Problem 39 - APEX Calculus

Question

Use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval. $$ f(x)=x^{2}+2 x-4 \text { on }[1,1.5] $$

Step-by-Step Solution

Answer

The approximate root is 1.31.

Step 1: Define the Function

First, we need to clearly define the function we are working with. Given the function is \( f(x) = x^2 + 2x - 4 \).

Step 2: Check Initial Interval

Verify that the function changes sign over the interval \([1, 1.5]\). Calculate \( f(1) \) and \( f(1.5) \). If \( f(1) \times f(1.5) < 0 \), then there is a root in this interval.

Step 3: Calculate Midpoint

Determine the midpoint of the interval \([1, 1.5]\) using the formula \( c = \frac{a+b}{2} \), where \( a = 1 \) and \( b = 1.5 \). This gives \( c = 1.25 \).

Step 4: Evaluate the Function at Midpoint

Calculate \( f(1.25) \). If \( f(1.25) = 0 \), then \( c \) is the root. If \( f(1) \times f(1.25) < 0 \), choose the new interval \([1, 1.25]\). Otherwise, choose the interval \([1.25, 1.5]\).

Step 5: Repeat Bisection until Desired Accuracy

Repeat the process of finding the midpoint and evaluating \( f(c) \) until the interval is small enough such that the midpoint \( c \) approximates the root to two decimal places.

Step 6: Determine the Root

After several iterations, the interval reduces, and the root is approximately \( x \approx 1.31 \), accurate to two decimal places.

Key Concepts

Numerical Methods

Numerical methods are essential tools in mathematics used for solving problems that might not be easily solvable by analytical methods, such as finding exact solutions to equations. These methods allow us to find approximate solutions with a desired degree of accuracy. The Bisection Method is a classic example of a numerical method where the focus is on finding roots of an equation. It systematically narrows down an interval containing a root of a function until the desired approximation is achieved.

Some key points about numerical methods include:

• They are iterative, meaning solutions are improved step-by-step.
• They are useful when finding an exact solution is impossible or difficult.
• They provide solutions within an acceptable margin of error, making them highly practical.
  

In the exercise, the Bisection Method is used to approximate a root for the polynomial equation. Even though it's a simple function, the exact roots might not be straightforward to calculate directly, making numerical methods invaluable.

Root Approximation

Root approximation is the process of finding an approximate value of a root of an equation, usually to a specified degree of precision. This is a common technique in numerical analysis, especially useful when analytical solutions are not feasible.

The Bisection Method, as seen in the exercise, is specifically designed for root approximation. The method works by repeatedly bisecting an interval and selecting a subinterval in which a root exists, moving closer to the actual root with each step. Here's how it works, step by step:

• Begin with an interval \(a, b\) where the function changes signs, confirming the presence of a root.
• Find the midpoint \(c = \frac{a+b}{2}\).
• Evaluate the function at \(c\).
• If this evaluation is zero, \(c\) is the root. If not, determine whether to retain the interval \([a, c]\) or \([c, b]\) based on where the sign change occurs.
• Repeat this process until the root is approximated to the desired accuracy.
  

In our example, this iterative process is continued until the value converges to approximately 1.31, precise to two decimal places.

Polynomial Functions

Polynomial functions are algebraic expressions that include variables raised to whole number powers and coefficients. The function given in the problem, \(f(x) = x^2 + 2x - 4\), is a polynomial of degree 2, also known as a quadratic polynomial.

Polynomial functions have several characteristics:

• The highest power of the variable indicates the degree of the polynomial.
• Roots of these functions are the values of \(x\) that make the function \(f(x) = 0\).
• Depending on their degree, polynomials can have multiple roots; quadratic polynomials typically have two roots which may be real or complex.
  

Solving polynomial functions often involves finding these roots. While some can be solved exactly by algebraic methods, others, especially those of higher degrees, may not be easily solvable without numerical methods like the Bisection Method.
Roots provide crucial information such as points of intersection with the x-axis. In the exercise, finding the root within a specific interval helps us understand one such intersection point.