Solve any maths problem

Step by step

1. f(x) = e^{x} - 3,\quad x_0 = 1

   Newton's method: x_{n+1} = x_n − f(x_n)/f′(x_n) — follow the tangent line down to the axis, repeat.

2. f'(x) = e^{x}

   Differentiate once; the same derivative serves every iteration.

3. x_{1} = 1.0 - \frac{-0.28171819}{2.71828175} = 1.10363832

   Iteration 1.

4. x_{2} = 1.10363832 - \frac{0.01511606}{3.01511621} = 1.0986249

   Iteration 2.

5. x_{3} = 1.0986249 - \frac{3.783 \cdot 10^{-5}}{3.00003767} = 1.09861229

   Iteration 3.

6. x_{4} = 1.09861229 - \frac{0.0}{3.0} = 1.09861229

   Iteration 4.

7. x_{5} = 1.09861229 - \frac{0.0}{3.0} = 1.09861229

   Iteration 5.

8. \text{converged}

   The iterates have stopped changing to 12 decimal places.

9. x \approx 1.09861229

   Quadratic convergence: the number of correct digits roughly doubles each step.