This small group activity using surfaces introduces a geometric interpretation of partial derivatives in terms of measured ratios of small changes. Students work in small groups to identify locations on their surface with particular properties. The whole class wrap-up discussion emphasizes the equivalence of multiple representations of partial derivatives.
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:
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:
- \(\frac{\partial T}{\partial x} < 0\) and \(\frac{\partial T}{\partial y} < 0\)
- \(\frac{\partial T}{\partial x} > 0\) and \(\frac{\partial T}{\partial y} < 0\)
- \(\frac{\partial T}{\partial x} < 0\) and \(\frac{\partial T}{\partial y} > 0\)
- \(\frac{\partial T}{\partial x} = 0\) 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.
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