = x5
− x3
+ 2x.
Solution.
dy = (5x4 − 3x2 + 2) dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 8 / 21
Local Linear Approximation and Differentials
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Local Linear Approximation and Differentials
Example
Find dy if
y =√ x3 + 3x2.
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Local Linear Approximation and Differentials
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pp
Example
Find dy if
y =√ x3 + 3x2.
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 9 / 21
Local Linear Approximation and Differentials
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pp
Example
Find dy if
y =√ x3 + 3x2.
Solution.
dy
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 9 / 21
Local Linear Approximation and Differentials
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pp
Example
Find dy if
y =√ x3 + 3x2.
Solution.
dy =1
2√ x3 + 3x2
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 9 / 21
Local Linear Approximation and Differentials
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pp
Example
Find dy if
y =√ x3 + 3x2.
Solution.
dy =1
2√ x3 + 3x2
· (3x2 + 6x)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 9 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
y =√ x3 + 3x2.
Solution.
dy =1
2√ x3 + 3x2
· (3x2 + 6x) dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 9 / 21
Local Linear Approximation and Differentials
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Example
Find dy if
xy2 = y + x.
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Local Linear Approximation and Differentials
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Example
Find dy if
xy2 = y + x.
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
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Example
Find dy if
xy2 = y + x.
Solution.
x
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
xy2 = y + x.
Solution.
x(2ydy)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
xy2 = y + x.
Solution.
x(2ydy) + y2
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
![Page 42: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/42.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
xy2 = y + x.
Solution.
x(2ydy) + y2dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
![Page 43: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/43.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
xy2 = y + x.
Solution.
x(2ydy) + y2dx = dy
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
![Page 44: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/44.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
xy2 = y + x.
Solution.
x(2ydy) + y2dx = dy + dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
xy2 = y + x.
Solution.
x(2ydy) + y2dx = dy + dx =⇒ dy =
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find dy if
xy2 = y + x.
Solution.
x(2ydy) + y2dx = dy + dx =⇒ dy =1 − y2
2xy
−1dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 10 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0=
⇒
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0=
⇒∆y
≈f (x0)∆x
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0=
⇒∆y
≈f (x0)∆x
=⇒ f (x) ≈ f (x0) + f (x0)(x − x0)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0=
⇒∆y
≈f (x0)∆x
=⇒ f (x) ≈ f (x0) + f (x0)(x − x0)
Equation of the
tangent line to the
graph of y = f (x) atthe point (x0, f (x0)):
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0=
⇒∆y
≈f (x0)∆x
=⇒ f (x) ≈ f (x0) + f (x0)(x − x0)
Equation of the
tangent line to the
graph of y = f (x) atthe point (x0, f (x0)):
y − f (x0) = f (x0)(x − x0)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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∆x ≈ 0=
⇒∆y
≈f (x0)∆x
=⇒ f (x) ≈ f (x0) + f (x0)(x − x0)
Equation of the
tangent line to the
graph of y = f (x) atthe point (x0, f (x0)):
y − f (x0) = f (x0)(x − x0)
∴
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0=
⇒∆y
≈f (x0)∆x
=⇒ f (x) ≈ f (x0) + f (x0)(x − x0)
Equation of the
tangent line to the
graph of y = f (x) atthe point (x0, f (x0)):
y − f (x0) = f (x0)(x − x0)
∴ y = f (x0) + f (x0)(x−x0)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
![Page 55: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/55.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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∆x ≈ 0=
⇒∆y
≈f (x0)∆x
=⇒ f (x) ≈ f (x0) + f (x0)(x − x0)
Equation of the
tangent line to the
graph of y = f (x) atthe point (x0, f (x0)):
y − f (x0) = f (x0)(x − x0)
∴ y = f (x0) + f (x0)(x−x0)
x0 x
f (x)
f (x) + f
(x0)(x−
x0)
y = f (x)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 11 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
L(x) = f (x0) + f (x0)(x − x0) is the local linear approximation of
f (x) at x0.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
L(x) = f (x0) + f (x0)(x − x0) is the local linear approximation of
f (x) at x0.
(The tangent line to the graph of f at x0 approximates the graph of f when
x is near x0.)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
L(x) = f (x0) + f (x0)(x − x0) is the local linear approximation of
f (x) at x0.
(The tangent line to the graph of f at x0 approximates the graph of f when
x is near x0.)
It can be shown that the local linear approximation is the “best” linear
approximation of f near x0.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
L(x) = f (x0) + f (x0)(x − x0) is the local linear approximation of
f (x) at x0.
(The tangent line to the graph of f at x0 approximates the graph of f when
x is near x0.)
It can be shown that the local linear approximation is the “best” linear
approximation of f near x0.
If dx = ∆x = x − x0,
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
L(x) = f (x0) + f (x0)(x − x0) is the local linear approximation of
f (x) at x0.
(The tangent line to the graph of f at x0 approximates the graph of f when
x is near x0.)
It can be shown that the local linear approximation is the “best” linear
approximation of f near x0.
If dx = ∆x = x − x0, then x = x0 + dx.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
L(x) = f (x0) + f (x0)(x − x0) is the local linear approximation of
f (x) at x0.
(The tangent line to the graph of f at x0 approximates the graph of f when
x is near x0.)
It can be shown that the local linear approximation is the “best” linear
approximation of f near x0.
If dx = ∆x = x − x0, then x = x0 + dx.
Since f (x) ≈ f (x0) + f
(x0)(x − x0), we have
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
L(x) = f (x0) + f (x0)(x − x0) is the local linear approximation of
f (x) at x0.
(The tangent line to the graph of f at x0 approximates the graph of f when
x is near x0.)
It can be shown that the local linear approximation is the “best” linear
approximation of f near x0.
If dx = ∆x = x − x0, then x = x0 + dx.
Since f (x) ≈ f (x0) + f
(x0)(x − x0), we have
f (x0 + dx) ≈ f (x0) + f (x0)dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 12 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
dx = ∆x ≈ 0
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
dx = ∆x ≈ 0=⇒
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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Remarks
dx = ∆x ≈ 0=⇒ dy = f (x)dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
dx = ∆x ≈ 0=⇒ dy = f (x)dx = f (x)∆x
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
dx = ∆x ≈ 0=⇒ dy = f (x)dx = f (x)∆x ≈ ∆y
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
dx = ∆x ≈ 0=⇒ dy = f (x)dx = f (x)∆x ≈ ∆y
∴ dy ≈ ∆y
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
dx = ∆x ≈ 0=⇒ dy = f (x)dx = f (x)∆x ≈ ∆y
∴ dy ≈ ∆y
dy is easier to compute than ∆y
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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Remarks
dx = ∆x ≈ 0=⇒ dy = f (x)dx = f (x)∆x ≈ ∆y
∴ dy ≈ ∆y
dy is easier to compute than ∆y
∴ dy is used to approximate ∆y when dx
≈0
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Remarks
dx = ∆x ≈ 0=⇒ dy = f (x)dx = f (x)∆x ≈ ∆y
∴ dy ≈ ∆y
dy is easier to compute than ∆y
∴ dy is used to approximate ∆y when dx
≈0
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 13 / 21
Local Linear Approximation and Differentials
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7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
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Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
![Page 76: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/76.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
![Page 77: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/77.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
![Page 78: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/78.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
![Page 79: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/79.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
L(x)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
![Page 80: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/80.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 80/158
Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
L(x) = f (8)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
l
![Page 81: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/81.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 81/158
Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
L(x) = f (8) + f (8)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
E l
![Page 82: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/82.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 82/158
Example
Find the local linear approximation of f (x) = 3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
L(x) = f (8) + f (8)(x
−8)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
E l
![Page 83: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/83.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 83/158
Example
Find the local linear approximation of f (x) =3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
L(x) = f (8) + f (8)(x
−8)
= 2
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
E l
![Page 84: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/84.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Example
Find the local linear approximation of f (x) =3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
L(x) = f (8) + f (8)(x
−8)
= 2 + 112
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
E l
![Page 85: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/85.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 85/158
Example
Find the local linear approximation of f (x) =3√ x at x0 = 8.
Solution.
We have f (x) =1
33√ x2
.
L(x) = f (x0) + f (x0)(x − x0)
∴ At x0 = 8:
L(x) = f (8) + f (8)(x
−8)
= 2 + 112
(x − 8).
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 14 / 21
Local Linear Approximation and Differentials
Example
![Page 86: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/86.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate 3√
27.027.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
![Page 87: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/87.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate 3√
27.027.
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example√
![Page 88: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/88.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate 3√
27.027.
Solution.
Let f (x) = 3√ x.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example3√
![Page 89: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/89.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate 3√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example3√
![Page 90: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/90.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 90/158
Approximate 3√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
A i 3√
7 7
![Page 91: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/91.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate 3√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Thus,
3√
27.027
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
A i 3√
27 027
![Page 92: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/92.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate 3√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Thus,
3√
27.027 = f (27 + 0.027)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
A i t 3√
27 027
![Page 93: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/93.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate 3√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Thus,
3√
27.027 = f (27 + 0.027)
≈f (27) + f (27)
·(0.027)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
A i t 3√
27 027
![Page 94: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/94.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 94/158
Approximate 3√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Thus,
3√
27.027 = f (27 + 0.027)
≈f (27) + f (27)
·(0.027)
= 3
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
Approximate 3√
27 027
![Page 95: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/95.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 95/158
Approximate√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Thus,
3√
27.027 = f (27 + 0.027)
≈f (27) + f (27)
·(0.027)
= 3 + 13 · 9
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
Approximate 3√
27 027
![Page 96: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/96.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Thus,
3√
27.027 = f (27 + 0.027)
≈f (27) + f (27)
·(0.027)
= 3 + 13 · 9
· (0.027)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
Approximate 3√
27 027
![Page 97: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/97.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
27.027.
Solution.
Let f (x) = 3√ x. Then f (x) =
1
33√ x2
.
f (x0 + dx)
≈f (x0) + f (x0)dx
Thus,
3√
27.027 = f (27 + 0.027)
≈f (27) + f (27)
·(0.027)
= 3 + 13 · 9
· (0.027)
= 3.01.
Instit te of Mathematics (UP Diliman) Local Linea A o and Diffe entials Mathematics 53 15 / 21
Local Linear Approximation and Differentials
Example
A i√
1
![Page 98: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/98.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
I tit t f M th ti (UP Dili ) L l Li A d Diff ti l M th ti 53 16 / 21
Local Linear Approximation and Differentials
Example
A i√
15 96
![Page 99: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/99.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
I tit t f M th ti (UP Dili ) L l Li A d Diff ti l M th ti 53 16 / 21
Local Linear Approximation and Differentials
Example
A i t√
15 96
![Page 100: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/100.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
Let f (x) =√ x.
I tit t f M th ti (UP Dili ) L l Li A d Diff ti l M th ti 53 16 / 21
Local Linear Approximation and Differentials
Example
A i t√
15 96
![Page 101: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/101.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
I tit t f M th ti (UP Dili ) L l Li A d Diff ti l M th ti 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15 96
![Page 102: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/102.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
http://slidepdf.com/reader/full/m53-lec232-local-linear-approx-and-differentialspdf 102/158
Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15 96
![Page 103: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/103.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96 = f (16 − 0.04)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15 96
![Page 104: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/104.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96 = f (16 − 0.04)
≈ f (16) + f (16) · (−0.04)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15 96
![Page 105: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/105.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96 = f (16 − 0.04)
≈ f (16) + f (16) · (−0.04)
= 4
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15 96
![Page 106: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/106.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96 = f (16 − 0.04)
≈ f (16) + f (16) · (−0.04)
= 4 +1
2
·4
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15.96
![Page 107: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/107.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96 = f (16 − 0.04)
≈ f (16) + f (16) · (−0.04)
= 4 +1
2
·4
· (−0.04)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15.96.
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Approximate√
15.96.
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96 = f (16 − 0.04)
≈ f (16) + f (16) · (−0.04)
= 4 +1
2
·4
· (−0.04)
= 4 − 0.005
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example
Approximate√
15.96.
![Page 109: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/109.jpg)
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pp√
5 96
Solution.
Let f (x) =√ x. Then f (x) =
1
2√ x
.
Thus,
√ 15.96 = f (16 − 0.04)
≈ f (16) + f (16) · (−0.04)
= 4 +1
2
·4
· (−0.04)
= 4 − 0.005= 3.995.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 16 / 21
Local Linear Approximation and Differentials
Example1
![Page 110: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/110.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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A ball 5 in in diameter is to be covered by a rubber material which is
1
16 inthick. Use differentials to estimate the volume of the rubber material that
will be used.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example1
![Page 111: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/111.jpg)
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A ball 5 in in diameter is to be covered by a rubber material which is
1
16 inthick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example1
![Page 112: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/112.jpg)
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A ball 5 in in diameter is to be covered by a rubber material which is 16 inthick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r:
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example1
![Page 113: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/113.jpg)
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A ball 5 in in diameter is to be covered by a rubber material which is 16 inthick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =
4
3πr
3
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example1
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A ball 5 in in diameter is to be covered by a rubber material which is 16 inthick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r
) =
4
3πr3
=⇒dV
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 115: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/115.jpg)
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A ball 5 in in diameter is to be covered by a rubber material which is16
in
thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r
) =
4
3πr3
=⇒dV
= 4πr2 dr
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 116: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/116.jpg)
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A ball 5 in in diameter is to be covered by a rubber material which is16
in
thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r
) =
4
3πr3
=⇒dV
= 4πr2 dr
Volume of rubber material
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 117: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/117.jpg)
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A ball 5 in in diameter is to be covered by a rubber material which is16
in
thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =4
3πr3 =
⇒dV = 4πr2 dr
Volume of rubber material = V ( 52 + 1
16 ) − V ( 52 )
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 118: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/118.jpg)
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y16thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =4
3πr3 =
⇒dV = 4πr2 dr
Volume of rubber material = V ( 52 + 1
16 ) − V ( 52 )
= ∆V
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 119: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/119.jpg)
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y16thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =4
3πr3 =
⇒dV = 4πr2 dr
Volume of rubber material = V ( 52 + 1
16 ) − V ( 52 )
= ∆V dr = 116
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 120: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/120.jpg)
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y16thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =4
3πr3 =
⇒dV = 4πr2 dr
Volume of rubber material = V ( 52 + 1
16 ) − V ( 52 )
= ∆V dr = 116
≈ dV
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 121: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/121.jpg)
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16thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =4
3πr3 =
⇒dV = 4πr2 dr
Volume of rubber material = V ( 52 + 1
16 ) − V ( 52 )
= ∆V dr = 116
≈ dV
= V ( 5
2)·
( 1
16)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 122: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/122.jpg)
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16thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =4
3πr3 =
⇒dV = 4πr2 dr
Volume of rubber material = V ( 52 + 1
16 ) − V ( 52 )
= ∆V dr = 116
≈ dV
= V ( 5
2)·
( 1
16)
= 4π( 52 )2 · ( 1
16 )
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A ball 5 in in diameter is to be covered by a rubber material which is 1 in
![Page 123: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/123.jpg)
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16thick. Use differentials to estimate the volume of the rubber material that
will be used.
Solution.
Volume of the ball with radius r: V (r) =4
3πr3 =
⇒dV = 4πr2 dr
Volume of rubber material = V ( 52 + 1
16 ) − V ( 52 )
= ∆V dr = 116
≈ dV
= V ( 5
2)·
( 1
16)
= 4π( 52 )2 · ( 1
16 )= 25π
16 in3
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 17 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
![Page 124: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/124.jpg)
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volume of the insulation.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
![Page 125: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/125.jpg)
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volume of the insulation.
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
![Page 126: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/126.jpg)
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volume of the insulation.
Solution.
Volume of the rod:
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
f
![Page 127: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/127.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
l f h i l i
![Page 128: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/128.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
l f h i l i
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
l f th i l ti
![Page 130: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/130.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
l f th i l ti
![Page 131: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/131.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
volume of the insulation
![Page 132: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/132.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation = V (4 + 0.001) − V (4)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
volume of the insulation
![Page 133: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/133.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation = V (4 + 0.001) − V (4)= ∆V
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
volume of the insulation
![Page 134: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/134.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation = V (4 + 0.001) − V (4)= ∆V dr = 0.001
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
volume of the insulation
![Page 135: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/135.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation = V (4 + 0.001) − V (4)= ∆V dr = 0.001
≈ dV
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
volume of the insulation
![Page 136: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/136.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation = V (4 + 0.001) − V (4)= ∆V dr = 0.001
≈ dV
= V (4) · (0.001)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
volume of the insulation
![Page 137: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/137.jpg)
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation = V (4 + 0.001) − V (4)= ∆V dr = 0.001
≈ dV
= V (4) · (0.001)
= 30π (4) · (0.001)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
A metal rod 15 cm long and 8 cm in diameter is to be insulated, except for
the ends, with a material 0.001 cm thick. Use differentials to estimate the
volume of the insulation.
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volume of the insulation.
Solution.
Volume of the rod: V = πr2h = 15πr2 =⇒ dV = 30πr dr
Volume of the insulation = V (4 + 0.001) − V (4)= ∆V dr = 0.001
≈ dV
= V (4) · (0.001)
= 30π (4) · (0.001)
= 0.12π cm3.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 18 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64
of an inch. Estimate the error in
the computed area of the square.
![Page 139: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/139.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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p q
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 140: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/140.jpg)
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p q
Solution.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 141: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/141.jpg)
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p q
Solution.
Area of square with side x:
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 142: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/142.jpg)
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p q
Solution.
Area of square with side x: A(x) = x2
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 143: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/143.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and DifferentialsExample
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 144: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/144.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dx
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and DifferentialsExample
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 145: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/145.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 146: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/146.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 147: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/147.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 148: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/148.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 149: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/149.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A|
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 150: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/150.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A| ≈ |dA|
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 151: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/151.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A| ≈ |dA|= 2x |dx|
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 152: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/152.jpg)
7/30/2019 M53 Lec2.3.2 Local Linear Approx and Differentials.pdf
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A| ≈ |dA|= 2x |dx| x = 8
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 153: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/153.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A| ≈ |dA|= 2x |dx| x = 8= 2(8)
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 154: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/154.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A| ≈ |dA|= 2x |dx| x = 8= 2(8)( 1
64 )
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 155: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/155.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A| ≈ |dA|= 2x |dx| x = 8= 2(8)( 1
64 )
= 14
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Local Linear Approximation and Differentials
Example
Suppose that the side of a square is measured with a ruler to be 8 inches
with a measurement error of at most
±1
64 of an inch. Estimate the error in
the computed area of the square.
![Page 156: M53 Lec2.3.2 Local Linear Approx and Differentials.pdf](https://reader031.vdocuments.site/reader031/viewer/2022021118/577cde201a28ab9e78ae716a/html5/thumbnails/156.jpg)
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Solution.
Area of square with side x: A(x) = x2 =⇒ dA = 2x dxMeasurement error of at most
±1
64 =
⇒ |dx
|= 1
64
Error in the computed area = ∆A
|∆A| ≈ |dA|= 2x |dx| x = 8= 2(8)( 1
64 )
= 14
∴ The propagated error in the computed area is at most ±14 of a square
inch.Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 19 / 21
Exercise
1 Find dydx if
i ( ) 2 2 3
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sin(xy) = xy2 − 2x3.
2 Determine D4x [ cos(4x) ].
3 Approximate 3√
8.03.
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 20 / 21
* * * The End * * *
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Next Meeting:
Rates of Change
Rectilinear Motion
Institute of Mathematics (UP Diliman) Local Linear Approx and Differentials Mathematics 53 21 / 21