Analysis III: Auditorium exercise class - Taylor Theorem ...
Transcript of Analysis III: Auditorium exercise class - Taylor Theorem ...
![Page 1: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/1.jpg)
Analysis III: Auditorium exercise classTaylor Theorem, Lagrange-remainder,Extrema of Multivariable FunctionsImplicit Function Theorem
Sofiya OnyshkevychNovember 22, 2021
![Page 2: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/2.jpg)
BITTE BEACHTEN SIE DIE 3G-REGEL!PLEASE OBEY THE 3G RULE!
−VOLLSTÄNDIG GEIMPFTE−GENESENE−GETESTETE
−FULLY VACCINATED−RECOVERED−TESTED
Zutritt zur Lehrveranstaltung haben nur:
Admission to the course is restrictedto persons who are:
(negative test result is valid for max. 24 hours)(negatives Testergebnis ist max. 24 Std. gültig)
Sollten Sie dies nicht nachweisen können, müssen Sie bitte den Raum jetzt verlassen.Andernfalls droht ein Hausverbot!
Vielen Dank für Ihr Verständnis.Schützen Sie sich und andere!
If you cannot prove this,please leave the room now. Otherwise you could be banned fromthe room!
Thank you for your understanding.Protect yourself and others!
1/21
![Page 3: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/3.jpg)
Taylor Theorem
Let D ⊂ Rn be open and convex. Let f : D→ R be a Cm+1–functionand x0 ∈ D. Then the Taylor–expansion in x ∈ D is well-defined
f(x) = Tm(x; x0) + Rm(x; x0),
where
Rm(x; x0) =∑
|α|=m+1
Dαf(x0 + θ(x−x0))α!
(x− x0)α , ∀θ ∈ (0, 1)
is a Lagrange–remainder.
2/21
![Page 4: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/4.jpg)
Error of a Taylor polynomial approximation
3/21
![Page 5: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/5.jpg)
Examples
R2((x, y); (x0, y0)) =
R3((x, y); (x0, y0)) =
4/21
![Page 6: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/6.jpg)
Exercise 1
Compute Taylor polynomial T2(x; x0) of the functionf(x, y, z) = xez − y2 centered around a point x0 = (1,−1, 0)T.
5/21
![Page 7: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/7.jpg)
Exercise 2
Compute the remainder in a Lagrange form R2(x; x0) off(x, y, z) = xez − y2 centered around a point x0 = (1,−1, 0)T.
6/21
![Page 8: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/8.jpg)
The estimate on the remainder
The Taylor approximation of a function reads as
f(x) = Tm(x; x0) + O(∥ x− x0 ∥m+1)
If Dαf, |α| = m+ 1 are bounded by C > 0 in a neighborhood of x0then the estimate holds
|Rm(x0; x)| ≤nm+1
(m+ 1)!C ∥ x− x0 ∥m+1
∞
7/21
![Page 9: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/9.jpg)
Exercise 3
Compute T2(x; x0) of a function f(x, y) = ex cos(y) at the pointx0 = (0, 0)T and the estimate for the associated remainder R2(x; x0)for (x, y) ∈ [−2, 2]× [−2, 2].
8/21
![Page 10: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/10.jpg)
Exercise 4
Compute T2(x; x0) of a function f(x, y) = cos(x2 + y2) at the pointx0 = (0, 0)T and the approximation error for (x, y) ∈ [0, π
4 ]× [0, π4 ].
9/21
![Page 11: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/11.jpg)
Extrema of multivariable function
Let D ⊂ Rn, f : D→ R and x0 ∈ D. Then at x0 the function f has
• a (strict) global maximum if ∀x ∈ D : f(x)(<)
≤ f(x0)• a (strict) local maximum if
∃ϵ > 0 ∀x ∈ D with ∥ x− x0 ∥< ϵ : f(x)(<)
≤ f(x0)
• analogously for minima
Note: x0 is called an extremum if it is maximum or minimum
10/21
![Page 12: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/12.jpg)
Stationary points
• The points x0 ∈ D for which it holds
grad f(x0) = 0
are called stationary points (critical points) of f.• Stationary points are not necessarily extrema.
11/21
![Page 13: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/13.jpg)
Necessary optimality conditions
• Let f ∈ D is C1, x0 ∈ D - local extremum =⇒ grad f(x0) = 0
• Let f ∈ D is C2, x0 ∈ D - stationary point• if x0 is local min (max)
=⇒ H f(x0) positive (negative) semi-definite.
12/21
![Page 14: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/14.jpg)
Sufficient optimality conditions
• Let f ∈ D is C2, x0 ∈ D - stationary point• H f(x0) positive (negative) definite
=⇒ x0 is strict local min (max)• H f(x0) indefinite =⇒ x0 is a saddle point
13/21
![Page 15: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/15.jpg)
Examples
Figure 1: f(x) = x3, f′(0) = 0 butx∗ = 0 isn’t extremum
Figure 2:f(x) = x4, f′′(x) = 12x2, f′′(0) = 0 butstill x∗ = 0 is minimum
14/21
![Page 16: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/16.jpg)
Exercise 5
Compute the stationary points of the following functions anddetermine whether it is min/max/saddle point
1. f(x, y) = xy+ x− 2y− 2
2. f(x, y) = 2x3 − 3xy+ 2y3 − 3
3. f(x, y) = (x2 + 2y2)e−x2−y2
4. f(x, y) = x5 − 3x3 + y2 + 15,
15/21
![Page 17: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/17.jpg)
Implicitly Defined Functionss
Consider a system of nonlinear equations
g(x) = 0,
with g : D ⊂ Rn → Rm,m < n, i.e more unknowns than equations. -underdetermined system of equations.
We want to solve such systems locally expressing some variables viaother.
16/21
![Page 18: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/18.jpg)
Implicit Function Theorem
Let g : D ⊂ Rn → Rm be a C1 - function. Let (x, y) ∈ D, wherex ∈ Rn−m, y ∈ Rm. Let (x0, y0) ∈ D - solution to g(x0, y0) = 0. If theJacobian matrix
∂g∂y (x0, y0) :=
∂g1∂y1 ... ∂g1
∂ym... ...∂gm∂y1 ... ∂gm
∂ym
is regular, then there exist neighbourhoods U of x0, V of y0,U× V ⊂ Dand a uniquely determined continuous differentiable functionf : U→ V : f(x0) = y0 and g(x, f(x)) = 0 for all x ∈ U and
J f(x) = −(∂g∂y (x, f(x))
)−1(∂g∂x (x, f(x))
)
17/21
![Page 19: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/19.jpg)
Exercise 7
Can the equation (x2 + y2 + 2z2) 12 = cos(z) be solved uniquely for y
in terms of x, z near (0, 1, 0)? For z in terms of x and y?
F(x, y, z) =
F(0, 1, 0) =
∂F∂y (0, 1, 0) =
∂F∂z (0, 1, 0) =
18/21
![Page 20: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/20.jpg)
Exercise 8
Consider the function F(x, y, z,u, v) : R5 → R2 given by
F(x, y, z,u, v) =(
xy2 + xzu+ yv2 − 3
u3yz+ 2xv− u2v2 − 2
)
Can we solve for u, v as functions of x, y, z near (1, 1, 1, 1, 1)?
Notice that F(1,1,1,1,1) = 0.
(∂F1∂u
∂F1∂v
∂F2∂u
∂F2∂v
)=
(∂F1∂u
∂F1∂v
∂F2∂u
∂F2∂v
)(1, 1, 1, 1, 1) =
19/21
![Page 21: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/21.jpg)
Some more exercises
Compute T2(x; x0) of a function
f(x, y) = cos(x) sin(y)ex−y
at the point x0 = (0, 0)T and the associated remainder R2(x; x0)
20/21
![Page 22: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/22.jpg)
Some more exercises
Compute Taylor polynomial of second degree T2(x; x0) of a function
f(x, y) = sin(x+ y) + yex−y
at the point x0 = (0, 0)T and the estimate for the Lagrange remainderfor |x| ≤ 0.1, |y| ≤ 0.1.
21/21
![Page 23: Analysis III: Auditorium exercise class - Taylor Theorem ...](https://reader034.fdocuments.in/reader034/viewer/2022042214/62599f66c5f75a34af689110/html5/thumbnails/23.jpg)
Thank you!
21/21