Glossary
Chain Rule
A fundamental differentiation rule used to find the derivative of a composite function. In the context of parametric equations, it's crucial for understanding how derivatives with respect to *x* are related to derivatives with respect to *t*.
Example:
When calculating from and , we are essentially applying the Chain Rule because , where .
Concavity
Describes the direction in which the graph of a function opens. For parametric equations, the sign of the second derivative ($d^2y/dx^2$) indicates whether the curve is concave up (positive) or concave down (negative).
Example:
If the second derivative of parametric equations for a diver's trajectory is negative, it means the path is concave down, indicating the diver is curving downwards.
First derivative of parametric equations (dy/dx)
Represents the slope of the tangent line to a parametric curve at a specific point. It is calculated by dividing the derivative of y with respect to t ($dy/dt$) by the derivative of x with respect to t ($dx/dt$).
Example:
If a car's position is given by and , its instantaneous vertical rate of change relative to its horizontal rate of change, or its first derivative of parametric equations, is .
Parametric functions
Functions where the independent variables x and y are both expressed in terms of a third, independent 'dummy' variable, typically *t*, which often represents time.
Example:
A drone's flight path might be described by its horizontal position x(t) and vertical position y(t) at any given time t, making it a parametric function.
Quotient Rule
A rule for differentiating a function that is expressed as the ratio of two other functions. It is frequently used when calculating the derivative of the first derivative ($dy/dx$) with respect to *t* for the second derivative formula.
Example:
If your first derivative is a fraction like , you'll need to use the Quotient Rule to find before calculating the second derivative.
Second derivative of parametric equations (d²y/dx²)
Measures the rate of change of the slope of a parametric curve, providing information about its concavity. It is found by taking the derivative of the first derivative ($dy/dx$) with respect to *t*, and then dividing that result by $dx/dt$.
Example:
To determine if a rollercoaster track defined parametrically is curving upwards or downwards at a certain point, you would calculate its second derivative of parametric equations; a positive value means it's concave up.