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Interpreting the Behavior of Accumulation Functions Involving Area AB & BC

Since $g(x) = \int_{a}^{x} f(t)\,dt$ has $g' = f$ and $g'' = f'$, every Unit 5 question about shape transfers directly: $g$ increases where $f$ is above the axis, has a maximum where $f$ crosses from positive to negative and a minimum where it crosses the other way, is concave up where $f$ is increasing, and has inflection points where $f$ turns around. Every one of those reads a sign or a turn of $f$, and none of them reads the size of $f$: a large positive $f$ means $g$ is climbing steeply, which is a reason for $g$ to keep rising rather than to peak.

Locating $g$'s features is a question about $f$'s sign; evaluating $g$ is a question about $f$'s signed area. For segments through $(0,0)$, $(2,4)$, $(4,0)$, $(6,-3)$, $(8,0)$, $(10,4)$, the three regions have areas $8$, $6$ and $4$ with the middle one below the axis, so $g(4) = 8$, $g(8) = 2$ and $g(10) = 6$, while the total geometric area is $18$. The maximum of $g$ sits at $x = 4$ and the maximum of $f$ at $x = 2$; the inflection points of $g$ sit at $x = 2$ and $x = 6$, where $f$ turns, and at neither of $f$'s zeros.

THE GRAPH IS f. EVERY LINE BELOW IS ABOUT g(x) = ∫ f FROM 0 TO x. +8 −6 +4 0 2 4 6 8 10 g(0) = 0 g(4) = 8 g(8) = 2 g(10) = 6 g RISES ON [0, 4], FALLS ON [4, 8], AND RISES AGAIN ON [8, 10]. g PEAKS AT x = 4, WHERE f CROSSES FROM ABOVE THE AXIS TO BELOW. g HAS INFLECTION POINTS AT x = 2 AND x = 6, WHERE f TURNS AROUND.
Drawn to scale at 40 px per unit across and 18 px per unit up. The peak of $f$ is at $x = 2$ and the peak of $g$ is at $x = 4$; the cyan dots mark $f$'s zeros, which are $g$'s critical points.
WHAT f DOES WHAT g DOES f IS POSITIVE → g IS INCREASING f IS NEGATIVE → g IS DECREASING f CROSSES + TO − → g HAS A MAXIMUM f CROSSES − TO + → g HAS A MINIMUM f IS INCREASING → g IS CONCAVE UP f TURNS AROUND → g HAS AN INFLECTION POINT g IS AN ANTIDERIVATIVE OF f, SO THIS IS UNIT 5 ONE LEVEL DOWN. f PLAYS THE ROLE f′ PLAYED THERE, AND f′ PLAYS f″. NOTHING HERE READS THE SIZE OF f. ONLY ITS SIGN AND ITS TURNS.
Six rows, and every left-hand entry is a sign or a turn. The one row students supply from memory instead of from this table is the last, which they place at $f$'s zeros rather than at $f$'s peaks.

The work

3 ways in · any order
Lesson
Interpreting the Behavior of Accumulation Functions Involving Area ›

Builds the translation table from g prime equals f and g double prime equals f prime, then works one graph all the way through: direction, critical points, values from signed areas, concavity and inflection points, with the peak of the integrand and the peak of the accumulation landing at different places on purpose.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items spanning the two failure modes of this topic: reading where the accumulation function rises, peaks or changes concavity off the integrand's size or its zeros rather than its sign and its turns, and reporting a total area where a signed accumulation was wanted.

Not yet available · 10 items
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Targeted Practice
Drill a single misconception ›

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.

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