Defining Average and Instantaneous Rates of Change at a Point AB & BC
The average rate of change of $f$ on $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$, the slope of the secant through the endpoints; written with an increment it is $\frac{f(a+h) - f(a)}{h}$. The instantaneous rate of change at $a$ is what those quotients approach as the interval collapses, $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, equivalently $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$. Simplify the quotient first, cancel the $h$, and only then let $h$ go to zero.
Two failures dominate. The first is reporting an interval's average as though it were a value at a point, so a secant slope gets called the rate at an endpoint or the midpoint. The second is a malformed quotient: a denominator that is an input rather than a change, a subtraction that runs the wrong way, or an expression with no limit attached, which computes an average and calls it an instant.
The work
3 ways in · any order
Lesson
Defining Average and Instantaneous Rates of Change at a Point
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Separates the secant slope on an interval from the limit of difference quotients at a point, drills writing the quotient so its numerator and denominator describe the same interval, and works the algebra that cancels the increment before the limit is taken.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: reporting an interval's average rate as though it were the rate at a point, and writing a difference quotient with the wrong increment, the wrong direction, or no limit at all.
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.