Glossary
Average Rate of Change
The measure of how much the radius *r* changes with respect to the angle *θ* over a specific interval, calculated as (Δr/Δθ).
Example:
If a polar function's radius changes from 5 to 3 as the angle goes from 0 to π/2, its average rate of change is (3-5)/(π/2 - 0) = -4/π.
Contracting Polar Function
A polar function where the radius *r* is negative and decreasing as the angle *θ* increases, causing the graph to move towards the origin.
Example:
If a polar function's r value goes from -2 to -1 as θ increases, it would be an example of a contracting polar function according to this definition, as it moves closer to the origin.
Expanding Polar Function
A polar function where the radius *r* is positive and increasing as the angle *θ* increases, causing the graph to move away from the origin.
Example:
A spiral defined by r = θ for θ > 0 is an expanding polar function because as θ grows, r also grows, making the curve move outwards.
Negative Slope (Average Rate of Change)
Indicates that the radius *r* is decreasing as the angle *θ* increases over the given interval.
Example:
If the average rate of change of r with respect to θ is -1.2, it means the function has a negative slope on that interval, and r is shrinking.
Polar Coordinates (r = f(θ))
A coordinate system where points are defined by a distance *r* from the origin and an angle *θ* from the positive x-axis. The function r = f(θ) describes how the radius changes with the angle.
Example:
Plotting the point (5, π/4) in polar coordinates means moving 5 units from the origin along a ray at a 45-degree angle.
Positive Slope (Average Rate of Change)
Indicates that the radius *r* is increasing as the angle *θ* increases over the given interval.
Example:
If the average rate of change of r with respect to θ is 0.5, it means the function has a positive slope on that interval, and r is growing.
Relative Extrema
Points on a function where it changes direction, specifically from increasing to decreasing or vice versa, indicating a local maximum or minimum value.
Example:
On a graph of a polar function, a point where the distance from the origin momentarily stops increasing and starts decreasing is a relative extremum.
Relative Maximum
A point where a function changes from increasing to decreasing, representing a local peak or the point relatively farthest from the origin in its immediate vicinity.
Example:
For the polar function r(θ) = 3 + 2cos(θ), the point (5, 0) is a relative maximum because r is at its largest value here before decreasing.
Relative Minimum
A point where a function changes from decreasing to increasing, representing a local valley or the point relatively closest to the origin in its immediate vicinity.
Example:
For the polar function r(θ) = 3 + 2cos(θ), the point (1, π) is a relative minimum because r is at its smallest value here before increasing.