Question

The estimate for the root of the function \( f(x) = e^{2x} + 2x \) after one iteration with an initial guess of \( x_0 = 0 \), using the Newton-Raphson method is \(\underline{\hspace{2cm}}\) (correct up to two decimal places).

Hint:

For the Newton-Raphson method, remember to compute both the function value and its derivative at the current guess to find the next approximation.

Answer 1

Correct Answer: -0.25

The Newton-Raphson formula is given by:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Where \( f(x) = e^{2x} + 2x \) and we need to calculate the first derivative \( f'(x) \):
\[ f'(x) = 2e^{2x} + 2 \] Now, using the initial guess \( x_0 = 0 \), we first calculate \( f(0) \) and \( f'(0) \):
\[ f(0) = e^{2 \times 0} + 2(0) = 1 \] \[ f'(0) = 2e^{2 \times 0} + 2 = 2 + 2 = 4 \] Now apply the Newton-Raphson formula:
\[ x_1 = 0 - \frac{1}{4} = -0.25 \] Thus, the estimate for the root is \( \boxed{-0.25} \).