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Chapter 3: Problem 5

Approximate the zero(s) of the function. Use Newton鈥檚 Method and continue the process until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results. \(f(x)=x^{3}+4\)

Short Answer

Expert verified
By a certain number of iterations using Newton's method, an approximate root of the given function is obtained. When this solution is compared with the root obtained from a graphing utility, they should be close within 0.001 error tolerance.

Step by step solution

01

Initial Approximation

Newton's method requires an initial guess for the root, in this case, let's start with an initial guess of \(x_0 = 0\).
02

Applying Newton's Method

Newton's Method formula is given by \(x_{n+1} = x_n - f(x_n)/f'(x_n)\). Based on the given function, the derivative \(f'(x)\) is \(3x^2\). So, the formula becomes \(x_{n+1} = x_n - (x_n^3+4)/(3x_n^2)\). By substituting the value of \(x_0\), compute \(x_1\), \(x_2\), and so forth, until the difference between two successive approximations \(|x_{n+1} - x_n|\) is less than 0.001.
03

Getting the Root from a Graphing Utility

By using a graphing utility or software, plot graph for \(y = x^3 + 4\). The point where the curve intersects the x-axis provides the root of the equation.
04

Comparing the Results

Compare the root obtained from Newton's method with the root obtained from the graphing utility. These should be reasonably close within the given error tolerance.

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Key Concepts

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

Roots of Equations
When we talk about finding the 'roots of equations,' we are referring to the values of the variable that make the equation equal to zero. In graphical terms, the roots of a function are the points at which the graph of the function intersects the x-axis.

In the given exercise, the function in question is a polynomial, described by the equation \(f(x)=x^{3}+4\). The roots are the solutions to the equation \(x^{3}+4=0\). Root-finding is not just an academic exercise; it has practical applications across physics, engineering, and economics, where determining specific quantifiables (like break-even points) is essential.
Numerical Approximation Methods
Numerical approximation methods are techniques used to find approximate solutions to equations when an exact analytical solution is difficult or impossible to ascertain. These methods, such as Newton's Method, use iteration to progressively tune an initial guess towards the solution.

The magic of numerical methods lies in their iterative nature, where each new approximation is built upon the previous one, using the structure of the function and its derivative to home in on the root. They are particularly useful for complex equations or systems of equations which do not have a straightforward analytical solution.
Convergence Criteria
Convergence criteria in numerical methods are essential for determining when the iterative process can stop. These criteria provide a measurable endpoint for the approximation process.

An example of a common convergence criterion is the difference between successive approximations, like the one mentioned in the exercise where the iterations continue until \(|x_{n+1} - x_n| < 0.001\). This specific value (0.001) is the threshold that dictates the desired accuracy of the solution. Convergence means that the method is producing successively closer approximations to the actual root, which is our goal.
Derivative Calculation
Derivative calculation is integral to Newton's Method. The derivative of a function at a point provides information about the slope of the function at that point. In effect, it helps predict how a small change in the input of the function will change the output.

In the context of the exercise's function \(f(x) = x^3 + 4\), the derivative \(f'(x) = 3x^2\) informs the Newton's Method formula on how to adjust our current guess to get closer to the root. It's this precise alteration, driven by derivative calculation, that allows for the iterative refinement towards the function's root.

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Most popular questions from this chapter

Maximum Area Twenty feet of wire is to be used to form two figures. In each of the following cases, how much wire should be used for each figure so that the total enclosed area is maximum? (a) Equilateral triangle and square (b) Square and regular pentagon (c) Regular pentagon and regular hexagon (d) Regular hexagon and circle What can you conclude from this pattern? \(\\{\text {Hint: The }\) area of a regular polygon with \(n\) sides of length \(x\) is \(A=(n / 4)[\cot (\pi / n)] x^{2} . \\}\)

Analyzing a Graph Using Technology In Exercises \(75-82,\) use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ g(x)=\frac{2 x}{\sqrt{3 x^{2}+1}} $$

Finding a Differential In Exercises \(11-20\) , find the differential \(d y\) of the given function. $$ y=\sqrt{x}+\frac{1}{\sqrt{x}} $$

Modeling Data A heat probe is attached to the heat exchanger of a heating system. The temperature \(T\) (in degrees Celsius) is recorded \(t\) seconds after the furnace is started. The results for the first 2 minutes are recorded in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline t & {0} & {15} & {30} & {45} & {60} \\\ \hline T & {25.2^{\circ}} & {36.9^{\circ}} & {45.5^{\circ}} & {51.4^{\circ}} & {56.0^{\circ}} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline t & {75} & {90} & {105} & {120} \\ \hline T & {59.6^{\circ}} & {62.0^{\circ}} & {64.0^{\circ}} & {65.2^{\circ}} \\\ \hline\end{array} $$ (a) Use the regression capabilities of a graphing utility to find a model of the form \(T_{1}=a t^{2}+b t+c\) for the data. (b) Use a graphing utility to graph \(T_{1} .\) (c) A rational model for the data is $$T_{2}=\frac{1451+86 t}{58+t}$$ Use a graphing utility to graph \(T_{2}\) (d) Find \(T_{1}(0)\) and \(T_{2}(0)\) (e) Find \(\lim _{t \rightarrow \infty} T_{2}\) . (f) Interpret the result in part (e) in the context of the problem. Is it possible to do this type of analysis using \(T_{1} ?\) Explain.

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=x^{2}, \quad(2,4) $$
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