Section 8–3 Speed as a derivative

(Mathematical notations / Limiting process / Differentiating functions)

In this section, the three interesting points discussed are mathematical notations, limiting process, and differentiation functions.

1. Mathematical notations:

“…The prefix Δ is not a multiplier, any more than sin θ means s i n θ — it simply defines a time increment, and reminds us of its special character (Feynman et al., 1963, section 8.3 Speed as a derivative).”

As a matter of convenience, special notations Δt and Δs have been assigned to replace the quantities ϵ and x respectively. According to Feynman, Δt means “an extra bit of t” and carries an implication that it can be made smaller. Next, he clarifies that the prefix Δ is not a multiplier and it simply means a time increment. In short, Δ is not a factor and it cannot be canceled in the ratio Δs/Δt to give s/t. To be precise, Δt may refer to infinitely small quantities that are smaller than any positive number of the real number system and it does not equal to zero. Moreover, the notation Δt can be replaced by dt, in which Δ and d are known as the “difference symbol” and “differential symbol” respectively.

On the other hand, Feynman elaborates that Δs has an analogous meaning for the distance s. However, physics teachers may explain that the notation Δs means a difference in displacement instead of distance. Furthermore, the notation s is related to the Latin word spatium, which means “a stretch or extent” in space. Importantly, velocity is equal to the limit of Δs/Δt as Δt approaches zero and it can be represented by ds/dt. Historically, Newton introduced the notation ḟ for the derivative of a function f, whereas Leibniz wrote the derivative of a function y with respect to the independent variable x as in dy/dx. One may prefer Leibniz’s notation because it tells us about the independent variable such as x, z, or t.

2. Limiting process:

“This statement is true only if the velocity is not changing during that time interval, and this condition is true only in the limit as Δt goes to 0 (Feynman et al., 1963, section 8.3 Speed as a derivative).”

In a sense, a philosophy of calculus is to make the best approximation. It is not surprising to see Feynman uses the word approximation three times in the paragraph of explaining the limiting process. Firstly, he mentions that to a good “approximation” we have another law, which states that the change in distance of a moving point is the velocity times the time interval, or Δs = vΔt. Secondly, Feynman clarifies that physicists like to write ds = vdt because by dt they mean Δt in circumstances in which it is very small and the expression is valid to a close “approximation.” Thirdly, he adds that if Δt is too long, the velocity might not be constant during the interval, and the “approximation” would become less accurate. Essentially, physicists write v = lim_Δt→0 Δs/Δt = ds/dt based on the “approximation” or limiting process.

A purpose of the limiting process in calculus is to avoid an embarrassing situation (or a definition problem): dy/dx = 0/0. In Principia, Newton (1999) first explains that “[t]hose ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity (p. 442).” However, Feynman did not like the symbol dy/dx. In his own words, “I didn’t like f(x) -- that looked to me like f times x. I also didn’t like dy/dx -- you have a tendency to cancel the d’s -- so I made a different sign, something like an & sign... I thought my symbols were just as good, if not better, than the regular symbols -- it doesn’t make any difference what symbols you use -- but I discovered later that it does make a difference (Feynman, 1997, p. 24).”

3. Differentiating functions:

“This is the fundamental process of calculus, differentiating functions. The process is even more simple than it appears (Feynman et al., 1963, section 8.3 Speed as a derivative).”

Feynman mentions that the terms ds or dt are called differentials, and clarifies that higher power of Δt may be dropped because they approach to 0 when the limit is taken. Simply put, the notation dx means a little bit of x and more precisely, the word differential refers to an infinitesimal (infinitely small) quantity. Interestingly, Bertrand Russell (1992) argues that “infinitesimals as explaining continuity must be regarded as unnecessary, erroneous and self-contradictory (p. 350).” Currently, there is no agreement on the usefulness of the concept of infinitesimal. Gardner writes that “[d]ebate over the infinitesimal versus the limit language goes nowhere because they are two ways of saying the same thing (Thompson & Gardner, 1998, p. 24).

Feynman states the quantity ds/dt as the “derivative of s with respect to t” and adds that the process of differentiating (or finding a derivative) is simpler than it appears. Furthermore, he suggests the rules for differentiating various types of functions can be memorized or can be found in tables. Nevertheless, one may prefer an intuitive or geometric explanation of differentiation. In general, physics teachers can explain the derivation of “power rule” (d/dx [xⁿ] = n x^n-1) involves binomial theorem. Better still, the process of differentiation can be visualized by using an applet that illustrates the slope of a tangent line at a point of a displacement-time graph (or position-time graph).

Questions for discussion:

1. What are the correct meanings of Δ and d in calculus?

2. How would you explain the limiting process?

3. How would you provide an intuitive or geometric explanation of differentiation?

The moral of the lesson: speed is a derivative of distance with respect to time.

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Newton, I. (1999/1687). The Principia: Mathematical principles of natural philosophy, translated by I. B. Cohen & A. Whitman. Berkeley: University of California Press.

4. Russell, B. (1992/1903). The Principles of Mathematics. London: Routledge.

5. Thompson, S. P. & Gardner, M. (1998). Calculus made easy. New York: St. Martin’s Press.