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.
1^∞ and 0⁰ forms
Limit of a sequence versus of a function
Why limits matter twice in BC
Three conditions for continuity at a
Growth hierarchy for limits
Special trig limits
One-sided limits and existence
Continuity of a composition
Intermediate Value Theorem
Repeated application of L'Hôpital
Why a table cannot prove a limit
L'Hôpital's Rule conditions
Short answer 1. Define or explain: Limit at infinity by degree
3 ptsShort answer 2. Define or explain: ∞ − ∞ form
3 ptsShort answer 3. Define or explain: 0·∞ requires rewriting
3 ptsShort answer 4. Define or explain: Limits producing horizontal asymptotes
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.
CALCULATOR PERMITTED. Let S be the shaded region bounded by the graph of the polar curve r(θ) = 3 + 2 sin(2θ) + cos(2θ) for 0 ≤ θ ≤ π, as shown in the figure (a two-lobed region above the x-axis: a large lobe to the right of the y-axis and a smaller lobe to the left). It can be shown that r′(θ) = 4 cos(2θ) − 2 sin(2θ). (Your calculator should be in radian mode.)
A. Find the area of region S. Show the setup for your calculations.
B. There is a point on the curve at which the slope of the line tangent to the curve is −3/7. At this point, dy/dθ = 3√2/2. Find dx/dθ at this point. Show the work that leads to your answer.
C(i). Find the value of θ in the interval 0 < θ < π/2 at which r has a critical point.
C(ii). Use a derivative test to determine whether the critical point is the location of a relative minimum, a relative maximum, or neither for r.
D. Find the average distance from the origin to a point on the polar curve r(θ) = 3 + 2 sin(2θ) + cos(2θ) for π/2 ≤ θ ≤ π. Show the setup for your calculations.