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Chapter 2: Problem 9

Use graphical approximation (a root finder or an intersection finder to find a solution of the equation in the given open interval. $$x^{4}+x-3=0 ; \quad(-\infty, 0)$$

Short Answer

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Question: Use a graphical approximation technique to find a root of the equation \(x^4 + x - 3 = 0\) in the interval \((-\infty, 0)\). Answer: After applying a graphical approximation technique, the approximate root in the specified interval is __________ (fill in the blank with the approximated root value).

Step by step solution

01

Write down the equation

The given equation is: $$x^{4}+x-3=0$$
02

Visualize the equation

Sketch the graph of the function $$f(x) = x^{4}+x-3$$ or use a graphing calculator to visualize the function. Observe the interval on the interval \((-\infty, 0)\) and locate any intersections with the x-axis, indicating possible roots.
03

Identify a subinterval for finding the root

Choose a subinterval where the function crosses the x-axis. Take note of two points close to the intersection, for example, \((-2, 0)\) and \((-1, 0)\), which will help in narrowing down the root between these points (you can visually identify this or use a graphing calculator).
04

Apply a graphical approximation technique

Use a root-finder or intersection finder (e.g., Bisection Method, Newton-Raphson Method, or any numerical method available on your calculator) to find the root in the identified subinterval. Make sure to use an appropriate level of accuracy/precision.
05

Write down the root

After applying the selected method, write down the root's approximate value found in the interval. The root found will be an approximation due to the graphical nature of the problem-solving method.

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Key Concepts

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

Polynomial Equations
Polynomial equations express relationships in which variables are raised to whole number powers. A polynomial equation has the form:
  • \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0\)
where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants, and \(n\) represents the highest power, or degree, of the equation. In the exercise provided, the polynomial equation is \(x^4 + x - 3 = 0\), which is a fourth-degree polynomial. Understanding the degree is crucial because it tells us about the possible number of roots the equation might have, which in this case can be up to four roots. Recognizing the form of a polynomial and analyzing its degree is the first step in tackling polynomial equations.
Root Finding Methods
Root finding methods are techniques used to find the solutions (or roots) of equations where the function equals zero. For polynomial equations, these roots correspond to the points where the graph of the function crosses the x-axis. Common root finding methods include:
  • Graphical methods: Involve sketching the graph to visually identify intercepts.
  • Analytical methods: For simpler polynomials, such as factoring or using the quadratic formula.
  • Numerical methods: Used where analytical methods are impractical.
Graphical approximation, as discussed in the solution provided, is a root finding method where you visually inspect a graph to pinpoint where it intersects the x-axis. Then a more precise numerical method, like the Bisection or Newton-Raphson Method, is often applied to hone in on a more exact value of the root.
Numerical Methods
Numerical methods are approaches used to approximate the roots of equations. They're particularly useful when equations are too complex to solve analytically. These methods include:
  • Bisection Method: Involves bracketing the root within an interval and then systematically narrowing it down.
  • Newton-Raphson Method: Uses tangents to approximate roots iteratively, offering faster convergence for well-selected starting points.
  • Secant Method: Similar to Newton-Raphson but uses two points for approximation rather than the tangent line.
The choice of method often depends on the function's properties and the required precision. In the exercise, a numerical method would refine the root estimation after visually locating a potential root through graphical methods.
Graphing Calculators
Graphing calculators are powerful tools for students and professionals dealing with polynomial equations. They combine graphical representation and numerical computation, enabling users to:
  • Plot functions to visually identify potential roots.
  • Use built-in functions to apply numerical methods like finding roots or intersections.
  • Adjust parameters for increased precision and better visualization.
When facing complex equations like \(x^4 + x - 3 = 0\), graphing calculators can help initially plot the function over the given interval. They offer functionalities to zoom in and locate roots more accurately without intensive manual calculation. This dual capability makes them an essential educational tool in mathematics.

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Most popular questions from this chapter

A mathematics book has 36 square inches of print per page. Each page has a left side margin of 1.5 inches and top, bottom, and right side margins of .5 inch. If a page cannot be wider than 7.5 inches, what should its length and width be to use the least amount of paper?

Use algebraic, graphical, or numerical methods to find all real solutions of the equation, approximating when necessary. $$\left|x^{3}+2\right|=5+x-x^{2}$$

The national average interest rate for a 30 -year fixed rate mortgage is approximated by $$ y=.015 x^{3}+.112 x^{2}-1.562 x+8.67 \quad(0 \leq x \leq 6) $$ where \(x=0\) corresponds to \(2000 .^{*}\) According to this model, when was the rate the lowest? What was the lowest rate?

In Exercises \(31-36\), determine which of the following viewing windows gives the best view of the graph of the given equation (a) \(-10 \leq x \leq 10 ; \quad-10 \leq y \leq 10\) (b) \(-5 \leq x \leq 25 ; \quad 0 \leq y \leq 20\) (c) \(-10 \leq x \leq 10 ;-100 \leq y \leq 100\) (d) \(-20 \leq x \leq 15 ; \quad-60 \leq y \leq 250\) (e) None of \(a, b, c, d\) gives a complete graph.. $$y=x^{4}+x-5$$

(a) Graph \(y=3 x^{3}-2 x^{2}+6\) in the standard window. (b) Use trace to move to a point whose \(x\) -coordinate is close to 1 (c) Set the zoom factors of your calculator to \(10 .\) Zoom-in once or twice. Does the graph appear to be a straight line near the point? (d) Repeat parts (a) \(-(c)\) at lowest point to the right of the \(y\) -axis. Is the result the same? If not, keep zooming in until it is (at each stage move the flashing cursor up or down, so it is on the graph). (e) What do parts (a) - (d) suggest about the graph?
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