The Newton-Raphson method is applied to compute the root of $f(x) = x^3 - 2x - 5 = 0$. Starting with an initial estimate $x_0 = 2.0$, what is the value of the next iterate $x_1$?

Answer Key:

2.1

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Function & Derivative:** $f(x) = x^3 - 2x - 5$, $f'(x) = 3x^2 - 2$. 2. **Stage 2: Evaluation at $x_0 = 2$:** $f(2) = 2^3 - 2(2) - 5 = -1$; $f'(2) = 3(2^2) - 2 = 10$. 3. **Stage 3: Newton Update:** $x_1 = 2 - (-1 / 10) = 2.1$. 4. **Stage 4: Final Value:** 2.1.

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