From secant rates to a derivative

Try

A position model is s(t)=t² metres, with t in seconds. Between t=2 and t=3, average velocity is (9−4)/(3−2)=5 metres per second. Between t=2 and t=2.1 it is (4.41−4)/0.1=4.1. Which number describes the instantaneous velocity at t=2?

Predict

Predict what happens when the second time approaches 2 from above, and then from below. A negative time increment does not automatically mean a negative velocity: both the position difference and the time difference can be negative.

Experiment

Write the second time as 2+h, where h is a nonzero increment in seconds. Expand before dividing:

[(2+h)²−2²]/h = [4+4h+h²−4]/h = 4+h.

h in seconds	Secant velocity in metres/second	Difference from 4
1	5	1
1/10	4.1	0.1
1/100	4.01	0.01
−1/10	3.9	−0.1
−1/100	3.99	−0.01

Draw the points (2,4) and (2+h,(2+h)²), and connect them. These are secant lines through two distinct points. The algebra shows their slopes approach 4 from both sides as h tends to zero.

Understand

For a function f and input a, the derivative f′(a) is the finite limit of [f(a+h)−f(a)]/h as h approaches zero, when that limit exists. The numerator is output change and the denominator is input change. The original quotient requires h≠0. First simplify under that condition; only then take a limit. Substituting h=0 at the start creates 0/0 and does not calculate a derivative.

For f(x)=x² at a general input a, expand (a+h)²=a²+2ah+h². The quotient is 2a+h for nonzero h, whose limit is 2a. Thus f′(a)=2a. Replacing the letter a by x gives the derivative function f′(x)=2x, which assigns a slope to each input.

The prime mark is notation for a derivative, not an exponent. The alternative notation ds/dt reads “the derivative of position with respect to time.” It has metres-per-second units in this example. These units help diagnose accidentally inverted quotients.

A finite h gives a secant estimate, not automatically an exact derivative. Here the signed estimate error is exactly h. For other functions, the error need not have this form. A tiny floating-point step can also cause subtraction error on a computer, so numerical agreement is evidence to check against a derivation, not the definition itself.

Apply

Derive the reciprocal example in the practice bank by combining fractions. Keep a≠0, h≠0 and a+h≠0 visible until the limit is taken. In the position problem, report velocity units and distinguish instantaneous velocity from average velocity over a nonzero interval.

Continue

The next lesson asks when the required finite limit fails. A graph can be continuous and still have conflicting one-sided slopes or a vertical tangent.