5.3 The Fundamental Theorem of Calculus

Lecture

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Lecture notes

1. FTC #1

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\left[\int_{a(x)}^{b(x)}f(t)\,dt\right]=b'(x)\cdot f(b(x))-a'(x)\cdot f(a(x))}

2. FTC #2

$\int _{a}^{b}f(x)\,dx=F(b)-F(a)$
Where ${\frac {d}{dx}}\left[F(x)\right]=f(x)$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x)} is called the antiderivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x)}

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