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.
g(a) = 0 always
Splitting at a sign change
Right sum with an increasing function
Choosing u in a substitution
FTC Part 2 with a chain rule
Net change theorem
Unequal subintervals are normal
Trapezoid on one subinterval
Definite integral as signed area
Net area versus total area
Accumulation function analysis
Do not integrate when asked to differentiate
Short answer 1. Define or explain: Over- or underestimate justification
3 ptsShort answer 2. Define or explain: Definite integral as a limit of Riemann sums
3 ptsShort answer 3. Define or explain: FTC Part 2 is for differentiating
3 ptsShort answer 4. Define or explain: g′ = f for an accumulation function
3 ptsFree response
9 ptsNO 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.