This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.

Students work in small groups to interpret level curves representing different concentrations of lead.

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This mini-lecture demonstrates the relationship between \(df\) on the tangent plane to its “components“ in coordinate directions, leading to the multivariable chain rule.

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.

Students draw the 3D graphs of equations using three variables. They make choices for drawing a stack of curves in parallel planes and a curve in a perpendicular plane (e.g. substituting in values for \(x\), \(y\), or \(z\). )

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.

This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the \(r\)-direction and derivatives in \(x\)- and \(y\)-directions using the chain rule.

This small group activity is designed to provide practice with the multivariable chain rule. Students determine a particular rate of change using given information involving other rates of change. The discussion emphasizes the equivalence of a variety of approaches, including the use of differentials. Good “review” problem; can also be used as a homework problem.

Students use chain rule diagrams to construct a multivariable chain rule in terms of differentials.