Q1: What is the correct derivative for $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = n , x^n$ for $n \in \mathbb{R}$?

A1: $f^\prime(x) = n x^n$

A2: $f^\prime(x) = n^2 x^n$

A3: $f^\prime(x) = n^2 x^{n-1}$

A4: $f^\prime(x) = n(n-1) x^{n-1}$

Q2: What is the correct derivative for the polynomial $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = x^5 + 3 x^2 + 2 x$?

A1: $f^\prime(x) = 5 x + 3 x^2 + 2$

A2: $f^\prime(x) = 5 x^4 + 6 x + 2$

A3: $f^\prime(x) = 5 x^4 + 6 x^2 + 2$

A4: $f^\prime(x) = 5 x^3 + 6 x + 2$

Q3: Let the power series $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \sum_{k=0}^\infty a_k x^k$. The radius of convergence is $\infty$. Which statement is not correct?

A1: For all $x \in \mathbb{R}$, the series $\sum_{k=0}^\infty a_k x^k$ is convergent.

A2: For all $x \in \mathbb{R}$, the series $\sum_{k=0}^\infty a_k x^k$ is absolutely convergent.

A3: The sequence of functions given by $g_n: \mathbb{R} \rightarrow \mathbb{R}$, $g_n(x) = \sum_{k=0}^n a_k x^k$ is uniformly convergent to $f$.

A4: For all $c \in \mathbb{R}$, the sequence of functions given by $g_n: [-c,c] \rightarrow \mathbb{R}$, $g_n(x) = \sum_{k=0}^n a_k x^k$ is uniformly convergent to $f$ on the interval $[-c,c]$.

Q4: Let the power series $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \sum_{k=0}^\infty a_k x^k$. The radius of convergence is $\infty$. Is $f$ differentiable?

A1: No, nowhere.

A2: Yes, but only for the point $x = 0$.

A3: Yes, everywhere.