Quadric surfaces

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We investigate quadric surfaces.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Consider $F(x,y)= 39 - 30 x + 5 x^2 + 2 y + y^2$ . Here is a plot of the quadric surface $z=F(x,y)$ :

Here we see a contour plot for $z = F(x,y)$ with the same domain:

Here we see a table of values for $z= F(x,y)$ with the same domain: $\renewcommand *{\arraystretch }{1.5} \begin {array}{c|c|c|c|c|c|c|c|}\hline \rule {.5cm}{0cm} & 77 & 42 & 17 & 2 & -3 & 2 & 17 \\ \hline & 74 & 39 & 14 & -1 & -6 & -1 & 14 \\ \hline & 73 & 38 & 13 & -2 & -7 & -2 & 13 \\ \hline & 74 & 39 & 14 & -1 & -6 & -1 & 14 \\ \hline & 77 & 42 & 17 & 2 & -3 & 2 & 17 \\ \hline & 82 & 47 & 22 & 7 & 2 & 7 & 22 \\ \hline & 89 & 54 & 29 & 14 & 9 & 14 & 29 \\ \hline & & & & & & & \end {array}$ The $x$ -coordinates should run along the bottom of this table and the $y$ coordinates should run along the left-hand side. Fill them in.

Estimate the height of each of the level curves in the contour plot for $F(x,y)$ above. Write the heights directly on the contour plot.

Pencil-in some gradient vectors on your set of level curves.

Where is the magnitude of the gradient vector large? Where is the magnitude of the gradient small?

Find $(x,y)$ such that $\grad F(x,y) = \vec {0}$ . Mark this position on the contour plot and on the table above.

Consider $G(x,y)= -7 - 2 x + x^2 - 12 y - 3 y^2$ . Here is a plot of the quadric surface $z=G(x,y)$ :

Here we see a contour plot for $z = G(x,y)$ with the same domain:

Here we see a table of values for $z= G(x,y)$ with the same domain:

$\renewcommand *{\arraystretch }{1.5} \begin {array}{c|c|c|c|c|c|c|c|}\hline \rule {.5cm}{0cm}& -19 & -22 & -23 & -22 & -19 & -14 & -7 \\\hline & -4 & -7 & -8 & -7 & -4 & 1 & 8 \\\hline & 5 & 2 & 1 & 2 & 5 & 10 & 17 \\\hline & 8 & 5 & 4 & 5 & 8 & 13 & 20 \\\hline & 5 & 2 & 1 & 2 & 5 & 10 & 17 \\\hline & -4 & -7 & -8 & -7 & -4 & 1 & 8 \\\hline & -19 & -22 & -23 & -22 & -19 & -14 & -7 \\\hline & & & & & & & \end {array}$

The $x$ -coordinates should run along the bottom of this table and the $y$ coordinates should run along the left-hand side. Fill them in.

Estimate the height of each of the level curves in the contour plot for $G(x,y)$ above. Write the heights directly on the contour plot.

Pencil-in some gradient vectors on your set of level curves.

Where is the magnitude of the gradient vector large? Where is the magnitude of the gradient small?

Find $(x,y)$ such that $\grad G(x,y) = \vec {0}$ . Mark this position on the contour plot and on the table above.