AP Precalc 1.2 Rates of ChangeAP Precalc 1.2 Rates of Change

A P P R E C A L C U L U S · UNIT1

Rates of Change

Topic 1.2 Study Guide

In one sentence: the average rate of change is the slope between two points, and we estimate the rate at a single point by shrinking the interval around it.

Average rate of change (AROC)

AROC on [a, b] = Δy / Δx = ( f(b) − f(a) ) / ( b − a )

Equals the slope of the secant line through (a, f(a)) and (b, f(b)).

Output on top. Order must match in top and bottom (b with b, a with a).

Units are output units per input unit. The word "per" signals a rate; the quantity before "per" is the dependent variable.

Function example. f(x) = x³ on [1, 3]: (27 − 1)/(3 − 1) = 13.

Table example. d(t) in meters: t = 10, 13, 19, 25 s gives d = 80, 76, 43, 30. On [13, 25]: (30 − 76)/(25 − 13) = −46/12 = −23/6 ≈ −3.83 m/s.

Rate of change at a point

Estimate with the AROC over a very small interval containing the point. The smaller the interval, the better the estimate. A symmetric interval is usually best (calculator active).

Example. g(x) = eˣ at x = 2. Use [1.999, 2.001]: (e2.001 − e1.999)/0.002 ≈ 7.389. Positive vs negative rate of change

Positive Negative

The two quantities move in the same direction (both up or both

down)

The two quantities move in opposite

directions

Less water used, shorter grass: positive More cats, fewer mice: negative

Traps. (1) Decreasing together is still positive. (2) Do not mix the order: (f(a) − f(b))/(b − a) flips the sign. (3) Keep units in the answer on word problems. (4) AROC is not f(b)/b; the slope uses a difference.

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Test-prep pattern

"For how many b is the AROC on [b, 5] equal to 0?" AROC = 0 means f(b) = f(5). Draw a horizontal line at the height of f(5) and count the other places it hits the graph (not b = 5 itself). Quick check

1. AROC of f(x) = x² on [0, 4]? 4

2. As savings decreases each semester, is the rate of change of savings with respect to semesters positive or negative? Negative

3. Why use a tiny interval to estimate the rate at a point? The secant slope gets close to the slope at that point

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