This homework is due on 4/3.

    1. Suppose we have $n$ known points $(x_i,y_i,z_i)$, and we want to find a plane $z = Ax+By+C$ that best approximates the relationship by minimizing the sum of the squared distances. Determine $A$, $B$, and $C$ using $x_i$, $y_i$, and $z_i$.
    2. Calculate the best fit line for this data.
  2. Choose one of the two problems below, each of which involves proving famous inequalities: the Cauchy-Schwarz Inequality and the Arithmetic-Geometric Inequality.
    1. Maximize $\sum_{i=1}^nx_iy_i$ subject to the constraints $\sum_{i=1}^n x_i^2 =1$ and $\sum_{i=1}^ny_i^2=1$.
    2. Find the maximum value of $f(x_1,\ldots,x_n)=\sqrt[n]{x_1\cdots x_n}$.
    1. Find volume of the solid that enclosed by $x=0$, $y=0$, $z=0$, and the plane $Ax+By+Cz=D$ where $A$, $B$, $C$, and $D$ are positive.
    2. Use the above result to demonstrate that the volume of a trirectangular tetrahedron with three equal sides that meet at the right angle, each of length $a$, is $\frac{a^3}{6}$.