In-class work for 26 June 2012
1.) Find the local maximum and minimum values and saddle points.
    a.) $f(x,y) = x^3y + 12x^2 - 8y$
    b.) $x^4 + y^4 - 4xy + 2$
    c.) $f(x,y) = xy + \frac{1}{x} + \frac{1}{y}$
    d.) $f(x,y) = e^x \cos(y)$
    e.) $f(x,y) = \sin(x) \sin(y)$

2.) Find the absolute maximum and absolute minimum on the set $D$.
    a.) $f(x,y) = 3 + xy - x - 2y$, $D$ is the closed triangular region with vertices $(1,0), (5,0)$, and $(1,4)$
    b.) $f(x,y) = xy^2$, $D = \{ (x,y) \colon x \geq 0, y \geq 0, x^2+y^2 \leq 3 \}$
    c.) $f(x,y) = x^3-3x-y^3+12y$, $D$ is the quadrilateral whose vertices are $(-2,3), (2,3), (2,2)$, and $(-2,-2)$.

3.) Find the shortest distance from the point $(2,1,-1)$ to the plane $x+y-z=1$.

4.) Find three positive numbers whose sum is $12$ and the sum of whose squares is as small as possible.

5.) A cardboard box without a lid is to have a volume of $32,000 cm^3$. Find the dimensions that minimize the amount of cardboard used.

6.) If the length of the diagonal of a rectangular box must be $L$, what is the largest possible volume? (note: this will be in terms of $L$!)

7.) Suppose that a scientist has reason to believe that two quantities $x$ and $y$ are related linearly, that is, $y=mx+b$, at least approximately, for values of $m$ and $b$. Experimental data comes in ordered pairs $(x_1,y_1), (x_2,y_2), \ldots, (x_n,y_n)$ and are plotted on the plane. The points do not lie exactly on a straight line, so the scientist wants to find constants $m$ and $b$ that fits the points as well as possible.

Let $d_i = y_i - (mx_i + b)$ be the vertical deviation ("signed distance") from the point $(x_i,y_i)$ to the line $y=mx+b$. The "method of least squares" is a way to determine the best values of $m$ and $b$ so as to minimize $\displaystyle\sum_{i=1}^n d_i^2$ (the sum of the squares of the deviations). Show that, according to this method, the line of best fit is obtained when $$m \displaystyle\sum_{i=1}^n x_i + bn = \displaystyle\sum_{i=1}^n y_i,$$ and $$m \displaystyle\sum_{i=1}^n x_i^2 + b \displaystyle\sum_{i=1}^n x_i = \displaystyle\sum_{i=1}^n x_i y_i.$$ The line of best fit is then found by solving these two equations in the unknowns $m$ and $b$.