Average to Instantaneous Rate of Change

Calculus 1 · Axiom Academy

LESSON Average to Instantaneous Rate How do you find the rate of change at a single instant? Shrink the interval until the secant line becomes the tangent. 1. Average Rate Is a Secant Slope For a function f , the average rate of change between x = a and x = a + h is the total change in f divided by the change in x . Geometrically, it is the slope of the secant line joining the two points and . average rate = rise over run = slope of the secant Now hold the first point fixed at x = a and slide the second point toward it by making h smaller. The two points close in on each other, and the secant line pivots about (a, f(a)) — rotating as its slope changes — until it settles onto a single line touching the curve at just that point: the tangent line . As the secant slope approaches one fixed number — the slope of the tangent line at x = a . That number is the instantaneous rate of change , and it is the value the difference quotient is heading toward: Let's make it concrete with f(x) = x^2 at x = 2 . The average rate over is , which simplifies to h + 4 . As h shrinks, that value marches straight down toward 4 : This limiting process is exactly the formal definition of the derivative. The instantaneous rate of change of f at x = a is the limit of the average rates as the interval shrinks to zero: The derivative f'(a) is three things at once — the same number seen three ways: How fast f is changing at the single point x = a — the speedometer reading.

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