
-
Title: Proof of the Fundamental Theorem of Calculus
-
Series: Real Analysis
-
Chapter: Riemann Integral
-
YouTube-Title: Real Analysis 56 | Proof of the Fundamental Theorem of Calculus
-
Bright video: Watch on YouTube
-
Dark video: Watch on YouTube
-
Ad-free video: Watch Vimeo video
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Exercise Download PDF sheets
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: ra56_sub_eng.srt missing
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Timestamps (n/a)
-
Subtitle in English (n/a)
-
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: $2\exp(x) - \exp(1)$
-
Last update: 2025-01