5.3 The Fundamental Theorem of Calculus: Difference between revisions
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: <math>\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))</math><br><br> | : <math>\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))</math><br><br> | ||
:2. FTC #2 <br> | :2. FTC #2 <br> | ||
: <math>\int_{a}^{b}f(x)dx=F(b)-F(a)</math><br> Where <math>\frac{d}{dx}\left[F(x)\right]=f(x)</math>. <math>F(x)</math> is called the <b>antiderivative</b> of <math>f(x)</math> <br> | : <math>\int_{a}^{b}f(x)\,dx=F(b)-F(a)</math><br> Where <math>\frac{d}{dx}\left[F(x)\right]=f(x)</math>. <math>F(x)</math> is called the <b>antiderivative</b> of <math>f(x)</math> <br> | ||
==Solutions== | ==Solutions== | ||
Revision as of 17:54, 25 August 2022
Lecture
Lecture notes
- 1. FTC #1
- Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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
- 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 \int_{a}^{b}f(x)\,dx=F(b)-F(a)}
Where 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[F(x)\right]=f(x)} . 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)}
Solutions
Mr. V solutions: 8, 20, 28
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