Taylor Series

The problem, from first principles

05-hessian computes a function's curvature at a point, but never connects that number to what it's actually for: predicting the function's value nearby without re-evaluating it. Taylor series is that connection — a way to approximate any smooth function, near a chosen point, using only its derivatives at that one point. It also explains, precisely, why gradient descent (08-gradient-descent) only ever needs the gradient: gradient descent is quietly using just the first-order Taylor approximation and discarding everything else.

From theory to code

Implement taylor_first_order(f, f_prime, a, x), the tangent-line approximation of f at x, centered at a, then taylor_second_order(f, f_prime, f_double_prime, a, x), adding the next term. The signatures and docstrings are already in the editor.

Constraints

• f, f_prime, f_double_prime are functions taking a single float and returning a float.
• a is the expansion point; x is where the approximation is evaluated (may equal a, or be arbitrarily far from it — accuracy degrades with distance, but the formula itself is defined everywhere).
• Both approximations must equal f(a) exactly when x == a.

Hints