Mr. Integral
Math notes/Calculus I/The Derivative

Linear Approximation and Newton’s Method — Cheat sheet

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Core rule

L(x)=f(a)+f′(a)(x−a),L(x)=f(a)+f'(a)(x-a), xn+1=xn−f(xn)/f′(xn).x_{n+1}=x_n-f(x_n)/f'(x_n).

Watch for

A local approximation needs a nearby base point. Newton requires a nonzero derivative and can leave the domain or cycle. Verify the residual and use a bracket when available.