Student Conversations

This activity is a good one for students to practice presenting different parts of their solutions. Valuable conversations that can be add on an individual group basis or as part of the presentations are:

• Does the“height” of the surface actually represent height or some other quantity?
• What are several different ways of observing whether a particular partial derivative is positive, negative, or zero?
• What are the advantages of the measuring tool? What are the limitations?
• How can you tell qualitatively when a partial derivative is large vs. small?
• Some students may conflate the value of the function with its derivative. Others may think that the derivative is large when the slope is changing by a lot. Ask students to distinguish between these concepts.
• If you are using handouts for the contour maps, students may discover that the contour maps can be folded to make an approximation of a surface. This can be tricky for the third contour map (using something made of cloth and using multiple people can help!).

On your Marks: Torches heat a thin \(10^{\prime\prime} \times 10^{\prime\prime}\) aluminum plate between \(100^\circ F\) and \(106^\circ F\). Your surface represents the plate's temperature. Label two points on the surface for each condition below:

1. \(\frac{\partial T}{\partial x} < 0\) and \(\frac{\partial T}{\partial y} < 0\)
2. \(\frac{\partial T}{\partial x} > 0\) and \(\frac{\partial T}{\partial y} < 0\)
3. \(\frac{\partial T}{\partial x} < 0\) and \(\frac{\partial T}{\partial y} > 0\)
4. \(\frac{\partial T}{\partial x} = 0\) and \(\frac{\partial T}{\partial y} > 0\)

Get Set: Pick a point on your surface satisfying the second condition above. Using the measurement tool, find the rates \(\frac{\partial T}{\partial x}\) and \(\frac{\partial T}{\partial y}\) at your point. (1 vertical inch = \(1^\circ F\).) Include units.

\[\frac{\partial T}{\partial x} = \underline{\hspace{3cm}} \hspace{2cm} \frac{\partial T}{\partial y}= \underline{\hspace{3cm}}\]

Go: For the contour map below, rank the points based on the value of \(\frac{\partial T}{\partial x}\) or \(\frac{\partial T}{\partial y}\).

Go: For each contour map below, rank the points based on the value of \(\frac{\partial T}{\partial x}\) or \(\frac{\partial T}{\partial y}\) at each point.

\(\hspace{0.25in}\) \(\hspace{0.25in}\)

\(\frac{\partial T}{\partial x}\) Neg. \(\underline{\hspace{.3in}}~ \underline{\hspace{.3in}}~ \underline{\hspace{.3in}}\) Pos. \(\hspace{1in}\) \(\frac{\partial T}{\partial x}\) Neg. \(\underline{\hspace{.3in}}~ \underline{\hspace{.3in}}~ \underline{\hspace{.3in}}\) Pos. \(\hspace{1in}\) \(\frac{\partial T}{\partial x}\) Neg. \(\underline{\hspace{.3in}}~ \underline{\hspace{.3in}}~ \underline{\hspace{.3in}}\) Pos.
\(\frac{\partial T}{\partial y}\) Neg. \(\underline{\hspace{.3in}}~ \underline{\hspace{.3in}}~ \underline{\hspace{.3in}}\) Pos. \(\hspace{1in}\) \(\frac{\partial T}{\partial y}\) Neg. \(\underline{\hspace{.3in}}~ \underline{\hspace{.3in}}~ \underline{\hspace{.3in}}\) Pos. \(\hspace{1in}\) \(\frac{\partial T}{\partial y}\) Neg. \(\underline{\hspace{.3in}}~ \underline{\hspace{.3in}}~ \underline{\hspace{.3in}}\) Pos.

Copyright 2014 by The Raising Calculus Group