Limits & Continuity 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.
Why limits matter twice in BC
1^∞ and 0⁰ forms
Intermediate Value Theorem
Squeeze theorem
Limits producing horizontal asymptotes
Why a table cannot prove a limit
Special trig limits
Three conditions for continuity at a
Oscillation as a failure mode
Removable vs jump discontinuity
0·∞ requires rewriting
Comparing growth rates
Short answer 1. Define or explain: Limit at infinity by degree
3 ptsShort answer 2. Define or explain: Growth hierarchy for limits
3 ptsShort answer 3. Define or explain: Recheck the form after each L'Hôpital step
3 ptsShort answer 4. Define or explain: ∞ − ∞ form
3 ptsFree response
9 ptsThis course has no free-response prompt tagged to this unit, so one from elsewhere in the course is used. It is still worth writing — the skill transfers.
NO CALCULATOR. The rate at which cars enter a parking garage is modeled by the function E, and the rate at which they leave is modeled by L. Both rates are measured in cars per hour, and selected values are given in the table. At time t = 0 there are 60 cars in the garage. t (hours) 0 2 5 8 12 E(t) 30 48 52 36 18 L(t) 10 22 40 54 30
A. Use a left Riemann sum with the four subintervals indicated by the table to approximate ∫₀¹² E(t) dt. Using correct units, interpret the meaning of this integral in the context of the problem.
B. Use the data to approximate E′(6.5). Show the computations that lead to your answer, and indicate units of measure.
C. Based on the data, is there a time t in the interval 5 < t < 8 at which the number of cars in the garage is neither increasing nor decreasing? Justify your answer.
D. Using a right Riemann sum with the four subintervals indicated by the table, approximate the number of cars in the garage at time t = 12.