AP Calculus AB 6.5 Accumulation

[Maximum number: 2]

The continuous function f is defined on the closed interval $-6 \leq x \leq 12$ . The graph of f, consisting of two semicircles and one line segment, is shown in the figure.

$Graph of $f$$

Graph of $f$

Let g be the function defined by $g(x)=\int_{6}^{x} f(t) d t$ .

Find all values of x in the open interval -6<x<12 at which the graph of g has a point of inflection. Give a reason for your answer.

B Find all values of x in the open interval -6<x<12 at which the graph of g has a point of inflection.

Give a reason for your answer.

The graph of g has a point of inflection where $g^{\prime \prime}=f^{\prime}$ changes sign, which is where $g^{\prime}=f$ changes from decreasing to increasing or vice versa.	Answer	Point 3 (P3)
The graph of g has points of inflection at x=-3 and x=6 because f changes from decreasing to increasing there.
The graph of g also has a point of inflection at x=3 because f changes from increasing to decreasing there.

Scoring Notes for Part B

- P3 is earned only for an answer of x=-3, x=3, and x=6. If any other/additional values of x in

-6<x<12 are declared to be points of inflection, the response does not earn either P3 or P4.

Consideration of x=-6 or of x=12 does not impact scoring.

- To earn P4, a response must tie the reason to the given graph of f.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because f changes from

increasing to decreasing or decreasing to increasing there" earns both P3 and P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because the slope of f

changes sign there" earns both P3 and P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because f attains relative

extrema there" earns both P3 and P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because g changes

concavity there" earns P3 but not P4.

○ A response of " g has a point of inflection at x=-3, x=3, and x=6 because $g^{\prime \prime}=f^{\prime}$

changes sign there" earns P3 but not P4.

○ A response that relies upon an ambiguous term such as "the function" or "the graph" does not

earn P4.

- Special case: A response with two of the three correct x-values with correct reasoning and no

other/additional values of x declared to be points of inflection earns P4 but not P3.

AP Calculus AB 6.5 Interpreting the Behavior of Accumulation Functions Involving Area Questions

Covered syllabus points

Covered syllabus points

Question 1