Estimating Derivatives of a Function at a Point AB & BC
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalWith only a table or a printed graph, a derivative at a point is estimated by an average rate of change over the tightest interval available. With rows on both sides of the point, the symmetric difference quotient $\frac{f(a+h) - f(a-h)}{2h}$ using the nearest neighbours is the preferred estimate; at the edge of a table, a one-sided quotient is all there is. From a graph the tangent is drawn first, and its slope is read from two grid points that lie on the tangent line.
The estimate goes wrong in two ways. Either the wrong pair of points is used, reaching past a nearer row to the ends of the table, using one side when both are available, or reading a height where a slope was asked for. Or the estimate is described as though it were exact, so an average rate over an interval gets reported as the instantaneous rate at a point rather than as an approximation to it.
The work
3 ways in · any order
Lesson
Estimating Derivatives of a Function at a Point
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Drills choosing the tightest pair of table rows around a point, running the symmetric difference quotient, reading a tangent slope off grid intersections rather than off the curve, and stating an estimate as an estimate with its units.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: estimating from the wrong pair of points, whether that is the ends of a table or two points on the curve, and reporting an interval's average rate as the exact instantaneous rate.
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 it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.