3.8.1 TryNewton’s method graphically for two steps starting from the given points on the figure. ... The point marked A.
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3.8.2 TryNewton’s method graphically for two steps starting from the given points on the figure. ... The point marked B.
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3.8.3 TryNewton’s method graphically for two steps starting from the given points on the figure. ... The point marked C.
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3.8.4 TryNewton’s method graphically for two steps starting from the given points on the figure. ... The point marked D.
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3.8.5 Use
Newton’s method for three steps to solve the following equations. Find
and sketch the tangent line for the first step, then find the Newton’s
method discrete-time dynamical system to check your answer for the first
step and compute the next two values. Compare with the result on your
calculator. ... (this can be solved exactly with the quadratic formula).
Get solution
3.8.6 Use
Newton’s method for three steps to solve the following equations. Find
and sketch the tangent line for the first step, then find the Newton’s
method discrete-time dynamical system to check your answer for the first
step and compute the next two values. Compare with the result on your
calculator. ...
Get solution
3.8.7 Use
Newton’s method for three steps to solve the following equations. Find
and sketch the tangent line for the first step, then find the Newton’s
method discrete-time dynamical system to check your answer for the first
step and compute the next two values. Compare with the result on your
calculator. ...
Get solution
3.8.8 Use
Newton’s method for three steps to solve the following equations. Find
and sketch the tangent line for the first step, then find the Newton’s
method discrete-time dynamical system to check your answer for the first
step and compute the next two values. Compare with the result on your
calculator. The point where cos(x) = x (in radians, of course).
Get solution
3.8.9 Many
equations can also be solved by repeatedly applying a discrete-time
dynamical system. Compare the following discrete time dynamical systems
with the Newton’s method discrete-time dynamical system. Show that each
has the same equilibrium, and see how close you get in three steps. Use the discrete-time dynamical system ... to solve ...
Get solution
3.8.10 Many
equations can also be solved by repeatedly applying a discrete-time
dynamical system. Compare the following discrete time dynamical systems
with the Newton’s method discrete-time dynamical system. Show that each
has the same equilibrium, and see how close you get in three steps. Use the discrete-time dynamical system ...to solve ... (based on Exercise 6).
Get solution
3.8.11
Many
equations can also be solved by repeatedly applying a discrete-time
dynamical system. Compare the following discrete time dynamical systems
with the Newton’s method discrete-time dynamical system. Show that each
has the same equilibrium, and see how close you get in three steps. Use
the discrete-time dynamical system ... to solve ... (based on Exercise
7).
Get solution
3.8.13 Find the value of the parameter r for
which the given discrete-time dynamical system will converge most
rapidly to its positive equilibrium. Follow the system for four steps
starting from the given initial condition. The logistic dynamical system ... Start from x0 = 0.75.
Get solution
3.8.13 Find the value of the parameter r for
which the given discrete-time dynamical system will converge most
rapidly to its positive equilibrium. Follow the system for four steps
starting from the given initial condition. The logistic dynamical system ... Start from x0 = 0.75.
Get solution
3.8.15 As
mentioned in the text, although Newton’s method works incredibly well
most of the time, it can perform poorly or even fail in many
circumstances. For each of the following, graph the function and
illustrate the following problems. Find two initial values from which Newton’s method fails to solve x(x − 1)(x + 1)= 0. Which starting points converge to a negative solution?
Get solution
3.8.15 As
mentioned in the text, although Newton’s method works incredibly well
most of the time, it can perform poorly or even fail in many
circumstances. For each of the following, graph the function and
illustrate the following problems. Find two initial values from which Newton’s method fails to solve x(x − 1)(x + 1)= 0. Which starting points converge to a negative solution?
Get solution
3.8.16 As
mentioned in the text, although Newton’s method works incredibly well
most of the time, it can perform poorly or even fail in many
circumstances. For each of the following, graph the function and
illustrate the following problems. Find two initial values from which Newton’s method fails to solve ...Graphically indicate a third such value.
Get solution
3.8.17 As
mentioned in the text, although Newton’s method works incredibly well
most of the time, it can perform poorly or even fail in many
circumstances. For each of the following, graph the function and
illustrate the following problems. Use Newton’s method to solve ... (the solution is 0). Why does it approach the solution so slowly?
Get solution
3.8.18
As
mentioned in the text, although Newton’s method works incredibly well
most of the time, it can perform poorly or even fail in many
circumstances. For each of the following, graph the function and
illustrate the following problems. Use Newton’s method to solve ...
(the solution is 0). This is the square root of the absolute value of x. Why does the method fail?
Get solution
3.8.19 Suppose we wish to solve the equation f (x) =0 but cannot compute the derivative f ′ (x) (this kind of problem arises when the function f (x) must be evaluated with a complicated computer program). One method approximates the derivative f ′ (x with f (x + 1) − f (x) (the
secant line). For each of the following cases, write an approximate
Newton’s method discrete- time dynamical system, illustrate the
procedure on a diagram, and try it for five steps to see how quickly it
approaches the solution. ...
Get solution
3.8.20 Suppose we wish to solve the equation f (x) =0 but cannot compute the derivative f ′ (x) (this kind of problem arises when the function f (x) must be evaluated with a complicated computer program). One method approximates the derivative f ′ (x with f (x + 1) − f (x) (the
secant line). For each of the following cases, write an approximate
Newton’s method discrete- time dynamical system, illustrate the
procedure on a diagram, and try it for five steps to see how quickly it
approaches the solution. ...
Get solution
3.8.21 There is an alternative way to approximate the slope of the function at the value ... : ... For
each of the following equations, use this estimate to write an
approximate Newton’s method discrete-time dynamical system and
illustrate the idea on a diagram. How is it different from an ordinary
discrete-time dynamical system? Run it for a few steps starting from x0 and x1
from the earlier problem. Does it converge faster than the earlier
approximation? Why? How does it compare with Newton’s method itself? ...
Get solution
3.8.22 There is an alternative way to approximate the slope of the function at the value ... : ... For
each of the following equations, use this estimate to write an
approximate Newton’s method discrete-time dynamical system and
illustrate the idea on a diagram. How is it different from an ordinary
discrete-time dynamical system? Run it for a few steps starting from x0 and x1
from the earlier problem. Does it converge faster than the earlier
approximation? Why? How does it compare with Newton’s method itself? ...
Get solution
3.8.23 Suppose the total amount of nectar that comes out of a flower after time t follows ... After
noting how this function differs from the forms studied in the text,
write the equation used to find the optimum time to remain, and then
solve it with Newton’s method (and algebraically if possible) when the
travel time τ takes on the following values. Draw the associated Marginal Value Theorem diagram. τ =0.
Get solution
3.8.24 Suppose the total amount of nectar that comes out of a flower after time t follows ... After
noting how this function differs from the forms studied in the text,
write the equation used to find the optimum time to remain, and then
solve it with Newton’s method (and algebraically if possible) when the
travel time τ takes on the following values. Draw the associated Marginal Value Theorem diagram. τ =1.
Get solution
3.8.25 Suppose a fish population follows the discrete-time dynamical system ... For the following values of r , find the equilibrium ...as a function of h, write the equation for the critical point of the payoff function ...and use Newton’s method to find the best h. r = 2.5.
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3.8.26 Suppose a fish population follows the discrete-time dynamical system ... For the following values of r , find the equilibrium ...and use Newton’s method to find the best h. r = 1.5.
Get solution
3.8.27 Consider a variant of the medication discrete-time dynamical system ... where the function ...represents the fraction used (see Section1.10, Exercise 39). Suppose that ...For the following values of α,
use the Intermediate Value Theorem to show that there is an
equilibrium, follow the solution of the discrete-time dynamical system
until it gets close to the equilibrium (about three decimal places), and
find the equilibrium with Newton’s method. α = 0.5.
Get solution
3.8.28 Consider a variant of the medication discrete-time dynamical system ... where the function ...represents the fraction used (see Section1.10, Exercise 39). Suppose that ...For the following values of α,
use the Intermediate Value Theorem to show that there is an
equilibrium, follow the solution of the discrete-time dynamical system
until it gets close to the equilibrium (about three decimal places), and
find the equilibrium with Newton’s method. α = 0.9.
Get solution
3.8.29 Thomas
Malthus predicted doom for the human species when he argued that
populations grow exponentially but their resources only grow linearly.
Find the time when the population runs out of resources in the following
cases. The population grows according to ..., and resources grow according to R(t)= 400 + 100t. The population starves when b(t) = R(t).
Get solution
3.8.30 Thomas
Malthus predicted doom for the human species when he argued that
populations grow exponentially but their resources only grow linearly.
Find the time when the population runs out of resources in the following
cases. The population grows according to ..., and resources grow according to R(t) = 4000 + 500t. The population starves when b(t) = R(t).
Get solution
3.8.31 A lung follows the discrete-time dynamical system ... where γ = 5.0 and the function ...is positive, decreasing, and α(0)=
1. We used the Intermediate Value Theorem (Exercise 31 in Section 3.4)
to show that there is an equilibrium for any such function α(c). Use Newton’s method to solve for the equilibrium for the following forms of α(c). ...
Get solution
3.8.32 A lung follows the discrete-time dynamical system ... where γ = 5.0 and the function ...is positive, decreasing, and α(0)=
1. We used the Intermediate Value Theorem (Exercise 31 in Section 3.4)
to show that there is an equilibrium for any such function α(c). Use Newton’s method to solve for the equilibrium for the following forms of α(c). ...
Get solution
3.8.33
An alternative method of solution, which is much slower but much
safer, is called bisection. The method is based on the Intermediate
Value Theorem. We will use it to solve ...
a. We know there is a solution between x = 0 and x = 2. Show that there is a solution between x = 1 and x = 2.
b. By computing g(1.5), show there is a solution between 1.0 and 1.5.
c. Compute g(1.25). There is either a solution between 1.0 and 1.25 or between 1.25 and 1.5. Which is it?
d. Compute the value of g at the midpoint of the previous interval, and find an interval half as big that contains a solution.
e. Continue bisecting the interval until your answer is right to three decimal places.
f. About how many more steps would it take to reach six decimal places?
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3.8.34 Solve the equation ...by
using the quadratic approximation for the function and solving each
step by using the quadratic formula. Compare how fast it converges with
Newton’s method. Which method do you think is better?
Get solution
