Considering surfaces

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We think about surfaces in different ways.

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

Considering tables

Let $F:\R ^2\to \R$ be a differentiable function that is roughly described by the following table of values:

Write the value of $F(-2,-1)$ .

Estimate $F^{(1,0)}(-2,-1)$ .

Estimate $F^{(0,1)}(-2,-1)$ .

Estimate $\grad F(-2,-1)$ .

If you were to leave the point $(-2,-1)$ in the direction of the gradient, what value would you find? Explain why this makes sense.

Use your computations above to estimate the formula for a plane tangent to $F(x,y)$ at the point $(-2,-1)$ .

Sketch the level curve $F(x,y) = 1$ on the table above.

Considering level sets

Consider the following contour plot for $z=G(x,y)$ :

What does a contour plot like this represent? Circle one:

A subset of $\R ^2$
A subset of $\R ^3$

Estimate the value of $G(0,-1)$ .

Estimate $G^{(1,0)}(0,-1)$ .

Estimate $G^{(0,1)}(0,-1)$ .

Estimate $\grad G(0,-1)$ .

If you were to leave the point $(0,-1)$ in the direction of the gradient, what value would you find? Explain why this makes sense.

Use your computations above to estimate the formula for a plane tangent to $G(x,y)$ at the point $(0,-1)$ .

Considering algebra

Let $H:\R ^2\to \R$ be described by: $H(x,y) = -1 + 2 x - x^2 - 12 y - 3 y^2$ As a gesture of friendship, we have included a graph $z = H(x,y)$ :

Compute:

(a): $H(-2,-1)$
(b): $H(0,-1)$

Compute:

(a): $H^{(1,0)}(-2,-1)$
(b): $H^{(0,1)}(-2,-1)$
(c): $H^{(1,0)}(0,-1)$
(d): $H^{(0,1)}(0,-1)$

Compare and contrast the functions $F$ , $G$ , and $H$ . Discuss.