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Title: Proof of the Fundamental Theorem of Calculus
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Series: Real Analysis
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Chapter: Riemann Integral
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YouTube-Title: Real Analysis 56 | Proof of the Fundamental Theorem of Calculus
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra56_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: [a,b] \rightarrow \mathbb{R}$ be a continuous function. What is the claim of the mean value theorem of integration?
A1: There is $\hat{x} \in [a,b]$ with $\int_a^b f(x) dx = f(\hat{x})$.
A2: There is $\hat{x} \in [a,b]$ with $\int_a^b f(\hat{x}) dx = f(\hat{x})$.
A3: There is $\hat{x} \in [a,b]$ with $\int_a^b f(x) dx = f(\hat{x}) (b-a)$.
A4: There is $\hat{x} \in [a,b]$ with $\int_a^b f(\hat{x}) dx = f(\hat{x})(a-b)$.
Q2: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function and $F: \mathbb{R} \rightarrow \mathbb{R}$ be given by $$ F(x) = \int_a^x f(t) dt ,.$$ What is correct for given real numbers $a,x$ and $h>0$?
A1: There is $\hat{x} \in [a,x]$ with $F(x+h) = F(\hat{x})$.
A2: There is $\hat{x} \in [x,x+h]$ with $F(x+h) - F(x) = f(\hat{x}) \cdot h$.
A3: There is $\hat{x} \in [a,x-h]$ with $F(x+h) - F(x) = f(\hat{x}) \cdot h$.
A4: There is $\hat{x} \in [x,x+h]$ with $F(x+h) - F(x) = f(\hat{x})$.
Q3: What is the integral $\int_1^x \exp(t) , dt$?
A1: $\exp(x) - 1$
A2: $\exp(x) - \exp(1)$
A3: $\exp(x)$
A4: $2\exp(x) - \exp(1)$
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Last update: 2025-01
