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3.15 · Signed Radius
What does 'signed radius' mean in polar functions? Why is the distance from origin NOT just r?
Distance = |r|, not r
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3.15 · 4 Distance Rules
State all four distance-from-origin rules based on sign and direction of r.
Positive/increasing, negative/decreasing, positive/decreasing, negative/increasing
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3.15 · Memory Shortcut
What is the shortcut for remembering when distance from origin is increasing vs decreasing?
Think about sign and direction 'matching'
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3.15 · Example 1 Table
f(x)=1+2sinx. On which interval is f negative and increasing? What does that mean for distance?
Zeros at x=7π/6 and x=11π/6. Min at x=3π/2.
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3.15 · Rectangular Sketch
What is the AP Exam strategy for analyzing distance behavior in polar functions?
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3.15 · Example 2
f(θ)=2−4cosθ. On (π, 5π/3), f is positive and decreasing. What is happening to the distance from origin?
Applies the core distance rule
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3.15 · Example 3 — ARC
f(θ)=3sin(2θ). On (3π/4, π), f is negative and increasing. What is the distance doing?
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3.15 · Relative Extremum
For polar functions, what condition produces a RELATIVE MAXIMUM? A RELATIVE MINIMUM?
Think about how r changes direction
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✓ Signed Radius
Distance = |r|, not r. A negative r still represents a positive distance from the origin — the curve plots in the opposite direction but is still |r| away.
r is the 'signed radius' — positive means direction of θ, negative means opposite direction. The actual distance from origin is always |r|.
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✓ 4 Distance Rules
r+ inc → dist INCREASING · r− dec → dist INCREASING · r+ dec → dist DECREASING · r− inc → dist DECREASING
The 'signed radius' r can be positive or negative. The actual distance is |r|. When r and its direction 'match' (both positive-direction), |r| grows. When they 'oppose', |r| shrinks.
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✓ Memory Shortcut
Same direction → distance increasing: r+ and inc (both go further), r− and dec (both go further negative). Oppose → decreasing.
Short: if the sign of r and direction of change 'point the same way', the curve moves farther from origin. If they oppose, it moves closer.
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✓ Example 1 — neg+inc
On (3π/2, 11π/6): f is negative and increasing → distance DECREASING (|r| shrinks as r approaches 0 from below).
f=1+2sinx. On (3π/2,11π/6): sin increasing, f goes from −1→0, negative but increasing. Distance decreasing.
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✓ Rectangular Sketch Strategy
Sketch f(θ) as a regular y=f(θ) rectangular graph. Read sign (above/below x-axis) and slope (increasing/decreasing) visually. Apply the 4 distance rules.
The rectangular sketch makes it easy to see intervals of positive/negative and increasing/decreasing. Much clearer than trying to read the polar graph directly.
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✓ Example 2 — Distance
Distance is DECREASING on (π, 5π/3). f is positive and decreasing → |r| shrinks.
On (π, 5π/3): f goes from 6→0, positive and decreasing. Apply rule: positive+decreasing = distance decreasing.
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✓ Example 3 — Distance
Distance DECREASING on (3π/4, π). f is negative and increasing (−3→0) → |r| shrinks.
Negative and increasing = distance decreasing (the curve moves toward the origin).
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✓ Relative Extremum Rules
Relative MAX: r changes from increasing to decreasing (|r| peaks). Relative MIN: r changes from decreasing to increasing (|r| troughs).
Extremum = change of direction. Max when r stops growing and starts shrinking. Min when r stops shrinking and starts growing.
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3.15 · Sign Change Trap
Does r changing from negative to positive mean a relative minimum for the polar curve? Explain.
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3.15 · ARC Formula
State the formula for the average rate of change of r=f(θ) over [a,b]. What does it represent geometrically?
Same formula as rectangular ARC
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3.15 · Example 5 ARC
f(θ)=3−3cosθ. Find the ARC over [π/2, π]. Show full computation.
f(π)=6, f(π/2)=3
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3.15 · ARC = 0 Meaning
For r=f(θ), what must be true for the ARC over [a,b] to equal zero?
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3.15 · Linear Approx
State the linear approximation formula for estimating f(θ) using the ARC over [a,b].
Point-slope form with ARC as slope
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3.15 · Example 8
f(π/6)=2, f(7π/6)=−1. Use ARC over [π/6, 7π/6] to approximate f(5π/6).
Step 1: find ARC. Step 2: apply point-slope.
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3.15 · Least ARC
For a polar graph where r values at points A, B, C, D are given, what does 'least ARC' mean?
Least = most negative
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✓ Sign Change Trap
NO. r changing sign just means the curve passed through the origin. Relative extremum requires r to change from increasing→decreasing (max) or decreasing→increasing (min).
Trap: if r goes from −1 to +1, the curve crossed r=0 (origin), but this is not an extremum. An extremum is about direction of change (slope), not sign.
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✓ ARC Formula
ARC = [f(b)−f(a)] / (b−a). Geometrically: the average rate of change of the radius r per radian of θ.
Same formula as rectangular average rate of change. Units: radius units per radian.
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✓ Example 5 — ARC
ARC = 6/π
f(π)=3−3(−1)=6. f(π/2)=3−3(0)=3. ARC=(6−3)/(π−π/2)=3/(π/2)=6/π.
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✓ ARC = 0 Meaning
ARC = 0 when f(a) = f(b) — the function returns to the same r value at both endpoints. The function can go up and down in between.
Example: f(3π/4)=−3 and f(7π/4)=−3 → ARC=0. The function went up and down between those angles but ended at the same value.
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✓ Linear Approx Formula
f(θ) ≈ f(θ₁) + [f(b)−f(a)]/(b−a) · (θ−θ₁)
This is point-slope form: y−y₁=m(x−x₁), where m=ARC, (θ₁,f(θ₁)) is a known endpoint, and θ is the target value.
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✓ Example 8 — Linear Approx
f(5π/6) ≈ 0
ARC=(−1−2)/(7π/6−π/6)=−3/π. f(5π/6)≈2+(−3/π)(5π/6−π/6)=2+(−3/π)(4π/6)=2−2=0.
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✓ Least ARC
Least = most negative. The interval where r decreased the most (largest drop) has the smallest (most negative) ARC.
ARC<0 when f decreased over the interval. 'Least' means the most negative number. The interval with the greatest drop in r has the least ARC.