# Concept derivation guides

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Concept guide



## [Fundamental theorem of calculus: proving G′=f from continuity, then ∫_a^b f=F(b)−F(a)](/en/derivations/derivation-mko-fundamental-theorem-calculus-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-fundamental-theorem-calculus)

Let f:ℝ→ℝ be continuous on all of ℝ, a∈ℝ, and G(x)=∫_a^x f(t)dt an oriented integral. (1) Prove that G is differentiable at every point with G′(x)=f(x); (2) if F:ℝ→ℝ is differentiable with F′=f, prove ∫_a^b f(t)dt=F(b)−F(a) for arbitrary real a, b; (3) with f(t)=6t²−2t+1 and the candidate F(x)=2x³−x²+x+1/2, compute ∫_2^{−1} f(t)dt, and explain why G(2)=0 does not give G′(2)=0 and why bounds coinciding at x=0 do not give a zero derivative.

Fundamental theorem of calculus: continuity and oriented endpoints

derivation-mko-fundamental-theorem-calculus-01





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