Glossary
Arc Length
The distance between two points along a curve. It represents the total length of the path traced by the curve over a given interval.
Example:
To find the arc length of a roller coaster track, you would calculate the total distance a car travels from the start to the end of the ride.
Cartesian Equations
Equations that define a curve using coordinates (x, y) in a rectangular coordinate system, typically expressing y as a function of x, or vice versa.
Example:
The equation y = x² + 3 is a Cartesian equation that describes a parabola.
Derivative (dx/dt, dy/dt)
The instantaneous rate of change of a function with respect to its independent variable. In parametric equations, dx/dt and dy/dt represent the rates of change of x and y with respect to the parameter t.
Example:
If x(t) represents a car's horizontal position, then dx/dt would be the car's horizontal velocity.
Integration
The process of finding the antiderivative of a function, used to calculate accumulated quantities like area, volume, or arc length by summing infinitesimal parts.
Example:
To find the total distance traveled by an object given its velocity function, you would perform integration over the time interval.
Parametric Equations
A set of equations that define the coordinates of points on a curve as functions of a single independent variable, often 't' (for time or another parameter).
Example:
The motion of a projectile can be described by parametric equations like x(t) = (v₀ cos θ)t and y(t) = (v₀ sin θ)t - (1/2)gt².
Pythagorean Theorem
A fundamental geometric theorem stating that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).
Example:
When calculating the distance between two points (x₁, y₁) and (x₂, y₂), the distance formula is derived directly from the Pythagorean Theorem.
Smooth Planar Curves
A curve in a plane that has a continuous first derivative, meaning it has no sharp corners, cusps, or breaks.
Example:
A circle or a parabola are examples of smooth planar curves, while a zigzag line is not.