• Derivatives as ratios of small changes

• Derivatives as measurable quantities determined experimentally

Estimated Time:

• Find $\frac{dx}{dF_{x}}$

• Knowledge of derivatives from previous math and physics courses

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

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. Covering the system may encourage students to test whether the derivative is constant.

• “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 10 grams 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 100 grams 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, $F_{weight}=m\ a_g$. The acceleration due to gravity varies at each location on Earth although is typically given in physics courses as $-9.8\ \text{m}/\text{s}^{2}$, 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: It may be natural for students to think about $x$ as their independent variable and $F_x$ as their dependent variable, as would be typical in a mathematics course. However, the derivative machine does not define which variable is dependent or independent. Both force and distance can be controlled when using the derivative machine which means dependent and independent variables are indistinguishable in this activity.

• 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.

This activity is the first of the Partial Derivative Machine sequence using the modified 1-D Partial Derivative Machine.

• Follow-up activity:
  • Internal Energy of the Derivative Machine: This small group activity introduces experimental measurements of integrals by determining the internal energy, $U$, of a nonlinear system at several locations.

This activity is the first of the Representations of Ordinary Derivatives sequence. It can be used in various physics contexts.

• Follow-up activities:
  • Recall the Derivative: This small whiteboard question prompts students to write something they know about derivatives.
  • Lecture on Multiple Representations of Derivatives: This lecture can cover many various representations of derivatives.
  • Lecture on Derivatives in Physics: This lecture describes derivatives as ratios of small changes which pertain to the size scale of interest.