• Derivatives as ratios of small changes

• Derivatives as measurable quantities determined experimentally

• Find $\frac{dx}{dF_{x}}$

Estimated Time:

• Knowledge of derivatives from previous math and physics courses

• Tabletop Whiteboard with markers
• “Derivative Machine” (Modified PDM)

This activity serves as an introduction to thinking about how to experimentally measure a derivative. The machine, which is a modified Partial Derivative Machine, has a spring system which is connected to two strings, one of which is clamped down in order to create a nonlinear one-dimensional machine. The other string has a marker for measuring the position and a hanger for attaching various masses. The derivative, $\frac{dx}{dF_{x}}$, is then measured by students using this machine. This derivative can be measured by taking a ratio of small changes of the position with respect to the mass placed on the hanger, $\frac{dx}{dF_{x}}\approx\frac{\Delta x}{\Delta F_{x}}$, where the change in the masses, $\Delta F_{x}$, is sufficiently small.

• Constant Derivative: Students may assume that this derivative will be constant for the system if they are able to see that the system consists of springs. This may be because during introductory physics courses, students are introduced to Hooke's law–that the force in a linear spring system is directly proportional to the change in position from the equilibrium point of the spring. However, this derivative is not constant for most of these systems which indicates that these are nonlinear systems where Hooke's law does not apply.

• “Small enough” $\Delta F_x$: Students may choose changes in mass which are too small or too large for an accurate estimate of the derivative. If students place 10g or less for each increase (or decrease) in $F_{x}$, there may not be measurable changes in the position, $x$, which will result in an inaccurate measurement of the derivative at that point. Choosing a step size of 100g or more for each increase (or decrease) in $F_{x}$ will cause a change in position, $x$, however, the step size is likely too large to provide an accurate measurement of the derivative. Since the derivative is the (instantaneous rate of change), measurements of a derivative should should use large enough increments of $F_{x}$ provide measurable changes but small enough increments of $F_{x}$ to provide a “narrow” estimate.

• Use of mass or force for $F_x$: $F_{x}$ typically represents a force in physics, however, the provided tools for measuring this derivative are masses. Weight, a force, is the mass multiplied by the acceleration due to gravity, $F_{weight}=ma$. The acceleration due to gravity varies at each location on Earth although is typically given in physics courses as $-9.8m/s^{2}$, however, this is an estimate and is not necessarily the correct value for the location that this experiment is completed. The masses are more precisely measured, and therefore, it is more accurate to use mass for $F_{x}$ calculations than it would be to use weight. It is not incorrect to use weight for $F_{x}$, but using mass will provide easier and more accurate measurements of the derivative.

• Identifying Dependent and Independent Variables: Which is the dependent and independent variable? (Do these correspond to intensive and extensive variables?)

• Is $\frac{dx}{dF_{x}}=\frac{1}{\frac{dF_{x}}{dx}}$? Can we only measure accurately, $\frac{dx}{dF_{x}}$ and not, $\frac{dF_{x}}{dx}$?

• Is this measuring the derivative or an approximation of the derivative?

A whole class discussion of this activity can follow about measuring derivatives and different representations of derivatives.