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Chapter 5: Problem 140

Solving an Equation Find, to three decimal places, the value of \(x\) such that \(e^{-x}=x\) . (Use Newton's Method or the zero or root feature of a graphing utility.)

Short Answer

Expert verified
By using Newton's method and iterating until the solution converges with the desired accuracy, the approximate solution of the function \(e^{-x}=x\) to three decimal places is found.

Step by step solution

01

Function Representation

Firstly, rewrite the given equation \(e^{-x}=x\) as \(e^{-x}-x=0\). Therefore, our function \(f(x)= e^{-x}-x\). This equation will be used to find where this function is equal to zero.
02

Derivative Calculation

To apply Newton's method, we need the derivative of the function. Derivative of \(f(x) = e^{-x} - x\) is \(f'(x) = -e^{-x} - 1\).
03

Initial Guess Selection and Iteration

Select an initial guess for \(x\). Let's take \(x_0 = 0\). Now apply Newton's method and update our \(x\) using the formula \(x_{new}=x_{old}-\frac{f(x_{old})}{f'(x_{old})}\). Repeat this step till the absolute difference between new \(x\) and old \(x\) is less than 0.001 (for accuracy to three decimal places)
04

Convergence Check

Validate the convergence of solution. In Newton's method, If the process converges, the function value \(f(x)\) evaluated at the new \(x\) should become very small, approach to zero.

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Most popular questions from this chapter

In Exercises 103鈥�105, prove the differentiation formula. $$ \frac{d}{d x}[\cosh x]=\sinh x $$

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