Problem 84 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the method of bisection (see Example 6 ) to find the root of the equation \(x^{3}-x+1=0\) accurate to two decimal places.

Step-by-Step Solution

Answer

Using the bisection method, the root of the equation \(x^3 - x + 1 = 0\) approximately lies between \(a = -1\) and \(b = 1\). Repeatedly compute midpoints and update interval endpoints according to function values until desired accuracy is reached. Verify the final root by substitution.

Step 1: Understand the Bisection Method

The bisection method is a root-finding algorithm that applies to any continuous functions. The method works by dividing an interval that contains a root of the function into two subintervals, picking the subinterval where the function changes sign, and repeating this process until the subinterval is small enough.

Step 2: Selecting Initial Values

Begin by choosing two initial values, \(a\) and \(b\), such that \(f(a)\) and \(f(b)\) have opposite signs. This is based on the Intermediate Value Theorem which guarantees that the function has at least one root in the interval \((a, b)\). In this case, a convenient choice may be \(a = -1\) and \(b = 1\) because \(f(-1) = -3\) and \(f(1) = 1\) have opposite signs.

Step 3: Compute Midpoint

The next step is to compute the midpoint \(c = (a + b)/2\). Check the value of the function at \(c\). If \(f(c)\) is very close to zero (within your desired precision), then stop - you've found your root.

Step 4: Check Sign

If \(f(c)\) is not sufficiently close to zero, then check whether \(f(a)\) and \(f(c)\) have opposite signs. If they do, then the root must be between \(a\) and \(c\), so set \(b = c\) and go back to Step 3. If \(f(a)\) and \(f(c)\) have the same sign, then the root must be between \(c\) and \(b\), so set \(a = c\) and go back to Step 3. Repeating Steps 3 and 4 should narrow down the range for possible roots to the required precision (two decimal places in this case). This method ensures accuracy and is quite systematic, making it a reliable way to find the root of the equation.

Step 5: Verification

Always verify your answer by substituting the root back into the original equation to confirm if it's indeed a root. If the equation holds true, you have the correct root. If not, there might be a mistake in the calculations. This is your solution using the bisection method for the given cubic equation \(x^3 - x + 1 = 0\).

Key Concepts

Root-Finding Algorithms

Root-finding algorithms are essential mathematical tools used to identify the roots of a function. A root of a function is a number for which the function evaluates to zero. These algorithms are crucial in various fields, including engineering and scientific research, where solving equations is a standard task.

There are several types of root-finding algorithms, each with its own characteristics:

• Bisection Method: This method is simple and reliable. It works by repeatedly bisecting an interval and selecting a subinterval in which the function changes sign.
• Newton's Method: This method uses derivatives to approximate the roots. It converges faster than the bisection method but requires the derivative of the function.
• Secant Method: A variant of Newton's method that does not require computing derivatives, making it useful for functions where derivatives are complex to obtain.

The bisection method is particularly favored for beginner-friendly exercises due to its straightforward implementation and guaranteed convergence, provided that the initial interval is chosen correctly.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that assures the existence of a root within a certain interval. According to IVT, if a continuous function changes signs over an interval, then there is at least one root within that interval.

For example, consider a function \( f(x) \) such that \( f(a) < 0 \) and \( f(b) > 0 \). The IVT guarantees that there exists a point \( c \) in the interval \((a, b)\) where \( f(c) = 0 \). This property is precisely what makes the bisection method viable, as it relies on identifying and halving intervals where sign changes indicate the presence of a root.

Understanding the IVT helps in selecting initial values for methods like bisection, as it clarifies that choosing \( a \) and \( b \) with function values of opposite signs guarantees the existence of a root in between.

Numerical Methods

Numerical methods are a collection of techniques used to obtain approximate solutions to complex mathematical problems. They are particularly useful when analytical solutions are difficult or impossible to find.

Key characteristics of numerical methods include:

• Approximations: Numerical methods provide an approximation rather than an exact answer, which is often sufficient for practical applications.
• Iterative Solutions: Methods like the bisection method involve iterative processes that converge gradually towards a solution.
• Algorithmic Structure: Most numerical methods can be systematically implemented and executed using algorithms, making them suitable for computer programming.

The bisection method, an example of a numerical method, uses these principles to solve equations like the cubic equation \( x^3 - x + 1 = 0 \). While it might not be the fastest method, it is methodical and ensures a solution with a desired degree of accuracy.

Cubic Equations

Cubic equations are polynomial equations of degree three, taking the general form \( ax^3 + bx^2 + cx + d = 0 \), where \( a eq 0 \). These equations can have one, two, or three real roots, and solving them can be more intricate than linear or quadratic equations.

Characteristics of cubic equations include:

• Nonlinearity: Cubic equations are nonlinear, making analytical solutions typically more complicated.
• Multiple Roots: They can have multiple real roots or one real and two complex roots, adding complexity to the root-finding process.
• Applications: Cubic equations appear in various real-world scenarios, such as physics problems involving motion and in financial calculations.

To solve these, numerical methods like the bisection method are often employed when an analytical approach is complex or impractical. This numerical analysis helps find approximate solutions, such as narrowing down the root of the equation \( x^3 - x + 1 = 0 \) to within a required precision.