Calculus AB · Unit 6 · Paper 2

Integration & Accumulation unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1 Paper 2 Paper 3

Each paper is built from this unit’s 47 terms and is the same for everyone, so a teacher can assign “Unit 6, Paper 2” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.

Suggested time 36 min 33 points0/17 attempted

Fundamental Theorem of Calculus part 2

Trapezoidal rule and concavity

Unequal subintervals are normal

No product rule for integrals

Definite integral with substitution

Accumulation function

g(a) = 0 always

Trapezoidal rule

Net area versus total area

u-substitution changes the limits

Definite integral is a number

Do not integrate when asked to differentiate

Short answer 1. Define or explain: Recombining given integral values

3 pts

Short answer 2. Define or explain: Choosing u in a substitution

3 pts

Short answer 3. Define or explain: Riemann sum from a table

3 pts

Short answer 4. Define or explain: Which property decides over or under

3 pts

Free response

9 pts

NO CALCULATOR. The continuous function f is defined on the closed interval −4 ≤ x ≤ 6. The graph of f consists of three line segments and a semicircle: • a line segment from (−4, 0) to (−2, 4); • a line segment from (−2, 4) to (0, 0); • a semicircle below the x-axis from (0, 0) to (4, 0), centered at (2, 0) with radius 2; • a line segment from (4, 0) to (6, 4). Let g be the function defined by g(x) = ∫₀ˣ f(t) dt.

A. Find g(−4) and g(4).

B. Find the x-coordinates of all points of inflection of the graph of g on the open interval −4 < x < 6. Justify your answer.

C. Find the absolute minimum value of g on the closed interval −4 ≤ x ≤ 6. Justify your answer.

D. Find the value of lim (x→0) [g(x)]/(x²), or explain why it does not exist.