Problem 12 - Calculus with Concepts in Calculus

Question

Find the slope \(m\) of the line joining \((a, f(a))\) and \((b, f(b))\). Then use the Newton-Raphson method to estimate the values of \(c\) for which \(f^{\prime}(c)=m .\) Continue the process until successive iterations obtained by the calculator are identical. $$ f(x)=\sin (\pi x / 2) ; a=-1, b=1 $$

Step-by-Step Solution

Answer

The slope \( m = 1 \), and iterative calculation is needed for \( c \) using the Newton-Raphson method starting at \( c = 0.5 \).

Step 1: Calculate the Function Values

First, evaluate the function \( f(x) = \sin(\frac{\pi x}{2}) \) at the points \( a = -1 \) and \( b = 1 \).\[ f(-1) = \sin\left(\frac{-\pi}{2}\right) = -1 \]\[ f(1) = \sin\left(\frac{\pi}{2}\right) = 1 \]

Step 2: Determine the Slope Formula

The slope \( m \) of the line passing through these points is given by the formula:\[ m = \frac{f(b) - f(a)}{b - a} \]

Step 3: Find the Slope

Substitute the values into the slope formula:\[ m = \frac{1 - (-1)}{1 - (-1)} = \frac{2}{2} = 1 \]

Step 4: Set Up the Newton-Raphson Method

The Newton-Raphson method formula for finding roots of \( f(x) = 0 \) is:\[ x_{n+1} = x_n - \frac{f'(x_n) - m}{f''(x_n)} \]

Step 5: Differentiate the Function

Find the first and second derivatives of \( f(x) \):\[ f'(x) = \frac{\pi}{2} \cos\left(\frac{\pi x}{2}\right) \]\[ f''(x) = -\left(\frac{\pi}{2}\right)^2 \sin\left(\frac{\pi x}{2}\right) \]

Step 6: Choose Initial Guess

Select an initial guess for \( c \). A sensible choice might be \( c = 0 \), since it is the midpoint of \( a \) and \( b \).

Step 7: Iteration 1

Calculate using the Newton-Raphson formula with \( x_0 = 0 \):\[ x_1 = 0 - \frac{\frac{\pi}{2} \cos(0) - 1}{-\left(\frac{\pi}{2}\right)^2 \sin(0)} \]Since \( \sin(0) = 0 \), the Newton-Raphson formula is indeterminate at this point. Choose another initial guess instead, like \( x_0 = 0.5 \).

Step 8: Iteration 2

Compute the next iteration with \( x_0 = 0.5 \):\[ f'(0.5) = \frac{\pi}{2} \cdot \cos\left(\frac{\pi \cdot 0.5}{2}\right) = \frac{\pi}{2} \cdot \cos\left(\frac{\pi}{4}\right) \]\[ f''(0.5) = -\left(\frac{\pi}{2}\right)^2 \sin\left(\frac{\pi \cdot 0.5}{2}\right) = -\left(\frac{\pi}{2}\right)^2 \sin\left(\frac{\pi}{4}\right) \]Calculate \( x_1 \) using these values.

Step 9: Check Successive Iterations

Continue the iteration process until the successive values of \( x_n \) and \( x_{n+1} \) are identical to the degree allowed by your calculator.

Key Concepts

Slope of a Line

When examining a line that connects two points on a plane, you might wonder how we determine the angle or steepness of that line. This is described by what we call the "slope" of the line. In this context, the slope helps us understand the direction and steepness, whether it's ascending upwards or descending down.
To find the slope of a line connecting two points, \((a, f(a))\) and \((b, f(b))\), we use the slope formula:

• \m = \frac{f(b) - f(a)}{b - a}\
• When the slope is positive, the line ascends as you move along; conversely, a negative slope indicates that the line descends.

Calculating the slope using the example from the exercise, we substitute the points \(a = -1\) and \(b = 1\), and find \m = 1\. This suggests a line that rises equally as it runs to the right, equivalent to a slope of 45 degrees.

Differentiation

Differentiation is a fundamental concept in calculus centered around finding the rate at which a function is changing at any given point. This rate of change is described by the function's derivative.
To apply the Newton-Raphson method effectively, we need both the first and second derivatives of the function. For the function \(f(x) = \sin\left(\frac{\pi x}{2}\right)\), the derivatives are:

• First derivative: \f'(x) = \frac{\pi}{2} \cos\left(\frac{\pi x}{2}\right)\
• Second derivative: \f''(x) = -\left(\frac{\pi}{2}\right)^2 \sin\left(\frac{\pi x}{2}\right)\

These derivatives are pivotal in the Newton-Raphson method, as they tell us both how the function is changing and also refine our estimates closer to the actual point where \(f'(x) = m\). Knowing how to differentiate correctly allows us to leverage these tools in the method, helping achieve the estimation faster.

Iteration Process

The iteration process encapsulates a sequence of steps that are repeated until a particular outcome is achieved. In the context of the Newton-Raphson method, this iterative process refines an estimate until successive numbers stabilize or converge.
The Newton-Raphson formula is iteratively applied: \x_{n+1} = x_n - \frac{f'(x_n) - m}{f''(x_n)}\

• Start with an initial guess. Here, choosing \(c = 0.5\) as an initial guess allows computation without encountering indeterminate solutions.
• Apply the formula to get a sequence of values, like getting from \(x_0\) to \(x_1\), \(x_2\) and so forth.
• Continue this process until the estimates converge, meaning \(x_{n+1}\) and \(x_n\) are sufficiently close.

This method is exceptionally powerful in all sorts of applications where precision and an exact solution are required. Proficiency in managing iterations is key to effectuating Newton-Raphson successfully.