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.
Accumulation function analysis
FTC Part 2 with a chain rule
FTC Part 2 is for differentiating
g′ = f for an accumulation function
Properties of definite integrals
Left sum with an increasing function
Variable in the lower limit
Trapezoid on one subinterval
Common antiderivatives
No product rule for integrals
Midpoint sums need midpoints
Recombining given integral values
Short answer 1. Define or explain: Fundamental Theorem of Calculus part 2
3 ptsShort answer 2. Define or explain: Accumulation function
3 ptsShort answer 3. Define or explain: u-substitution
3 ptsShort answer 4. Define or explain: Riemann sum from a table
3 ptsFree response
9 ptsCALCULATOR PERMITTED. Water flows into a storage tank at a rate modeled by R(t) = 40 + 12t − t² gallons per hour, and flows out at a rate modeled by L(t) = 25 + 8t gallons per hour, for 0 ≤ t ≤ 10 hours. At time t = 0 the tank contains 100 gallons of water.
A. Find the total number of gallons of water that flow into the tank over the interval 0 ≤ t ≤ 10. Show the setup for your calculations.
B. Is the amount of water in the tank increasing or decreasing at time t = 4? Give a reason for your answer.
C. Find the time t, for 0 ≤ t ≤ 10, at which the amount of water in the tank is a maximum. Justify your answer.
D. Find the number of gallons of water in the tank at time t = 10.