Question 1:
Consider the 2x2 symmetric matrix
$$ A = \left[ \begin{array}{cc} a & b; \ c & d \end{array} \right] $$
and the 2x1 vector
$$ x = \left[ \begin{array}{c} u; \ v \end{array} \right]. $$
(Here a semicolon indicates that the following entries belong to the next row of the matrix.) Show that the derivative of $x^T A x$ with respect to $x$ can be expressed as
$$ x^T A + A^T x. $$
(Hint: write out the derivatives of the scalar quantity $x^T A x$ with respect to the individual components $u$ and $v$, then think about how you might express these two scalar quantities in terms of matrix-vector operations.) How can the derivative be simplified in the special case where $A$ is symmetric ($A^T = A$)? Can you argue that this expression for the derivative works for any $n \times n$ matrix (not just 2x2)?
Question 2:
Suppose you are given a triangle in the plane as three vertices $p_1, p_2, p_3$. Write an algorithm for determining whether a given query point $x$ is in the triangle. (You can write this in pseudocode, or "real" code; however you like!)