Activity: The Hot Plate

Prerequisite concepts

Differential Calculus The derivative is a function

One of Zandieh's process-object layers for derivatives is that derivatives are functions. This means that the value of a derivative depends on where in the domain of the function one is looking. The derivative of a function is itself a function, with the same domain as the original function. Both the (derivative) function and the value of the derivative at a point are commonly referred to as "the derivative."

"With respect to what" matters

Taking derivatives of the same function with respect to different variables will produce different results. In physics, these different results might have different dimensions and thus the derivative can have wildly different physical interpretations based off of "'with respect to' what".

The Hot Plate

The value of a partial derivative depends on the value(s) of what is held constant

FIXME: Is this really a statement about paths? I don't think we have path-related concepts.

There is a partial derivative in every direction at any point There is a tangent line in every direction at every point The magnitude of the gradient is proportional to the density of contour lines

Instructor guide mvhotplate

The Hot Plate

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.

Representations used

$\frac{\partial f}{\partial x}$

Contour Maps

Surface

Inclinometer

Concepts taught

There is a partial derivative in every direction at any point

For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.

There is a tangent line in every direction at every point

For a smooth, two dimensional function, a tangent line exists at any point and in any direction.

The value of a partial derivative depends on the value(s) of what is held constant

FIXME: Is this really a statement about paths? I don't think we have path-related concepts.

The magnitude of the gradient is proportional to the density of contour lines

Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.