When does decreasing radius mean moving toward the pole?
You will be able to: Distinguish signed radial change from change in distance to the origin.
When does decreasing radius mean moving toward the pole?
A point can pass through the origin and continue along the opposite ray. Its signed radius becomes negative, even while its distance from the origin grows.
A useful starting point: How do tangents and concavity work in polar form? →
Words and symbols before equations
- Pole distance ρ
- abs(r), always nonnegative.
- Signed radial rate
- r′=dr/dθ.
- Distance rate
- d(abs(r))/dθ, when it exists.
- Angular parameter
- Here θ increases; this is not automatically physical time.
What this picture assumes
Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. r=2−θ for increasing θ. Distance is abs(r); its θ derivative is undefined at θ=2. No angular time rule is supplied, so rates are per radian.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- θ=1 rad; r=1 units; point (0.540302,0.841471); pole distance 1 units. dr/dθ=−1; distance derivative -1 units/rad.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Where r>0, pole distance is r and its derivative is r′. Where r<0, pole distance is −r and its derivative is −r′.
Thus decreasing r means moving closer only when r is positive. For r=2−θ and θ>2, r<0 and r′=−1, but distance increases at 1 unit per radian.
At θ=2 this model passes through the pole, and abs(2−θ) has a corner: its θ derivative does not exist there. One-sided distance behavior still makes sense.
To convert these statements to change per time, an angular time law is needed; apply the chain rule with dθ/dt.
A worked example, step by step
For r=2−θ at θ=3, find signed radius, distance to the pole and its rate per angle.
- r=2−3=−1.
- Distance is abs(−1)=1.
- Since r<0, d(abs(r))/dθ=−r′=1.
- The point is moving farther from the pole as θ increases, despite r′=−1.
Do not call r′ the derivative of distance when r is negative. Also distinguish per-radian rates from per-second rates.
At a negative r, is a more negative value closer to the pole?
Compare with an explanation
No. Its absolute value is larger, so it is farther away.
Predict. Change one thing. Explain.
Move θ through 2 radians for r=2−θ. Compare signed radius and distance. Explain the sign change in the distance rate.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
θ=1 rad; r=1 units; point (0.540302,0.841471); pole distance 1 units. dr/dθ=−1; distance derivative -1 units/rad.
Read the representation: Teal shows a curve; orange marks the selected point, tangent, ray or swept region described above. Dashed segments are auxiliary comparisons. Read the model conditions and units before comparing lengths.
Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. r=2−θ for increasing θ. Distance is abs(r); its θ derivative is undefined at θ=2. No angular time rule is supplied, so rates are per radian.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the units, coordinate rates, parameter direction, tracing count or radial boundaries. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor r=1−2θ, compare pole distance and its derivative at θ=0 and θ=1, and identify the pole crossing.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: At θ=0, r=1 and distance=1.
- 1 point: There r>0, so distance derivative is −2.
- 1 point: At θ=1, r=−1 and distance=1, but its derivative is +2.
- 1 point: The pole crossing is θ=1/2; abs(1−2θ) has no derivative there.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What is physical pole distance?
The absolute value of r.
RECALL 2What happens to its derivative for negative r?
It is the negative of r′.
RECALL 3What additional information gives a time rate?
The angular rate dθ/dt.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When does decreasing radius mean moving toward the pole?
- ρ=abs(r).
- For r≠0, ρ′=sign(r)r′.
- At r=0, inspect one-sided behavior.
Remember: Do not call r′ the derivative of distance when r is negative. Also distinguish per-radian rates from per-second rates.
Conditions: Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. r=2−θ for increasing θ. Distance is abs(r); its θ derivative is undefined at θ=2. No angular time rule is supplied, so rates are per radian.
Refresh Kid · AP Calculus BC Unit 9 · Objectives FUN-3.G · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.7, FUN-3.G. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. This unit covers BC topics 9.1–9.9. Parametric and vector motion is planar. Polar arc length, surface area and three-dimensional vector calculus are not assigned as required Unit 9 material.
Parametric slope and second derivatives retain their nonzero-denominator conditions. Speed is the magnitude of velocity; displacement and distance use different integrals. Polar coordinates allow signed radii, with distance abs(r). Polar tangents use Cartesian angular rates. Areas use squared radial boundaries with correct angular intervals, tracing counts and boundary switches; the pole is checked separately.
All focused explanations, examples, practice and models are original Refresh Kid work. OpenStax was consulted for mathematical cross-checking. Khan Academy’s destination was checked, but JavaScript lesson content was not fully readable by the research tool. Organic Chemistry Tutor video titles, creator and destinations were checked; full videos were not reviewed. No questions, diagrams or provider scripts were copied. Resources are optional; Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional camera and spatial inspection. The original parameter-lift model uses self-hosted Three.js with its MIT license. The teal projection is the physical circle. The vertical axis of the orange lift is time, not spatial height, and its scale is explicitly stated. No spatial arc-length calculation is made from that lift. Keyboard controls and labeled 2D alternatives remain available, without autoplay or required WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this implementation has not been evaluated with learners.
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