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: $2\exp(x) - \exp(1)$