Finite difference - Executable Engineering: Interactive Numerical Methods in Python

Approach from the secant¶

Numerical differentiation approximates the continuous derivative of a function using discrete function evaluations. Geometrically, the true tangent slope $f'(x_0)$ is the limiting slope of secant lines connecting $(x_0, f(x_0))$ to $(x_0 + h, f(x_0 + h))$ as the step size $h$ approaches zero:

f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

(1)

As the step size $h$ contracts toward zero, the secant line rotates continuously into the true analytical tangent line.

What issues arise in finite precision?

import numpy as np
try:
    import executable_engineering as exe
except ImportError:
    %pip install -q executable_engineering
    import executable_engineering as exe

def f_geom(x):
    return np.sin(x) + 0.1 * x**2

fig_secant = exe.plot_secant_tangent_limit(
    f_geom, x0=1.5, h_values=(1.2, 0.6, 0.3, 0.1), x_range=(0.2, 3.2),
    title="Secant to Tangent Convergence as Step Size h → 0"
)
fig_secant.show()