Question

More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=\frac{x}{6}-\sec x \quad \text { on } \quad[0,8]$$

Short answer

a) Bisection method b) Newton's method c) Secant method d) Fixed-point iteration Answer: b) Newton's method

Step-by-step solution

Graph the functions \(g(x)=\frac{x}{6}\) and \(h(x)=\cos x\)

Sketch these functions for the given range [0,8]. We will find the roots of the function \(f(x)\) (the intersection points), and refine our approximations in the next steps.

Identify intersections in the range [0,8]

Check for in the graph, where \(g(x)\) intersects \(h(x)\). Select initial approximations for the intersecting points by visually observing the graph.

Apply a root-finding algorithm

Choose a root-finding algorithm such as the bisection method or Newton's method to refine the approximations of the intersections. To use Newton's method: - Start with the approximations from Step 2 - For each approximation, iterate the Newton's method formula: $$x_{new} = x_{old} - \frac{f(x_{old})}{f'(x_{old})},$$ where: $$f(x)=\frac{x}{6}-\sec x$$ and $$f'(x) = \frac{1}{6} + \sec x\tan x$$ Iterate until the desired level of accuracy is reached.

State the roots of the function within the interval [0,8]

After applying the root-finding algorithm, report the final roots of the function \(f(x)\) within the interval [0,8]. These roots correspond to the points where \(f(x) = 0\) (the intersections of \(g(x)\) and \(h(x)\)).

Key concepts

Graphical Analysis

Graphical analysis is a vital tool in understanding the behavior of functions and finding their roots. It involves sketching the function on a coordinate plane to visually identify points of interest such as intercepts, peaks, troughs, and particularly, points where the function crosses the x-axis, which are the roots. In the context of the exercise, this means plotting both the functions \(g(x)=\frac{x}{6}\) and \(h(x)=\cos x\) for the interval \[0,8\].

To find the roots of \(f(x)=\frac{x}{6}-\sec x\), one needs to locate the intersection points of \(g(x)\) and \(h(x)\) on the graph, since these points satisfy \(g(x)=\cos x\), which is equivalent to \(f(x)=0\). Graphical analysis provides a preliminary estimate for these roots, which will later be refined through numerical methods. Visual observation of the graph can reveal not just the approximate locations of the roots, but also give insight into the number of roots to expect within the given range.

Newton's Method

Newton's method, also known as Newton-Raphson method, is a powerful numerical technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. Starting from an initial guess, which could be given by graphical analysis, the method iterates a process that converges to the actual root.

The iteration involves computing a sequence of approximations using the formula: \[x_{\text{new}} = x_{\text{old}} - \frac{f(x_{\text{old}})}{f'(x_{\text{old}})}\], where \(f'(x)\) is the derivative of the function. For the function \(f(x)\) in the exercise, this is \(f'(x) = \frac{1}{6} + \sec x\tan x\). With every iteration, \(x_{\text{old}}\) is updated to \(x_{\text{new}}\) until the value of \(f(x)\) at the updated point is sufficiently close to zero, indicating that we have found a root.

It's crucial to note that Newton's method works well if the initial approximation is close to the actual root and if the function behaves nicely (is differentiable and its derivative doesn't equal zero at the root). If these conditions are not met, the method might fail to converge or might converge to the wrong root.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in mathematics, describing properties of angles and the relationships between the sides of triangles. In calculus, these functions often appear in many contexts, including root finding problems. The function \(f(x)\) in our exercise features the secant function, \(\sec x\), which is the reciprocal of the cosine function.

Understanding the behavior of trigonometric functions is critical when dealing with root finding. For example, knowing that \(\cos x\) varies between -1 and 1 can inform us that \(\sec x\), having values that are the reciprocals of those of \(\cos x\), will display vertical asymptotes where \(\cos x = 0\). These characteristics must be taken into account when graphing these functions and when applying numerical methods like Newton's method, as these features can affect convergence and the accuracy of our root approximations.

When tackling calculus problems involving trigonometric functions, it's essential to recall identities, inverse functions, periodicity, and their derivatives. These properties all come into play in understanding and solving for the roots of trigonometric equations.