Parametric, Polar & Vector-Valued Functions 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 34 terms and is the same for everyone, so a teacher can assign “Unit 9, Paper 3” 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

Finding the limits for a polar area

Vector-valued position

Second derivative parametrically

Area between polar curves

Displacement versus distance

Petal width

Common polar curves

Area in polar coordinates

Vertical and horizontal tangents parametrically

Polar intersections can miss the pole

Parametric derivative

Parametric equations describe motion, not just a curve

Short answer 1. Define or explain: Finding position from velocity

3 pts

Short answer 2. Define or explain: Area between two polar curves

3 pts

Short answer 3. Define or explain: Speed in the plane

3 pts

Short answer 4. Define or explain: Speeding up in the plane

3 pts

Free response

9 pts

CALCULATOR PERMITTED. Curve C is defined by the polar equation r(θ) = 2 sin²θ for 0 ≤ θ ≤ π. Curve C and the semicircle r = 1/2 (also for 0 ≤ θ ≤ π) are shown in the xy-plane: C is a large oval-shaped loop reaching up to about y = 2.1, and the small semicircle of radius 1/2 sits over the origin, crossing C near θ = π/6 and θ = 5π/6. (Your calculator should be in radian mode.)

A. Find the rate of change of r with respect to θ at the point on curve C where θ = 1.3. Show the setup for your calculations.

B. Find the area of the region that lies inside curve C but outside the graph of the polar equation r = 1/2. Show the setup for your calculations.

C. It can be shown that dx/dθ = 4 sin θ cos²θ − 2 sin³θ for curve C. For 0 ≤ θ ≤ π/2, find the value of θ that corresponds to the point on curve C that is farthest from the y-axis. Justify your answer.

D. A particle travels along curve C so that dθ/dt = 15 for all times t. Find the rate at which the particle’s distance from the origin changes with respect to time when the particle is at the point where θ = 1.3. Show the setup for your calculations.