Are the following statements true or false?
? 1. If \(\vec{u}\) is a unit vector, then \(D_\vec{u}f(a, b)\) is a vector.
? 2. The gradient vector \(\nabla f(a, b)\) is tangent to contour curves of \(f\) at \((a, b)\).
? 3. \(D_\vec{u}f(a, b)\) is parallel to \(\vec{u}\).
? 4. \(\nabla f(a, b)\) must be a vector in 3-dimensional space.
? 5. Suppose \(f_x(a, b)\) and \(f_y(a, b)\) both exist and are continuous. Then there is always a direction in which the rate of change of \(f\) at \((a, b)\) is zero.
? 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.
? 7. \(D_\vec{u}f(a, b) = ||\nabla f(a, b)||\).
? 8. If \(\vec{u}\) is perpendicular to \(\nabla f(a, b)\), then \(D_\vec{u}f(a, b) = (0, 0)\).
