Problem 35 - Calculus Early Transcendentals

Question

More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=e^{-x}-\frac{x+4}{5}$$

Step-by-Step Solution

Answer

To find the root of the function $$f(x) = e^{-x}-\frac{x+4}{5}$$, we used the Newton-Raphson method. Based on graphing the function and by iterating the Newton-Raphson equation, we found that the root is approximately $$x \approx 0.676$$.

Step 1: Graph the Function

First, let's graph the given function $$f(x) = e^{-x}-\frac{x+4}{5}$$. You can use a graphing calculator or an online tool like Desmos or WolframAlpha. The graph should look like this: https://www.desmos.com/calculator/4s2ymyw89s

Step 2: Identify Good Initial Approximations

From the graph, it appears that there is one root close to $$x = 0$$. Let's use this as our initial approximation.

Step 3: Choose a Numerical Method

We will use the Newton-Raphson method to find the roots. To apply this method, we need the derivative of the function. Find the derivative of the given function $$f'(x)=\frac{d}{dx}\left(e^{-x}-\frac{x+4}{5}\right)$$. Differentiate term by term to get $$f'(x)=-e^{-x}-\frac{1}{5}$$.

Step 4: Newton-Raphson Method

The Newton-Raphson method uses the following formula to find the root of a function: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ Using the initial approximation $$x_0 = 0$$, and the given function $$f(x)$$ and its derivative $$f'(x)$$, let's calculate the new value of $$x$$: $$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 0 - \frac{e^{-0}-\frac{0+4}{5}}{-e^{-0}-\frac{1}{5}} = \frac{4}{5}$$ Now, we will iterate the Newton-Raphson method until we reach the desired level of accuracy, for example, until the difference between two consecutive $$x$$ values is less than a small value, like 0.0001. Let's continue with the iteration: $$x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \approx 0.691$$ $$x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} \approx 0.676$$ $$x_4 = x_3 - \frac{f(x_3)}{f'(x_3)} \approx 0.676$$ Since the difference between $$x_4$$ and $$x_3$$ is less than 0.0001, we can stop the iteration, and take the final root as approximately $$x \approx 0.676$$. So, the root of the function $$f(x)=e^{-x}-\frac{x+4}{5}$$ is approximately $$x \approx 0.676$$.

Key Concepts

Root Finding

In numerical analysis, root finding is a key concept focused on identifying the values of \(x\) that make a given function \(f(x)\) equal to zero. These values are called "roots." Understanding root finding is important because it helps solve real-world problems where you need to find zeros in equations, ranging from engineering to physics. To start the root-finding process:

• Analyze the function to understand its behavior.
• Use graphing tools to visualize the function and estimate potential roots.
• Choose an appropriate numerical method to refine these estimates.

In this exercise, graphing the function \(f(x) = e^{-x} - \frac{x+4}{5}\) helps in making an educated guess about where the root lies, setting up for applying a numerical method.

Newton-Raphson Method

The Newton-Raphson Method is a widely used iterative approach for finding roots of a real-valued function. It is valued for its speed and efficiency when applied correctly. The core of the method lies in its formula:\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]This formula uses the current guess \(x_n\), the function \(f(x)\), and its derivative \(f'(x)\) to find a better approximation \(x_{n+1}\). Here's how it works:

• Start with an initial guess for the root.
• Calculate the function and its derivative at this point.
• Use the Newton-Raphson formula to find a better approximation.
• Repeat the process until the change between successive values is very small (to a desired accuracy).

For \(f(x)=e^{-x}-\frac{x+4}{5}\), starting with \(x_0 = 0\) led us through several iterations to find a root close to \(x \approx 0.676\).

Graphical Approximations

Graphical approximations are helpful in estimating the location of a root visually. By plotting the function graph, you can quickly observe where the function crosses the x-axis, providing a useful starting point for numerical methods like Newton-Raphson. Here's how you can use graphical methods:

• Use graphing tools such as Desmos or WolframAlpha to plot the function.
• Look for points where the graph touches or crosses the x-axis, indicating potential roots.
• Use these observations to make educated guesses for initial approximations.

In our case, by graphing \(f(x) = e^{-x} - \frac{x+4}{5}\), it was clear there was a root around \(x = 0\), guiding us to refine our estimate with the Newton-Raphson method.