Glossary
Angle (θ)
In polar coordinates, 'θ' represents the angle measured counter-clockwise from the positive x-axis to the line segment connecting the pole to the point.
Example:
A point on the negative y-axis would have an angle (θ) of 3π/2 in polar coordinates.
Cartesian Equations
Equations that define curves or functions using the rectangular coordinate system (x and y axes).
Example:
The equation x² + y² = 9 is a Cartesian equation for a circle centered at the origin with radius 3.
Chain Rule (in polar context)
A fundamental differentiation rule used to find the derivative of a composite function; in polar coordinates, it's crucial for calculating dy/dx from expressions involving r and θ.
Example:
When finding dy/dx for a polar function, the chain rule is applied to x = rcosθ and y = rsinθ, treating r as a function of θ.
Differentiate (in polar context)
To find the derivative of a polar function, either with respect to θ (dr/dθ) or with respect to x (dy/dx), to analyze its rate of change or slope.
Example:
To find how quickly the radius of a spiral changes with angle, you would differentiate r = θ with respect to θ.
Distance (r)
In polar coordinates, 'r' represents the directed distance from the pole (origin) to a point.
Example:
For the polar point (5, π/6), the distance (r) from the origin is 5 units.
First Derivative (r'(θ) / Radial Component)
The instantaneous rate of change of the distance 'r' from the origin with respect to the angle 'θ', also known as the radial component of the curve.
Example:
A positive first derivative (r'(θ)) means the curve is moving away from the pole as θ increases.
Parametric Equations
Equations that define coordinates (like x and y) in terms of a third independent variable, often 't' (for time) or 'θ' (for angle).
Example:
To describe the path of a projectile, you might use parametric equations like x(t) = (v₀cosα)t and y(t) = (v₀sinα)t - (1/2)gt².
Polar Coordinates
A two-dimensional coordinate system where each point is determined by a distance 'r' from a fixed point (the pole) and an angle 'θ' from a fixed direction (the positive x-axis).
Example:
Instead of (3, 4) in Cartesian, a point might be (5, π/3) in polar coordinates, representing a distance of 5 units at an angle of 60 degrees.
Polar Functions
Functions defined in terms of polar coordinates, typically in the form r = f(θ), which describe curves in the polar coordinate system.
Example:
The equation r = 2sin(3θ) describes a rose curve, a common type of polar function.
Pole
The fixed origin point in a polar coordinate system, analogous to the origin (0,0) in the Cartesian system.
Example:
When graphing a polar function, all distances 'r' are measured from the pole.
Second Derivative (r''(θ) / Radial Curvature)
The rate of change of the first derivative of 'r' with respect to 'θ', indicating the rate of change of the curvature and providing information about concavity in the radial direction.
Example:
Analyzing the sign of the second derivative (r''(θ)) can help determine if the curve is 'bending' towards or away from the pole.
Slope of the Tangent Line (in polar context)
The instantaneous rate of change of y with respect to x (dy/dx) for a curve defined by a polar function, representing the slope of the line tangent to the curve at a given point.
Example:
Calculating the slope of the tangent line for r = 1 + cosθ at θ = π/2 helps determine the steepness of the cardioid at that specific point.
Vector-Valued Functions
Functions whose output is a vector, typically representing position, velocity, or acceleration in space, with components defined by a single independent variable.
Example:
The position of a particle moving in a plane can be described by a vector-valued function r(t) = <x(t), y(t)>.
dr/dθ
The derivative of the radial distance 'r' with respect to the angle 'θ', indicating the rate at which the distance from the pole changes as the angle changes.
Example:
If r = θ², then dr/dθ = 2θ, showing that the radius increases linearly with the angle.
dy/dx (Slope of Tangent Line for Polar Functions)
The derivative of y with respect to x, calculated using the chain rule (dy/dθ) / (dx/dθ), which gives the slope of the tangent line to a polar curve in the Cartesian plane.
Example:
To find the horizontal tangents of a polar curve, you would set the numerator of dy/dx equal to zero.