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Chapter 6: Problem 21

Approximate the real zeros of \(g(x):=x^{4}-x-3\)

Short Answer

Expert verified
The exact solution is impossible to obtain analytically. Use numerical methods like the Bisection method or Newton's method to approximate the zeros. The initial approximations are obtained by analyzing the graph of the function. For more accurate results, repeat the approximation process with finer calculations until acceptable precision is reached.

Step by step solution

01

Understanding the Function

The function given is \( g(x)=x^{4}-x-3 \). The goal is to find the x-values at which the function equals zero.
02

Initial Approximations

Plot the function or analyze the polynomial function using a graphing utility. By doing so, one can obtain the approximate zeros visually. This can be used as initial approximations for the following steps.
03

Apply Bisection Method

Once we have initial approximations, use the bisection method to refine these approximations. Bisection method is a root-finding algorithm that divides the interval into two halves and selects the half where a function changes sign. Hence, applying this method should provide a better approximation of the zeros.
04

Apply Newton鈥檚 Method

For an even more accurate approximation, apply Newton's method. This involves finding the derivative of the original function, choosing a preliminary approximation, and using the Newton-Raphson formula to refine this value. Repeat this process until the approximation is accurate enough.

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Key Concepts

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

Polynomial Function
A polynomial function is a mathematical expression that involves sums of powers of a variable, typically written as sums of terms of the form \(a_n x^n\) where \(a_n\) represents coefficients, and \(n\) represents non-negative integers as their powers. These functions are very important in algebra and calculus due to their versatile nature.
  • Polynomial functions can have different degrees, determined by the highest power of the variable.
  • The structure allows polynomials to take many shapes when graphed, such as straight lines, parabolas, and more complex curves.
  • The function given in the exercise, \(g(x) = x^4 - x - 3\), is a fourth-degree polynomial because the highest power of \(x\) is 4.
Understanding polynomial functions helps us analyze their characteristics such as intercepts, turning points, and the behavior at the boundaries. These features are crucial when approximating roots as they provide insights on where the function鈥檚 graph crosses the x-axis.
Bisection Method
The bisection method is a reliable algorithm used to find real zeros of continuous functions, especially when an initial approximation is known. It works by repeatedly dividing an interval into two halves and choosing the subinterval where the function changes sign.
  • This method is particularly effective when the function is continuous on a closed interval \([a, b]\) and \(f(a)\) and \(f(b)\) have opposite signs.
  • By narrowing down the interval iteratively, the method gets closer to the root, making it more precise with each iteration.
  • For the function \(g(x)=x^4-x-3\), you'd start by identifying an interval where the function transitions from positive to negative or vice versa, indicating a root exists in the interval.
The beauty of the bisection method lies in its simplicity and ability to converge to a root if it indeed exists within the interval.
Newton's Method
Newton's Method, often called the Newton-Raphson method, is another powerful tool for finding approximations of the roots of a real-valued function. It is an iterative method that improves upon a previous approximation by considering the function鈥檚 derivative.
  • The formula used is \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), where \(x_n\) is the current approximation, and \(f'(x_n)\) is the derivative at that point.
  • It's important to choose a good initial approximation, often derived from methods like the bisection method, to ensure quick convergence.
  • This method can be more efficient than the bisection method as it generally converges faster, assuming an appropriate starting point.
For our polynomial \(g(x) = x^4 - x - 3\), evaluating the derivative and iterating helps in pinpointing the real zeros with high accuracy.
Real Zeros
Real zeros of a function are the x-values for which the function equals zero. These points are critical because they are where the graph of a function touches or cuts through the x-axis.
  • Finding real zeros is essential in understanding the full behavior of a function as they often represent points of equilibrium.
  • For polynomial functions like \(g(x)=x^4-x-3\), real zeros can be located by solving \(g(x) = 0\).
  • Techniques such as the bisection and Newton's method come into play to refine and accurately identify these zeros.
Real zeros are not only crucial for understanding the geometric representation of polynomial functions but also have practical applications in science and engineering for solving real-world problems.

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Most popular questions from this chapter

Calculate \(e\) correct to seven decimal places.

The function \(f(x)=8 x^{3}-8 x^{2}+1\) has two zeros in \([0,1]\). Approximate them, using Newton's Method, with the starting points (a) \(x_{1}:=\frac{1}{8}\), (b) \(x_{1}:=\frac{1}{4}\). Explain what happens.

Approximate the real zeros of \(g(x):=x^{4}-x-3\)

Let \(a>b>0\) and let \(n \in \mathbb{N}\) satisfy \(n \geq 2\). Prove that \(a^{1 / n}-b^{1 / n}<(a-b)^{1 / n} \cdot[\) Hint \(:\) Show that \(f(x):=x^{1 / n}-(x-1)^{1 / n}\) is decreasing for \(x \geq 1\), and evaluate \(f\) at 1 and \(a / b\).]

Give an example of a uniformly continuous function on \([0,1]\) that is differentiable on \((0,1)\) but whose derivative is not bounded on \((0,1)\).
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