Fundamental Theorem of Calculus Part 2

Theorem

1. If $a,b\in \mathbb{R}$, $a<b$
2. If $f$ is Continuous on $[a,b]$
3. Define $F(x)={\int }_a^{x}f(t)dt$ where $x\in [a,b]$ Then:

• $F(x)$ is continuous on $[a,b]$
• $F(x)$ is differentiable on $(a,b)$
• $\forall x\in [a,b],F^{\prime }(x)=f(x)$
  • Endpoint derivatives are one-sided

Proof

• FToC Part 2 Proof

Examples

• Derivatives of Functions Defined by Integrals