Applications to Error Analysis

Calculus 3 · Axiom Academy

LESSON Taylor Series: Error Analysis Using Taylor expansions to estimate errors, propagate uncertainty through functions, and understand higher-order error terms in practical calculations. 1. Taylor Series and Truncation Error When we approximate a function f(x) near x = a using the first few terms of its Taylor series, the difference between the true value and our approximation is called the truncation error or remainder term R_n(x) . where R_n(x) is the remainder after n terms. For some c between a and x (Lagrange form): 2. Estimating Computational Errors Taylor series let us quantify how errors propagate through a calculation. Suppose we compute , where is a small measurement error. Keeping just the linear term turns the input error into an output error. Example: a square-root calculation We measure x = 4 , but with an uncertainty . What is the resulting error in ? Expand around x = 4 , where gives slope : For small the first-order (linear) term dominates — that is what makes so useful. Real problems have several uncertain inputs at once. The total differential is the multivariable version of : it adds up each variable's contribution to the output error. Each input error dx, dy is scaled by that variable's partial derivative, and the pieces add. The single-variable case is just one term: When the errors are independent and random (not worst-case), they combine in quadrature instead of adding directly: Physical example: a pendulum's period

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