This concept combines the second-degree Taylor (quadratic) approximation formula with the Second Fundamental Theorem of Calculus to express the local quadratic approximation of an accumulation function F(x) = ∫₀ˣ f(t) dt near x = 0 entirely in terms of the integrand f and its derivative. It relies on the formal definitions of the quadratic approximation (F(x) ≈ F(0) + F′(0)x + F″(0)x²/2), the Second Fundamental Theorem of Calculus (F′(x) = f(x)), and the additional differentiability assumption on f required to obtain F″. This belongs to single-variable calculus, bridging the theory of local polynomial approximation with the fundamental theorem relating integration and differentiation.
This concept combines the second-degree Taylor (quadratic) approximation formula with the Second Fundamental Theorem of Calculus to express the local quadratic approximation of an accumulation functi…