Question

The Newton-Raphson method is used to find the root of the equation
\[ f(x) = e^{-x} - x \] If the initial guess for the root is 0, then the estimate of the root after the first iteration is ............

• A. 0.56
• B. -0.50
• C. 0.50
• D. -0.56

Hint:

The Newton-Raphson method is an iterative method to approximate the roots of a function. If the initial guess is close to the actual root, the method converges quickly.

Answer 1

The Correct Option is C

We are given the function: \[ f(x) = e^{-x} - x \] and the initial guess for the root is $x_0 = 0$. The Newton-Raphson method is given by the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] First, we need to compute the derivative of the function: \[ f'(x) = \frac{d}{dx}(e^{-x} - x) = -e^{-x} - 1 \] Now, we use the Newton-Raphson formula for the first iteration with $x_0 = 0$: \[ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \] Substitute the values into the formula: \[ f(0) = e^{-0} - 0 = 1 \text{and} f'(0) = -e^{0} - 1 = -1 - 1 = -2 \] Now, calculate $x_1$: \[ x_1 = 0 - \frac{1}{-2} = 0 + 0.5 = 0.50 \] Therefore, the estimate of the root after the first iteration is $0.50$.