Are the following statements true or false?

? True False  1. $f_{\vec{u}} \, (a,b) = || \nabla f(a,b) ||$.

? True False  2. $f_{\vec{u}} \, (a,b)$ is parallel to $\vec{u}$.

? True False  3. If $\vec{u}$ is a unit vector, then $f_{\vec{u}} (a,b)$ is a vector.

? True False  4. If $\vec{u}$ is perpendicular to $\nabla f(a,b)$, then $f_{\vec{u}} \, (a,b) = \langle 0, 0 \rangle$.

? True False  5. The gradient vector $\nabla f(a,b)$ is tangent to the contour of $f$ at $(a,b)$.

? True False  6. If $f(x,y)$ has $f_x(a,b) = 0$ and $f_y(a,b) = 0$ at the point $(a,b)$, then $f$ is constant everywhere.

? True False  7. $\nabla f(a,b)$ is a vector in 3-dimensional space.

? True False  8. Suppose $f_x(a,b)$ and $f_y(a,b)$ both exist. Then there is always a direction in which the rate of change of $f$ at $(a,b)$ is zero.