Q1: Let $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ and $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be two totally differentiable functions where $f\circ \gamma$ is constant. What is the derivative $\frac{d}{dt}( f(\gamma(t)))$?

A1: 0

A2: 1

A3: One needs more information.

Q2: Let $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ be given by $\gamma(t) = \binom{t}{t}$. How does the curve look in $\mathbb{R}^2$?

A1: It’s a straight line.

A2: It’s a circle.

A3: It’s a parabula.

Q3: Let $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ be given by $\gamma(t) = \binom{t}{t}$. Assume that the image of $\gamma$ lies on a contour line of a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. What can one say about the gradient $\mathrm{grad}f(\gamma(t))$?

A1: It’s perpendicular to the vector $\binom{1}{1}$.

A2: It’s perpendicular to the vector $\binom{1}{0}$.

A3: It’s parallel to the vector $\binom{1}{0}$.

A4: It’s parallel to the vector $\binom{1}{1}$.