Glossary
Cartesian graph
A graph that uses the Cartesian coordinate system, where points are located by their perpendicular distances from two fixed, perpendicular axes (x and y).
Example:
Plotting y = x² on a standard x-y plane creates a parabola on a Cartesian graph.
Derivative
The instantaneous rate of change of a function with respect to its independent variable, representing the slope of the tangent line at any point.
Example:
The derivative of a car's position function gives its instantaneous velocity.
Dummy variable (t)
An independent variable, typically 't' for time, used in parametric equations to relate x and y coordinates without being explicitly plotted on the graph.
Example:
In x(t) = t² and y(t) = 2t, 't' is the dummy variable that dictates the position (x, y) at any given moment.
Instantaneous rate of change
The rate at which a quantity is changing at a specific moment in time, represented by the derivative of the quantity's function.
Example:
The speedometer in a car shows its instantaneous rate of change of position, which is its speed.
Limit definition (of derivative)
The formal definition of a derivative as the limit of the difference quotient, representing the instantaneous rate of change of a function.
Example:
Using the limit definition to find the derivative of f(x) = x² involves evaluating lim (h→0) [(x+h)² - x²]/h.
Parameters
Variables that define a system or function, often remaining constant within a specific context but can change between different contexts. In parametric equations, 't' is the parameter.
Example:
In the equation of a circle (x-h)² + (y-k)² = r², h, k, and r are parameters that define the circle's center and radius.
Parametric curve
A curve whose points (x, y) are defined by functions of a single independent parameter, typically 't'.
Example:
The path traced by a projectile launched into the air can be modeled as a parametric curve, with x(t) and y(t) describing its horizontal and vertical positions over time.
Parametric derivative
The process of finding the derivative dy/dx for a curve defined by parametric equations, typically by calculating (dy/dt) / (dx/dt).
Example:
Calculating the slope of a cycloid's path at a specific point requires using the parametric derivative formula.
Parametric equations
A set of equations where the x and y coordinates of a point are expressed as functions of a third independent variable, often 't' (representing time).
Example:
Imagine a roller coaster's path described by x(t) = 5cos(t) and y(t) = 5sin(t) + 2t, where 't' tracks the time since the ride started, showing its parametric equations.
Polar coordinates
A two-dimensional coordinate system that specifies a point's position by its distance from the origin (r) and the angle (θ) it makes with the positive x-axis.
Example:
A point located at (3, π/2) in polar coordinates means it's 3 units from the origin along the positive y-axis.
Power rule (of derivative)
A rule for differentiating functions of the form x^n, stating that the derivative is nx^(n-1).
Example:
To find the derivative of f(x) = x⁵, apply the power rule to get f'(x) = 5x⁴.
Product rule (of derivative)
A rule for differentiating the product of two functions, (uv)' = u'v + uv'.
Example:
If f(x) = x²sin(x), use the product rule to find f'(x) = 2xsin(x) + x²cos(x).
Quotient rule (of derivative)
A rule for differentiating the quotient of two functions, (u/v)' = (u'v - uv')/v².
Example:
To differentiate f(x) = sin(x)/x, apply the quotient rule to get f'(x) = (xcos(x) - sin(x))/x².
Real-valued functions
Functions whose output (range) consists of real numbers, typically mapping real numbers to real numbers.
Example:
f(x) = x² is a real-valued function because for any real input x, the output f(x) is also a real number.
Tangent line
A straight line that touches a curve at a single point and has the same slope as the curve at that point.
Example:
At the peak of a parabola, the tangent line is horizontal, indicating a slope of zero.
Vector-valued functions
Functions that output a vector, where each component of the vector is a function of one or more independent variables.
Example:
The position of a drone in 3D space might be given by r(t) = <t², sin(t), e^t>, which is a vector-valued function describing its trajectory over time.
dx/dt
The derivative of the x-component of a parametric equation with respect to the parameter 't', representing the instantaneous rate of change of x with respect to t.
Example:
If x(t) = t² - 1, then dx/dt = 2t, indicating the rate at which the x-coordinate changes over time.
dy/dt
The derivative of the y-component of a parametric equation with respect to the parameter 't', representing the instantaneous rate of change of y with respect to t.
Example:
If y(t) = 3t², then dy/dt = 6t, showing how the y-coordinate changes as time 't' progresses.
dy/dx
The derivative of y with respect to x, representing the slope of the tangent line to a curve in the Cartesian plane. For parametric equations, it's calculated as (dy/dt) / (dx/dt).
Example:
If a particle's path is given by x(t) and y(t), then dy/dx tells you the slope of its path at any given point.