Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
3rd Edition
ISBN: 9780840064189
Author: Frederick R. Adler
Publisher: Cengage Learning
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Chapter 1.5, Problem 20E
To determine
The solutions of the given discrete-time dynamical systems at t = 20
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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
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Chapter 1 Solutions
Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
Ch. 1.2 - Identify the variables and parameters in the...Ch. 1.2 - Prob. 2ECh. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Compute the values of the following functions at...Ch. 1.2 - Graph the given points and say which point does...Ch. 1.2 - Graph the given points and say which point does...Ch. 1.2 - Graph the given points and say which point does...Ch. 1.2 - Graph the given points and say which point does...
Ch. 1.2 - Evaluate the following functions at the given...Ch. 1.2 - Evaluate the following functions at the given...Ch. 1.2 - Prob. 13ECh. 1.2 - Prob. 14ECh. 1.2 - Prob. 15ECh. 1.2 - Prob. 16ECh. 1.2 - Prob. 17ECh. 1.2 - Prob. 18ECh. 1.2 - Prob. 19ECh. 1.2 - Prob. 20ECh. 1.2 - Prob. 21ECh. 1.2 - Prob. 22ECh. 1.2 - Prob. 23ECh. 1.2 - Prob. 24ECh. 1.2 - Prob. 25ECh. 1.2 - Find the inverses of each of the following...Ch. 1.2 - Prob. 27ECh. 1.2 - Prob. 28ECh. 1.2 - Prob. 29ECh. 1.2 - Graph each of the following functions and its...Ch. 1.2 - Graph each of the following functions and its...Ch. 1.2 - Graph each of the following functions and its...Ch. 1.2 - Prob. 33ECh. 1.2 - Find the compositions of the given functions....Ch. 1.2 - Prob. 35ECh. 1.2 - Prob. 36ECh. 1.2 - Prob. 37ECh. 1.2 - Prob. 38ECh. 1.2 - Prob. 39ECh. 1.2 - Prob. 40ECh. 1.2 - Prob. 41ECh. 1.2 - Prob. 42ECh. 1.2 - Prob. 43ECh. 1.2 - Prob. 44ECh. 1.2 - Prob. 45ECh. 1.2 - Prob. 46ECh. 1.2 - Prob. 47ECh. 1.2 - Prob. 48ECh. 1.2 - Prob. 49ECh. 1.2 - Prob. 50ECh. 1.2 - Prob. 51ECh. 1.2 - Prob. 52ECh. 1.2 - The following series of functional compositions...Ch. 1.2 - The following series of functional compositions...Ch. 1.2 - The following series of functional compositions...Ch. 1.2 - Prob. 56ECh. 1.2 - Prob. 57ECh. 1.2 - Prob. 58ECh. 1.2 - Prob. 59ECh. 1.2 - Prob. 60ECh. 1.2 - Prob. 61ECh. 1.2 - Prob. 62ECh. 1.2 - Prob. 63ECh. 1.2 - Prob. 64ECh. 1.2 - Prob. 65ECh. 1.2 - Prob. 66ECh. 1.2 - Prob. 67ECh. 1.2 - Prob. 68ECh. 1.2 - Prob. 69ECh. 1.2 - Prob. 70ECh. 1.3 - Prob. 1ECh. 1.3 - Prob. 2ECh. 1.3 - Prob. 3ECh. 1.3 - Prob. 4ECh. 1.3 - Prob. 5ECh. 1.3 - Prob. 6ECh. 1.3 - Prob. 7ECh. 1.3 - Prob. 8ECh. 1.3 - Prob. 9ECh. 1.3 - Prob. 10ECh. 1.3 - Prob. 11ECh. 1.3 - Prob. 12ECh. 1.3 - Prob. 13ECh. 1.3 - Prob. 14ECh. 1.3 - Prob. 15ECh. 1.3 - Prob. 16ECh. 1.3 - Prob. 17ECh. 1.3 - Prob. 18ECh. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.3 - Prob. 21ECh. 1.3 - Prob. 22ECh. 1.3 - Prob. 23ECh. 1.3 - Prob. 24ECh. 1.3 - Prob. 25ECh. 1.3 - Prob. 26ECh. 1.3 - Prob. 27ECh. 1.3 - Prob. 28ECh. 1.3 - Prob. 29ECh. 1.3 - Prob. 30ECh. 1.3 - Prob. 31ECh. 1.3 - Prob. 32ECh. 1.3 - Find the mass in kilograms of the following...Ch. 1.3 - Prob. 34ECh. 1.3 - Change the units in the following functions, and...Ch. 1.3 - Prob. 36ECh. 1.3 - Prob. 37ECh. 1.3 - Prob. 38ECh. 1.3 - Prob. 39ECh. 1.3 - Prob. 40ECh. 1.3 - Prob. 41ECh. 1.3 - Prob. 42ECh. 1.3 - Prob. 43ECh. 1.3 - Prob. 44ECh. 1.3 - Prob. 45ECh. 1.3 - Prob. 46ECh. 1.3 - Prob. 47ECh. 1.3 - Prob. 48ECh. 1.3 - Prob. 49ECh. 1.3 - Prob. 50ECh. 1.4 - For the following lines, find the slopes between...Ch. 1.4 - Prob. 2ECh. 1.4 - Prob. 3ECh. 1.4 - Prob. 4ECh. 1.4 - Prob. 5ECh. 1.4 - Prob. 6ECh. 1.4 - Prob. 7ECh. 1.4 - Prob. 8ECh. 1.4 - Prob. 9ECh. 1.4 - Prob. 10ECh. 1.4 - Prob. 11ECh. 1.4 - Prob. 12ECh. 1.4 - Prob. 13ECh. 1.4 - Prob. 14ECh. 1.4 - Prob. 15ECh. 1.4 - Prob. 16ECh. 1.4 - Prob. 17ECh. 1.4 - Prob. 18ECh. 1.4 - Prob. 19ECh. 1.4 - Prob. 20ECh. 1.4 - Prob. 21ECh. 1.4 - Prob. 22ECh. 1.4 - Prob. 23ECh. 1.4 - Prob. 24ECh. 1.4 - Prob. 25ECh. 1.4 - Prob. 26ECh. 1.4 - Prob. 27ECh. 1.4 - Prob. 28ECh. 1.4 - Prob. 29ECh. 1.4 - Prob. 30ECh. 1.4 - Prob. 31ECh. 1.4 - Prob. 32ECh. 1.4 - Prob. 33ECh. 1.4 - Prob. 34ECh. 1.4 - Prob. 35ECh. 1.4 - Prob. 36ECh. 1.4 - Prob. 37ECh. 1.4 - Prob. 38ECh. 1.4 - Prob. 39ECh. 1.4 - Prob. 40ECh. 1.4 - Prob. 41ECh. 1.4 - Prob. 42ECh. 1.4 - The following data give the elevation of the...Ch. 1.4 - Prob. 44ECh. 1.4 - Prob. 45ECh. 1.4 - Prob. 46ECh. 1.4 - Prob. 47ECh. 1.4 - Prob. 48ECh. 1.4 - Prob. 49ECh. 1.4 - Prob. 50ECh. 1.4 - Prob. 51ECh. 1.4 - Prob. 52ECh. 1.4 - Prob. 53ECh. 1.4 - Prob. 54ECh. 1.4 - Prob. 55ECh. 1.4 - Prob. 56ECh. 1.4 - Prob. 57ECh. 1.4 - Prob. 58ECh. 1.4 - Prob. 59ECh. 1.4 - Prob. 60ECh. 1.5 - Write the updating function associated with each...Ch. 1.5 - Write the updating function associated with each...Ch. 1.5 - Write the updating function associated with each...Ch. 1.5 - Write the updating function associated with each...Ch. 1.5 - Compose the updating function associated with each...Ch. 1.5 - Compose the updating function associated with each...Ch. 1.5 - Prob. 7ECh. 1.5 - Prob. 8ECh. 1.5 - Prob. 9ECh. 1.5 - Prob. 10ECh. 1.5 - Prob. 11ECh. 1.5 - Prob. 12ECh. 1.5 - Prob. 13ECh. 1.5 - Prob. 14ECh. 1.5 - Prob. 15ECh. 1.5 - Prob. 16ECh. 1.5 - Prob. 17ECh. 1.5 - Prob. 18ECh. 1.5 - Prob. 19ECh. 1.5 - Prob. 20ECh. 1.5 - Prob. 21ECh. 1.5 - Prob. 22ECh. 1.5 - Prob. 23ECh. 1.5 - Prob. 24ECh. 1.5 - Prob. 25ECh. 1.5 - Prob. 26ECh. 1.5 - Prob. 27ECh. 1.5 - Prob. 28ECh. 1.5 - Prob. 29ECh. 1.5 - Prob. 30ECh. 1.5 - Use the formula for the solution to find the...Ch. 1.5 - Prob. 32ECh. 1.5 - Prob. 33ECh. 1.5 - Prob. 34ECh. 1.5 - Prob. 35ECh. 1.5 - Prob. 36ECh. 1.5 - Prob. 37ECh. 1.5 - Prob. 38ECh. 1.5 - Prob. 39ECh. 1.5 - Prob. 40ECh. 1.5 - Prob. 41ECh. 1.5 - Prob. 42ECh. 1.5 - Prob. 43ECh. 1.5 - Prob. 44ECh. 1.5 - Prob. 45ECh. 1.5 - Prob. 46ECh. 1.5 - Prob. 47ECh. 1.5 - Prob. 48ECh. 1.5 - Prob. 49ECh. 1.5 - Prob. 50ECh. 1.5 - Prob. 51ECh. 1.5 - Prob. 52ECh. 1.5 - Prob. 53ECh. 1.5 - Prob. 54ECh. 1.5 - Prob. 55ECh. 1.5 - Prob. 56ECh. 1.5 - Prob. 57ECh. 1.5 - Prob. 58ECh. 1.5 - Prob. 59ECh. 1.5 - Prob. 60ECh. 1.5 - Prob. 61ECh. 1.5 - Prob. 62ECh. 1.6 - The following steps are used to build a cobweb...Ch. 1.6 - Prob. 2ECh. 1.6 - Prob. 3ECh. 1.6 - Prob. 4ECh. 1.6 - Prob. 5ECh. 1.6 - Prob. 6ECh. 1.6 - Prob. 7ECh. 1.6 - Prob. 8ECh. 1.6 - Prob. 9ECh. 1.6 - Prob. 10ECh. 1.6 - Prob. 11ECh. 1.6 - Prob. 12ECh. 1.6 - Prob. 13ECh. 1.6 - Find the equilibria of the following discrete-time...Ch. 1.6 - Prob. 15ECh. 1.6 - Prob. 16ECh. 1.6 - Prob. 17ECh. 1.6 - Prob. 18ECh. 1.6 - Prob. 19ECh. 1.6 - Prob. 20ECh. 1.6 - Prob. 21ECh. 1.6 - Prob. 22ECh. 1.6 - Prob. 23ECh. 1.6 - Prob. 24ECh. 1.6 - Prob. 25ECh. 1.6 - Prob. 26ECh. 1.6 - Prob. 27ECh. 1.6 - Prob. 28ECh. 1.6 - Prob. 29ECh. 1.6 - Prob. 30ECh. 1.6 - Prob. 31ECh. 1.6 - Prob. 32ECh. 1.6 - Prob. 33ECh. 1.6 - Prob. 34ECh. 1.6 - Prob. 35ECh. 1.6 - Prob. 36ECh. 1.6 - Prob. 37ECh. 1.6 - Prob. 38ECh. 1.6 - Prob. 39ECh. 1.6 - Prob. 40ECh. 1.6 - Prob. 41ECh. 1.6 - Prob. 42ECh. 1.6 - Cobweb and find the equilibrium of the following...Ch. 1.6 - Prob. 44ECh. 1.6 - Prob. 45ECh. 1.6 - Prob. 46ECh. 1.6 - Prob. 47ECh. 1.6 - Prob. 48ECh. 1.6 - Prob. 49ECh. 1.6 - Prob. 50ECh. 1.7 - Prob. 1ECh. 1.7 - Prob. 2ECh. 1.7 - Prob. 3ECh. 1.7 - Prob. 4ECh. 1.7 - Prob. 5ECh. 1.7 - Prob. 6ECh. 1.7 - Prob. 7ECh. 1.7 - Prob. 8ECh. 1.7 - Prob. 9ECh. 1.7 - Prob. 10ECh. 1.7 - Prob. 11ECh. 1.7 - Prob. 12ECh. 1.7 - Prob. 13ECh. 1.7 - Prob. 14ECh. 1.7 - Prob. 15ECh. 1.7 - Prob. 16ECh. 1.7 - Prob. 17ECh. 1.7 - Use the laws of logs to rewrite the following if...Ch. 1.7 - Prob. 19ECh. 1.7 - Prob. 20ECh. 1.7 - Prob. 21ECh. 1.7 - Prob. 22ECh. 1.7 - Prob. 23ECh. 1.7 - Prob. 24ECh. 1.7 - Prob. 25ECh. 1.7 - Prob. 26ECh. 1.7 - Prob. 27ECh. 1.7 - Prob. 28ECh. 1.7 - Prob. 29ECh. 1.7 - Prob. 30ECh. 1.7 - Prob. 31ECh. 1.7 - Prob. 32ECh. 1.7 - Prob. 33ECh. 1.7 - Prob. 34ECh. 1.7 - Prob. 35ECh. 1.7 - Prob. 36ECh. 1.7 - Prob. 37ECh. 1.7 - Prob. 38ECh. 1.7 - Prob. 39ECh. 1.7 - Prob. 40ECh. 1.7 - Prob. 41ECh. 1.7 - Prob. 42ECh. 1.7 - Prob. 43ECh. 1.7 - Prob. 44ECh. 1.7 - Prob. 45ECh. 1.7 - Prob. 46ECh. 1.7 - Prob. 47ECh. 1.7 - Prob. 48ECh. 1.7 - Prob. 49ECh. 1.7 - Prob. 50ECh. 1.7 - Prob. 51ECh. 1.7 - Prob. 52ECh. 1.7 - Prob. 53ECh. 1.7 - Prob. 54ECh. 1.7 - Prob. 55ECh. 1.7 - Prob. 56ECh. 1.7 - Prob. 57ECh. 1.7 - Prob. 58ECh. 1.7 - Prob. 59ECh. 1.7 - Prob. 60ECh. 1.7 - Prob. 61ECh. 1.7 - Prob. 62ECh. 1.7 - Prob. 63ECh. 1.7 - Prob. 64ECh. 1.7 - Prob. 65ECh. 1.7 - Prob. 66ECh. 1.7 - Prob. 67ECh. 1.7 - Prob. 68ECh. 1.7 - Prob. 69ECh. 1.7 - Prob. 70ECh. 1.7 - Prob. 71ECh. 1.7 - Prob. 72ECh. 1.7 - Prob. 73ECh. 1.8 - Prob. 1ECh. 1.8 - Prob. 2ECh. 1.8 - Prob. 3ECh. 1.8 - Prob. 4ECh. 1.8 - Prob. 5ECh. 1.8 - Prob. 6ECh. 1.8 - Prob. 7ECh. 1.8 - Prob. 8ECh. 1.8 - Prob. 9ECh. 1.8 - Prob. 10ECh. 1.8 - Prob. 11ECh. 1.8 - Prob. 12ECh. 1.8 - Prob. 13ECh. 1.8 - Prob. 14ECh. 1.8 - Prob. 15ECh. 1.8 - Prob. 16ECh. 1.8 - Prob. 17ECh. 1.8 - Prob. 18ECh. 1.8 - Prob. 19ECh. 1.8 - Prob. 20ECh. 1.8 - Prob. 21ECh. 1.8 - Prob. 22ECh. 1.8 - Prob. 23ECh. 1.8 - Prob. 24ECh. 1.8 - Prob. 25ECh. 1.8 - Prob. 26ECh. 1.8 - Prob. 27ECh. 1.8 - Prob. 28ECh. 1.8 - Prob. 29ECh. 1.8 - Prob. 30ECh. 1.8 - Prob. 31ECh. 1.8 - Prob. 32ECh. 1.8 - Prob. 33ECh. 1.8 - Prob. 34ECh. 1.8 - Prob. 35ECh. 1.8 - Prob. 36ECh. 1.8 - Prob. 37ECh. 1.8 - Prob. 38ECh. 1.8 - Prob. 39ECh. 1.8 - Prob. 40ECh. 1.8 - Prob. 41ECh. 1.8 - Prob. 42ECh. 1.8 - Prob. 43ECh. 1.8 - Prob. 44ECh. 1.8 - Prob. 45ECh. 1.8 - Prob. 46ECh. 1.8 - Prob. 47ECh. 1.8 - Prob. 48ECh. 1.8 - Prob. 49ECh. 1.8 - Prob. 50ECh. 1.8 - Prob. 51ECh. 1.9 - Prob. 1ECh. 1.9 - Prob. 2ECh. 1.9 - Prob. 3ECh. 1.9 - Prob. 4ECh. 1.9 - Prob. 5ECh. 1.9 - Prob. 6ECh. 1.9 - Prob. 7ECh. 1.9 - Prob. 8ECh. 1.9 - Prob. 9ECh. 1.9 - Prob. 10ECh. 1.9 - Prob. 11ECh. 1.9 - Prob. 12ECh. 1.9 - Prob. 13ECh. 1.9 - Prob. 14ECh. 1.9 - Prob. 15ECh. 1.9 - Prob. 16ECh. 1.9 - Prob. 17ECh. 1.9 - Prob. 18ECh. 1.9 - Prob. 19ECh. 1.9 - Prob. 20ECh. 1.9 - Prob. 21ECh. 1.9 - Prob. 22ECh. 1.9 - Prob. 23ECh. 1.9 - Prob. 24ECh. 1.9 - Prob. 25ECh. 1.9 - Prob. 26ECh. 1.9 - Prob. 27ECh. 1.9 - Prob. 28ECh. 1.9 - Prob. 29ECh. 1.9 - Prob. 30ECh. 1.9 - Prob. 31ECh. 1.9 - Prob. 32ECh. 1.9 - Prob. 33ECh. 1.9 - Prob. 34ECh. 1.9 - Prob. 35ECh. 1.9 - Prob. 36ECh. 1.9 - Prob. 37ECh. 1.9 - Prob. 38ECh. 1.9 - Prob. 39ECh. 1.9 - Prob. 40ECh. 1.9 - Prob. 41ECh. 1.9 - Prob. 42ECh. 1.9 - Prob. 43ECh. 1.9 - Prob. 44ECh. 1.9 - Prob. 45ECh. 1.9 - Prob. 46ECh. 1.9 - Prob. 47ECh. 1.9 - Prob. 48ECh. 1.9 - Prob. 49ECh. 1.9 - Prob. 50ECh. 1.9 - Prob. 51ECh. 1.10 - A population consists of 200 red birds and 800...Ch. 1.10 - A population consists of 200 red birds and 800...Ch. 1.10 - A population consists of 200 red birds and 800...Ch. 1.10 - Prob. 4ECh. 1.10 - Prob. 5ECh. 1.10 - Prob. 6ECh. 1.10 - Prob. 7ECh. 1.10 - Prob. 8ECh. 1.10 - Prob. 9ECh. 1.10 - Prob. 10ECh. 1.10 - Prob. 11ECh. 1.10 - Prob. 12ECh. 1.10 - Prob. 13ECh. 1.10 - Prob. 14ECh. 1.10 - Prob. 15ECh. 1.10 - Prob. 16ECh. 1.10 - Prob. 17ECh. 1.10 - Prob. 18ECh. 1.10 - Prob. 19ECh. 1.10 - Prob. 20ECh. 1.10 - Prob. 21ECh. 1.10 - Prob. 22ECh. 1.10 - Prob. 23ECh. 1.10 - Prob. 24ECh. 1.10 - Prob. 25ECh. 1.10 - Prob. 26ECh. 1.10 - Prob. 27ECh. 1.10 - Prob. 28ECh. 1.10 - Prob. 29ECh. 1.10 - Prob. 30ECh. 1.10 - Prob. 31ECh. 1.10 - Prob. 32ECh. 1.10 - Prob. 33ECh. 1.10 - Prob. 34ECh. 1.10 - Prob. 35ECh. 1.10 - The model of selection studied in this section is...Ch. 1.10 - Prob. 37ECh. 1.10 - Prob. 38ECh. 1.10 - Prob. 39ECh. 1.10 - Prob. 40ECh. 1.10 - Prob. 41ECh. 1.10 - Prob. 42ECh. 1.10 - Prob. 43ECh. 1.10 - Prob. 44ECh. 1.10 - Prob. 45ECh. 1.10 - Prob. 46ECh. 1.10 - Prob. 47ECh. 1.10 - Prob. 48ECh. 1.10 - Prob. 49ECh. 1.11 - Prob. 1ECh. 1.11 - Prob. 2ECh. 1.11 - Prob. 3ECh. 1.11 - Prob. 4ECh. 1.11 - Prob. 5ECh. 1.11 - Prob. 6ECh. 1.11 - Prob. 7ECh. 1.11 - Prob. 8ECh. 1.11 - Prob. 9ECh. 1.11 - Prob. 10ECh. 1.11 - Prob. 11ECh. 1.11 - Prob. 12ECh. 1.11 - Prob. 13ECh. 1.11 - Prob. 14ECh. 1.11 - Prob. 15ECh. 1.11 - Prob. 16ECh. 1.11 - Prob. 17ECh. 1.11 - Prob. 18ECh. 1.11 - Prob. 19ECh. 1.11 - Prob. 20ECh. 1 - Suppose you have a culture of bacteria, where the...Ch. 1 - Prob. 2SPCh. 1 - Prob. 3SPCh. 1 - A lab has a culture of a new kind of bacteria...Ch. 1 - Prob. 5SPCh. 1 - Prob. 6SPCh. 1 - Prob. 7SPCh. 1 - Prob. 8SPCh. 1 - Prob. 9SPCh. 1 - Prob. 10SPCh. 1 - A person develops a small liver tumor. It grows...Ch. 1 - Prob. 12SPCh. 1 - Prob. 13SPCh. 1 - Prob. 14SPCh. 1 - Prob. 15SPCh. 1 - Prob. 16SPCh. 1 - Prob. 17SPCh. 1 - Prob. 18SPCh. 1 - Prob. 19SPCh. 1 - Prob. 20SPCh. 1 - Prob. 21SPCh. 1 - Prob. 22SPCh. 1 - Prob. 23SPCh. 1 - Prob. 24SPCh. 1 - Prob. 25SPCh. 1 - Prob. 26SPCh. 1 - Prob. 27SPCh. 1 - Prob. 28SPCh. 1 - Prob. 29SP
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