Rates of Change and the Calculus Bridge

Objective: Explain average rate and secant lines, apply the rule, and justify relevant restrictions.

The average rate of change from u to v, u≠v, is [f(v)−f(u)]/(v−u). It is the slope of a secant line through two graph points and carries output units per input unit. It need not equal the rate at either endpoint. Reversing both endpoints leaves the ratio unchanged because both numerator and denominator reverse signs.

Objective: Explain difference quotients and tangent rates, apply the rule, and justify relevant restrictions.

At a fixed input a, [f(a+h)−f(a)]/h with h≠0 is a secant slope over a shrinking interval. Its limit as h→0, if finite and existing, defines the instantaneous rate f′(a). Simplify before taking the limit; setting h=0 in the original quotient gives 0/0. This lesson introduces the derivative as a limit rather than a table of differentiation rules.