Rates of Change
The average rate of change of a function over an interval is the change in output divided by the change in input: the slope of the secant line through the interval’s endpoints. The rate of change at a point is a different object, a local reading that precalculus estimates by shrinking the interval around the point. Only linear functions collapse the two into one number.
The traps here are all identity confusions: reading an average over an interval as the speed at its endpoint, reporting a total change as if it were a rate, or letting the function’s values speak for its rate. The lesson drills the formula, the secant picture, the shrinking-interval estimate, and the sign conventions until the objects stay separate.
The work
3 ways in · any order
Lesson
Rates of Change
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One formula, two ideas: the average rate of change over an interval and the rate at a single point. The lesson builds the secant picture, the shrinking-interval estimate, and the sign conventions, then closes with ten scenarios that keep averages, instants, totals, and values from blurring together.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 1.2 misconceptions: treating an interval average as the rate at a point, and collapsing the rate story into the value or concavity story. The bank ships with the Unit 1 data drop.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.