Numerical Methods

If $f(a)$ and $f(b)$ have opposite signs and $f$ is continuous, there's a root between $a$ and $b$.

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Converges quickly but can fail with poor starting point or $f'(x_n) \approx 0$.

Example

$f(x) = x^3 - 2$. $f'(x) = 3x^2$. $x_0 = 1$.

$x_1 = 1 - \frac{-1}{3} = 1.333...$ $x_2 = 1.333 - \frac{0.370}{5.333} \approx 1.264$

Converges to $\sqrt[3]{2} \approx 1.2599$.

$\int_a^b f(x)\,dx \approx \frac{h}{2}[y_0 + 2(y_1 + y_2 + ... + y_{n-1}) + y_n]$

where $h = \frac{b-a}{n}$.

When It Works Well

Better with more strips. Overestimates for concave-up curves, underestimates for concave-down.