Paper 1, Section II, D

Analysis I | Part IA, 2011

State and prove the Fundamental Theorem of Calculus.

Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be integrable, and set F(x)=0xf(t)dtF(x)=\int_{0}^{x} f(t) \mathrm{d} t for 0<x<10<x<1. Must FF be differentiable?

Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be differentiable, and set g(x)=f(x)g(x)=f^{\prime}(x) for 0x10 \leqslant x \leqslant 1. Must the Riemann integral of gg from 0 to 1 exist?

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