Forward, Backward, Central Differences

Numerical Analysis · Axiom Academy

LESSON Forward, Backward, and Central Differences Deriving the formulas from Taylor series The Taylor series expands a function around a point x: This gives us the exact value of f(x+h) in terms of f and its derivatives at x. The key is that derivatives appear naturally! Start with the Taylor expansion for f(x+h): 3 Deriving Backward Difference Now expand f(x-h) using Taylor series (substitute -h for h): Notice the error term has the same order O(h), but with opposite sign compared to forward difference! 4 Deriving Central Difference (The Magic!) The clever trick: subtract the f(x-h) expansion from f(x+h): Let's visualize how error decreases as h gets smaller:

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