List of practice Questions

Let \( (x, y) \in \mathbb{R}^2 \). The rate of change of the real-valued function \[ V(x, y) = x^2 + x + y^2 + 1 \] at the origin in the direction of the point \( (1, 2) \) is __________ (round off to the nearest integer).

Consider ordinary differential equations given by \[ \frac{dx_1(t)}{dt} = 2x_2(t), \quad \frac{dx_2(t)}{dt} = r(t) \] with initial conditions \( x_1(0) = 1 \) and \( x_2(0) = 0 \). If \[ r(t) = \begin{cases} 1, & t \geq 0 \\ 0, & t < 0 \end{cases} \] then at \( t = 1 \), \( x_1(t) = \underline{2cm} \) (round off to the nearest integer).