Q1: What is not a correct definition of $f^\prime(x_0)$ for a function $f: \mathbb{R} \rightarrow \mathbb{R}$?

A1: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0}$.

A2: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x_0) - f(x)}{x_0 - x}$.

A3: $\displaystyle \lim_{z \rightarrow x_0} \frac{f(x_0) - f(z)}{x_0 - z}$.

A4: $\displaystyle \lim_{x_0 \rightarrow x} \frac{f(x_0) - f(x)}{x - x_0}$.

A5: $\displaystyle \lim_{n \rightarrow \infty} \frac{f(x_n) - f(x_0)}{x_n - x_0}$ if we get the same value for each sequence $x_n \xrightarrow{n \rightarrow \infty} x_0$.

Q2: Which of the following functions is not a linear function (= affine function = linear polynomial) $f: \mathbb{R} \rightarrow \mathbb{R}$?

A1: $f(x) = 3 x + 8$

A2: $f(x) = 3x$

A3: $f(x) = 8$

A4: $f(x) = 0$

A5: $f(x) = |x|$

Q3: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = |x|$ and $x_0 = 1$. What is $f^\prime(x_0)$?

A1: 0

A2: -1

A3: 1

A4: It does not exist.

Q4: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = |x|$ and $x_0 = 0$. What is $f^\prime(x_0)$?

A1: 0

A2: -1

A3: 1

A4: It does not exist.