Glossary
Arc Length
The arc length of a curve is the measure of the distance along the curve, representing the total length if the curve were straightened out.
Example:
Imagine a drone flying along a parabolic path given by y = x^2. The total distance the drone travels from x=0 to x=5 is its arc length.
Derivative (f'(x))
The derivative of a function, denoted as f'(x), represents the instantaneous rate of change of the function with respect to its independent variable.
Example:
To find the arc length of y = sin(x), you first need to calculate its derivative, which is cos(x).
Distance Traveled
Distance traveled is the total length of the path covered by an object in motion, always a non-negative value, irrespective of the direction of movement.
Example:
If a remote-controlled car drives along a winding path described by a function, the distance traveled is the total length of that path, even if the car ends up near its starting point.
Integral
An integral is a mathematical operation that calculates the accumulation of quantities, often used to find areas, volumes, or total change from a rate.
Example:
To find the total arc length of a curve, you must evaluate a definite integral of the square root of one plus the derivative squared.
Integrand
The integrand is the function or expression that is being integrated in a definite or indefinite integral.
Example:
In the arc length formula, S = ∫_a^b √(1+[f'(x)]^2) dx, the term √(1+[f'(x)]^2) is the integrand.
Smooth, Planar Curve
A smooth, planar curve is a continuous curve in a two-dimensional plane that has no sharp corners or cusps, meaning its derivative is continuous over the interval.
Example:
The path of a perfectly thrown baseball, modeled as a parabola, is an example of a smooth, planar curve because it has no sudden changes in direction.