Exercise I.1. Consider F := B := {v € R" : ||v|| ≤ 1} and fix xo € R". (a) Suppose to, t₁> 0 and vo, V₁ € F. Consider the trajectory x(): [0, to +₁] → R" given by x(t) = xo + tvo xo+tovo + (t-to)v₁ if 0 ≤ t ≤ to if to ≤t≤to +ti Show there exists w€ F so that x(to + ₁) = xo + (to +t₁)w. Illustrate this graphically in dimension 2. (b) Generalize part (a) from just two velocities to any finite number m € N. (c) Show that RT) (xo) = B(xo; T) := {y € R" : ||y-xo|| ≤T}.
Exercise I.1. Consider F := B := {v € R" : ||v|| ≤ 1} and fix xo € R". (a) Suppose to, t₁> 0 and vo, V₁ € F. Consider the trajectory x(): [0, to +₁] → R" given by x(t) = xo + tvo xo+tovo + (t-to)v₁ if 0 ≤ t ≤ to if to ≤t≤to +ti Show there exists w€ F so that x(to + ₁) = xo + (to +t₁)w. Illustrate this graphically in dimension 2. (b) Generalize part (a) from just two velocities to any finite number m € N. (c) Show that RT) (xo) = B(xo; T) := {y € R" : ||y-xo|| ≤T}.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Exercise I.1. Consider F:= B = {veR": ||v|| ≤ 1} and fix xo € Rn.
(a) Suppose to, t₁ > 0 and Vo, V₁ € F. Consider the trajectory x(): [0, to +t₁] →
R" given by
x(t) =
xo + tvo
xo+tovo + (tto)v₁
if 0 ≤ t ≤ to
if to ≤ t ≤to + ti
Show there exists w€ F so that x(to + t₁) = xo + (to +t₁)w. Illustrate this
graphically in dimension 2.
(b) Generalize part (a) from just two velocities to any finite number mE N.
(c) Show that R(¹)(xo) = B(xo; T) := {y € R" : y - xo|| ≤T}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe87a569b-7721-4104-9b29-fedcf0a2817b%2F96054225-4665-4d80-8b2a-9e027912bf20%2Ffnn35m_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise I.1. Consider F:= B = {veR": ||v|| ≤ 1} and fix xo € Rn.
(a) Suppose to, t₁ > 0 and Vo, V₁ € F. Consider the trajectory x(): [0, to +t₁] →
R" given by
x(t) =
xo + tvo
xo+tovo + (tto)v₁
if 0 ≤ t ≤ to
if to ≤ t ≤to + ti
Show there exists w€ F so that x(to + t₁) = xo + (to +t₁)w. Illustrate this
graphically in dimension 2.
(b) Generalize part (a) from just two velocities to any finite number mE N.
(c) Show that R(¹)(xo) = B(xo; T) := {y € R" : y - xo|| ≤T}.
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