Everyday usage and use in mathematics
In everyday language, the term "rate" refers to the speed or frequency of an event over a specific period of time. For instance, your heart-rate is the number of times your heart beats every minute, while an annual percentage rate (APR) is the percentage of a debt that that debt increases by every year.
The word "change" refers to the act or process of becoming different. In mathematics, "rate of change" has a precise meaning that's intuitive based on the everyday use of those words. It quantifies how much an output quantity varies relative to an input quantity.
Importance in mathematics
Rate of change is often represented by the slope of a line in a graph, which shows how much one variable changes on relative to the change in another variable. The Greek letter delta (Δ) is commonly used to denote change. For example, \( Δy / Δx \) represents the change in \(y\) (output) relative to the change in \(x\) (input).
Slope
In everyday language, the term slope describes the steepness or incline of a surface, such as a hill or roof. For example, a "steep slope" indicates a sharp incline, while a "gentle slope" refers to a more gradual incline.
In mathematics, the term slope specifically refers to a measure of the steepness or incline of a line on a graph. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, the slope \( m \) is given by the formula: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.
Importance in mathematics
Slope is used to describe the rate of change of a function, which indicates how a quantity changes in relation to another. In the context of a linear equation, the slope tells us how much the output (\( y \)) changes for a unit change in the input (\( x \)).
For example, in the equation of a straight line, \( y = mx + b \), the slope \( m \) represents the rate of change of \( y \) with respect to \( x \). If \( m \) is positive, the line slopes upward, indicating that as \( x \) increases, \( y \) also increases. If \( m \) is negative, the line slopes downward, showing that as \( x \) increases, \( y \) decreases. A zero slope means