Question

Fixed Point In Exercises 23-26, approximate the fixed point of the function to two decimal places. [A fixed point \(x_{0}\) of a function \(f\) is a value of \(x\) such that \(\left.f\left(x_{0}\right)=x_{0} .\right]\) $$ f(x)=e^{x / 10} $$

Short answer

The fixed point of the function \(f(x) = e^{x / 10}\) to two decimal places is approximately \(x = 1.58\).

Step-by-step solution

Initial Guess

Start with an initial guess for a value of \(x\). This initial guess can be arbitrary, but it is better if it is close to the actual solution. Let's start with \(x = 0\).

Iteration Process

Compute the function \(f(x)\) for our initial \(x\). Our new \(x\) value will be \(f(x)\) which would be \(e^{0 / 10} = 1\). Therefore, our new estimate for \(x\) is \(x=1\). Now replace \(x=1\) into the equation and repeat this process until the value of \(x\) doesn't change by a significant amount.

Further Iteration

For the next iterations, the value of \(x\) are as follows - \(\begin{array}{ll} \text { Iteration } & x \ 1 & 1 \ 2 & e^{1 / 10} \ 3 & e^{e^{1 / 10} / 10} \ 4 & e^{e^{e^{1 / 10} / 10} / 10} \end{array}\). Continue this process until the value of \(x\) stops changing to two decimal places.

Convergence

After several iterations, the solution won't change by more than a hundredth place, i.e., the procedure will continue to produce 1.58 in the hundredths digit. Hence, the fixed point of the function \(f(x) = e^{x / 10}\) to two decimal places is approximately \(x = 1.58\).

Key concepts

Fixed Point Iteration

Fixed point iteration is a method used in numerical analysis to find a point, referred to as a 'fixed point,' where a function maps back to itself. In simpler terms, it's looking for an answer that doesn't change when you apply the function. Imagine performing the same action repeatedly and ending up at the same point where you started—this is what fixed point iteration seeks to find.

Convergence of Sequences

Convergence is crucial to the concept of fixed point iteration. It's the property that ensures the process leads you closer to the fixed point with each iteration. A sequence is convergent if the numbers in the sequence get closer and closer to a single number, called the 'limit'. In our case, the iterations of applying the function repeatedly should ideally converge to the fixed point. When the sequence no longer changes significantly, e.g., differences are less than 0.01, we say it has converged.

Exponential Functions

Exponential functions data rapidly increasing (or decreasing) phenomena. The function we're dealing with, f(x) = e^{x / 10}, is an exponential function where the base e (Euler's number, approximately 2.71828) is raised to a power that is a fraction of x. These functions are unique as their rate of change is proportional to the value of the function itself—meaning they change faster as they grow larger. In our case, it affects how quickly the sequence of fixed point iterations approaches its limit.