Rate (mathematics)

Rate of change

Consider the case where the numerator ${\displaystyle f}$ of a rate is a function ${\displaystyle f(a)}$ where ${\displaystyle a}$ happens to be the denominator of the rate ${\displaystyle \delta f/\delta a}$. A rate of change of ${\displaystyle f}$ with respect to ${\displaystyle a}$ (where ${\displaystyle a}$ is incremented by ${\displaystyle h}$) can be formally defined in two ways:

${\displaystyle \begin{array}{cc}\text{Average rate of change}& =\frac{f(x+h)-f(x)}{h}\\ \text{Instantaneous rate of change}& =\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}\end{array}}$

where f(x) is the function with respect to x over the interval from a to a+h. An instantaneous rate of change is equivalent to a derivative.

For example, the average speed of a car can be calculated using the total distance traveled between two points, divided by the travel time. In contrast, the instantaneous velocity can be determined by viewing a speedometer.

List of temporal rates

In chemistry and physics:

• Speed, the rate of change of position, or the change of position per unit of time
• Acceleration, the rate of change in speed, or the change in speed per unit of time
• Power, the rate of doing work, or the amount of energy transferred per unit time
• Frequency, the number of occurrences of a repeating event per unit of time
  • Angular frequency and rotation speed, the number of turns per unit of time
• Reaction rate, the speed at which chemical reactions occur
• Volumetric flow rate, the volume of fluid which passes through a given surface per unit of time; e.g., cubic meters per second

Counts-per-time rates

• Radioactive decay, the amount of radioactive material in which one nucleus decays per second, measured in becquerels

In computing:

• Bit rate, the number of bits that are conveyed or processed by a computer per unit of time
• Symbol rate, the number of symbol changes (signaling events) made to the transmission medium per second
• Sampling rate, the number of samples (signal measurements) per second

Miscellaneous definitions:

• Rate of reinforcement, number of reinforcements per unit of time, usually per minute
• Heart rate, usually measured in beats per minute

Economics/finance rates/ratios

• Exchange rate, how much one currency is worth in terms of the other
• Inflation rate, the ratio of the change in the general price level during a year to the starting price level
• Interest rate, the price a borrower pays for the use of the money they do not own (ratio of payment to amount borrowed)
• Price–earnings ratio, market price per share of stock divided by annual earnings per share
• Rate of return, the ratio of money gained or lost on an investment relative to the amount of money invested
• Tax rate, the tax amount divided by the taxable income
• Unemployment rate, the ratio of the number of people who are unemployed to the number in the labor force
• Wage rate, the amount paid for working a given amount of time (or doing a standard amount of accomplished work) (ratio of payment to time)

Other rates

• Birth rate, and mortality rate, the number of births or deaths scaled to the size of that population, per unit of time
• Literacy rate, the proportion of the population over age fifteen that can read and write
• Sex ratio or gender ratio, the ratio of males to females in a population