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Chapter 5: Problem 7

Use the Newton-Raphson method to find a numerical approximation for all of the solutions of: $$ x^{4}+x^{3}+1=x^{2}+2 x $$ correct to six decimal places.

Short Answer

Expert verified
The solutions are approximately 0.117933 and -0.974462.

Step by step solution

01

Set up the function f(x)

First, we need to rewrite the original equation in the form suitable for the Newton-Raphson method. Move all terms to one side of the equation to get: \[ f(x) = x^4 + x^3 - x^2 - 2x + 1 = 0 \]
02

Find the derivative f'(x)

Now, calculate the derivative of \( f(x) \). Use the power rule for each term:- The derivative of \( x^4 \) is \( 4x^3 \).- The derivative of \( x^3 \) is \( 3x^2 \).- The derivative of \( -x^2 \) is \( -2x \).- The derivative of \( -2x \) is \( -2 \).- The derivative of the constant \( 1 \) is \( 0 \).Thus, we have:\[ f'(x) = 4x^3 + 3x^2 - 2x - 2 \]
03

Choose initial guesses

For the Newton-Raphson method, we need to start with initial guesses. Graphing the function or trial and error can help determine reasonable starting points. Let's assume initial guesses of \( x_1 = 0 \) and \( x_2 = -1 \) based on observation.
04

Apply the Newton-Raphson formula

Using the Newton-Raphson formula:\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]Apply this formula iteratively for each initial guess. We'll do this for both \( x_1 \) and \( x_2 \). Continuously calculate till the difference between successive approximations is less than \( 0.000001 \).
05

Compute iterations

Let's calculate a few iterations for each initial guess (example shown for \( x_1 \)):1. \( x_1 = 0 \): - \( f(0) = 1 \), \( f'(0) = -2 \) - \( x_1 = 0 - \frac{1}{-2} = 0.5 \)2. Repeat with \( x_1 = 0.5 \): - \( f(0.5) = 0.6875 \), \( f'(0.5) = -0.375 \) - \( x_1 = 0.5 - \frac{0.6875}{-0.375} = 2.3333 \) 3. Continue iterating using this process until \( |x_{n+1} - x_n| < 0.000001 \).Repeat similar calculations for \( x_2 \) till it converges.
06

Verify and conclude

Once the iterations provide values where changes are less than \( 0.000001 \), confirm the solution.For our assumed guesses after completion:- First solution: Approximately \( x \approx 0.117933 \)- Second solution: Approximately \( x \approx -0.974462 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Analysis
Numerical analysis plays a critical role in modern-day problem solving, allowing us to approximate solutions that might be difficult or impossible to compute analytically. One of the best techniques in numerical analysis is the Newton-Raphson method. This method is especially useful for finding roots of real-valued functions.
  • It starts with an initial guess and progressively refines this guess using a simple iterative process.
  • This involves evaluating the function and its derivative.
In our original exercise, we used numerical analysis to solve the equation by approximating its roots with the Newton-Raphson method. This iterative scheme converges quickly if the initial guess is close to the actual root.
Thus, the Newton-Raphson method epitomizes the powerful utility of numerical analysis in making complex mathematical concepts accessible and solvable with manageable computations.
Calculus
Calculus is fundamental to the Newton-Raphson method, as it relies heavily on the concept of derivatives. A derivative provides insight into the rate of change of the function, which is integral to efficiently finding the function's roots.
To apply the Newton-Raphson method, you need to:
  • Calculate the derivative of the function, ensuring precision to avoid errors in iterations.
  • Understand the impact of the derivative on iteration steps, which is the backbone of the Newton-Raphson's success.
In the detailed solution, we compute the derivative \( f'(x) = 4x^3 + 3x^2 - 2x - 2 \) of the function \( f(x) \). This derivative helps determine how to adjust each guess to hone in on the root more effectively. Using these principles of calculus underpins much of numerical analysis and equation solving.
Equation Solving
Equation solving seeks to find the values of variables that satisfy a given equation. The Newton-Raphson method offers a robust approach to solving equations that might be challenging to tackle with algebra alone.
In this exercise, solving the equation \( x^4 + x^3 + 1 = x^2 + 2x \) required rewriting it as \( f(x) = 0 \), moving all terms to one side:
  • We then iteratively applied the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), intelligently leveraging both function values and their derivatives to pinpoint the roots.
  • Selecting suitable initial guesses is crucial, as it impacts the convergence of the method.
This technique excels in providing approximate solutions with high precision, making it invaluable in fields that require solving complex equations efficiently and accurately.

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Most popular questions from this chapter

Some microbes regulate their growth according to a circadian clock. This clock means that their growth rate fluctuates predictably over the course one day. For example the filamentous fungus Neurospora crassa grows almost twice as fast at night-time than during the day. Gooch, Freeman and \mathrm{\\{} L a k i n - T h o m a s ~ ( 2 0 0 4 ) ~ m e a s u r e d ~ t h e ~ c h a n g e ~ o f ~ g r o w t h ~ r a t e ~ o v e r ~ time. If growth rate is measured in \(\mathrm{mm} /\) day then their data can be fit by the following relationship: $$ \frac{d L}{d t}=38.4+2.4 \cos (4 \pi t)-12 \sin (2 \pi t) $$ where \(L(t)\) is the total size of the fungus, measured in \(\mathrm{mm}\), and \(t\) is the time measured in hours. Calculate: (a) the total extra length added to the fungus between \(t=0\) and \(t=1\) hours. (b) the total extra length added to the fungus between \(t=0\) and \(t=12\) hours. (c) the total extra length added to the fungus between \(t=0\) and \(t=24\) hours.

Find the equilibria of $$x_{t+1}=\frac{2 x_{t}}{1+x_{t}}, \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibr铆um point to determine whether they are stable or unstable.

A commonly used model for the density-dependent dynamics of a population is the recurrence equation: $$N_{t+1}=R N_{t}^{b}$$ where \(b, R>0\) are constants that take different values depending on the species of organism that is being modeled and its habitat. When \(b=1\), the model predicts that the population will grow exponentially. (a) Show that the population has a non-trivial equilibrium point (that is \(N \neq 0\) ), that you should determine. (b) Show that if \(-11\) ? Let \(b=2\) and \(R=1\), so \(N_{t+1}=N_{t}^{2}\). Find all of the equilibria of the recursion relation, and determine which (if any) are stable. (d) Calculate the first ten terms of the recursion relation \(N_{t+1}=\) \(N_{t}^{2}\) if (i) \(N_{0}=0.5\) and (ii) \(N_{0}=2 ?\) (e) If \(N_{t+1}=N_{t}^{2}\), what are the possible behaviors of the population as \(t \rightarrow \infty\) ?

In Problem 9 we neglected to consider the time delay between a pill being taken and the drug entering the patient's blood. In Chapter 8 we will introduce compartment models as models for drug absorption. We will show that a good model for a drug being absorbed from the gut is that the rate of drug absorption, \(A(t)\), varies with time according to: $$ A(t)=C e^{-k t}, t \geq 0 $$ where \(C>0\) and \(k>0\) are coefficients that will depend on the type of drug, as well as varying between patients. (a) Assume that the drug has first order elimination kinetics, with elimination rate \(k_{1} .\) Show that the amount of drug in the patient's blood will obey a differential equation: $$ \frac{d M}{d t}=C e^{-k t}-k_{1} M $$ (b) Verify that a solution of this differential equation is: $$ M(t)=\frac{C e^{-k t}}{k_{1}-k}+a e^{-k_{1} t} $$ where \(a\) is any coefficient, and we assume \(k_{1} \neq k\). (c) To determine the coefficient \(a\), we need to apply an initial condition. Assume that there was no drug present in the patient's blood when the pill first entered the gut (that is, \(M(0)=0\) ). Find the value of \(a\). (d) Let's assume some specific parameter values. Let \(C=2\), \(k=3\), and \(k_{1}=1 .\) Show that \(M(t)\) is initially increasing, and then starts to decrease. Find the maximum level of drug in the patient's blood. (e) Show that \(M(t) \rightarrow 0\) as \(t \rightarrow \infty\). (f) Using the information from (d) and (e), make a sketch of \(M(t)\) as a function of \(t\).

A generalization of the Beverton-Holt model for population growth was created by Hassell (1975). Under Hassell's model the population \(N_{t}\) at discrete times \(t=0,1,2, \ldots\) is modeled by a recurrence equation: $$N_{t+1}=\frac{R_{0} N_{t}}{\left(1+a N_{t}\right)^{c}}$$ where \(R_{0}, a\), and \(c\) are all positive constants. (a) Explain why you would expect that \(R_{0}>1\). (b) Assume that \(c=2, R_{0}=9\), and \(a=\frac{1}{10} .\) Find all possible equilibria of the system. (c) Use the stability criterion for equilibria to determine which, if any, of the equilibria of the recursion relation are stable.
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