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3.15 · Signed Radius
What does it mean for r to be negative in a polar function?
Think about direction from origin
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3.15 · Distance from Origin
The distance from the origin to a polar point (r, θ) is always ___?
|r|, not r
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3.15 · Rule 1 — r increasing
If r is positive and increasing, what is happening to the curve?
Getting farther from origin
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3.15 · Rule 2 — r decreasing
If r is positive and decreasing toward 0, what is happening to the curve?
Approaching the pole
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3.15 · Rule 3 — r negative
If r is negative, where is the curve relative to the angle θ?
Opposite side of the origin
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3.15 · Rule 4 — r → more negative
If r is negative and becoming more negative, what happens to distance?
Distance from origin is increasing
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3.15 · Relative Extrema
When does a polar function f(θ) have a relative maximum distance from the origin?
When |r| is at a local max
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3.15 · ARC Formula
State the Average Rate of Change formula for f on [a,b].
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✓ Signed Radius
Negative r means the point plots in the direction OPPOSITE to θ — on the other side of the origin (pole).
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✓ Distance from Origin
Always |r| — the absolute value. Even when r is negative, the distance from origin is positive: |−3| = 3.
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✓ Rule 1 — r increasing
The curve is moving AWAY from the origin — the distance from the pole is growing.
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✓ Rule 2 — r decreasing
The curve is spiraling or curving toward the pole (origin). At r=0 the curve touches the origin.
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✓ Rule 3 — r negative
The curve plots on the OPPOSITE side of the origin from where θ points — reflected across the pole.
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✓ Rule 4 — r → more negative
Distance = |r| is increasing. Even though r is getting more negative, the curve is moving farther from the origin.
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✓ Relative Extrema
When |r| = |f(θ)| reaches a local maximum — i.e. where f(θ) has a local max or local min value (both give max distance).
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✓ ARC Formula
ARC = (f(b) − f(a)) / (b − a)
The average rate of change over [a,b]. For polar functions, this measures how fast r changes per unit of θ.
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3.15 · ARC Polar
For r=f(θ), what does ARC represent geometrically?
ARC = (f(b)−f(a))/(b−a)
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3.15 · Linear Approximation
State the linear approximation formula for f near θ=a.
Uses ARC as slope
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3.15 · Example 1 — ARC
f(θ)=1+2sinθ on [0, π/2]. Find the ARC.
f(0)=1, f(π/2)=3
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3.15 · Example 2 — Lin. Approx
Use ARC from [0,π/2] of f(θ)=1+2sinθ to estimate f(π/6).
ARC = 4/π from Example 1
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3.15 · Increasing |r|
r = sinθ on [π, 3π/2]. Is the distance from the origin increasing or decreasing?
sinθ is negative on [π,3π/2]
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3.15 · Pinch Point
When r=0 in a polar function, what is happening geometrically?
The curve passes through the pole
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3.15 · Canvas Worked
f(x)=1+2sinx. At x=3π/2 the graph has a local min. What is f(3π/2) and what does this mean for distance?
sin(3π/2)=−1
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✓ ARC Polar
How fast r is changing per radian of θ — the average slope of the r=f(θ) graph over the interval [a,b].
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✓ Linear Approximation
f(θ) ≈ f(a) + ARC · (θ − a)
Uses the ARC as the slope of the tangent approximation near θ=a.
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✓ Example 1 — ARC
ARC = (f(π/2)−f(0))/(π/2−0) = (3−1)/(π/2) = 2/(π/2) = 4/π
f(0)=1+2sin(0)=1. f(π/2)=1+2(1)=3.
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✓ Example 2 — Lin. Approx
f(π/6) ≈ f(0) + (4/π)·(π/6) = 1 + 2/3 = 5/3 ≈ 1.667
Actual: f(π/6)=1+2·(1/2)=2. The approximation is close for small intervals.
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✓ Increasing |r|
Distance is INCREASING. sinθ is negative on [π,3π/2], so r<0 and |r|=|sinθ| grows from 0 to 1. The curve moves away from origin.
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✓ Pinch Point
The curve passes through the POLE (origin). r=0 means the point is at the origin regardless of θ.
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✓ Canvas Worked
f(3π/2)=1+2(−1)=−1. Distance from origin = |−1| = 1. The curve is at distance 1 from origin, plotting on the opposite side of θ=3π/2.