A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter I, Problem 3E
a.
To determine
To determine whether the set is finite or not.
b.
To determine
To determine whether the set is finite or not.
c.
To determine
To determine whether the set is finite or not.
d.
To determine
To determine whether the set is finite or not.
e.
To determine
To determine whether the set is finite or not.
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(b) Let I[y] be a functional of y(x) defined by
[[y] = √(x²y' + 2xyy' + 2xy + y²) dr,
subject to boundary conditions
y(0) = 0,
y(1) = 1.
State the Euler-Lagrange equation for finding extreme values of I [y] for this prob-
lem. Explain why the function y(x) = x is an extremal, and for this function,
show that I = 2. Without doing further calculations, give the values of I for the
functions y(x) = x² and y(x) = x³.
Please use mathematical induction to prove this
L
sin 2x (1+ cos 3x) dx
59
Chapter I Solutions
A Transition to Advanced Mathematics
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