Examine individual changes

[ 0 => '{{ref improve|date=February 2015}}', 1 => 'In [[mathematics]], a '''rate''' is a [[ratio]] between two [[measurement]]s with different units.<ref>{{cite web |url=http://www.mathpropress.com/glossary/glossary.html#R |title=On-line Mathematics Dictionary |publisher=MathPro Press |date=January 14, 2006 |accessdate=2009-03-01}}</ref> If the unit or quantity in respect of which something is changing is not specified, usually the rate is ''per unit time''. However, a rate of change can be specified per unit [[time]], or per unit of [[length]] or [[mass]] or another quantity. The most common type of rate is "per unit time", such as [[speed]], [[heart rate]] and [[flux]]. Ratios that have a non-time denominator include [[exchange rate]]s, [[literacy rate]]s and [[electric flux]].', 2 => false, 3 => 'In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a [[heart rate]] is expressed "beats per minute"). A rate defined using two numbers of the same [[Units of measurement|units]] (such as [[tax rate]]s) or counts (such as [[literacy rate]]) will result in a [[dimensionless quantity]], which can be expressed as a [[percentage]] (for example, the global [[literacy rate]] in 1998 was 80%) or [[fraction (mathematics)|fraction]] or as a [[multiple (mathematics)|multiple]].', 4 => false, 5 => 'Often ''rate'' is a synonym of [[rhythm]] or [[frequency]], a count per second (i.e., [[Hertz]]); e.g., [[radio frequencies]] or [[heart rate]] or [[sample rate]].', 6 => false, 7 => '==Rate of change==', 8 => '{{main|Derivative}}', 9 => 'A rate of change can be formally defined in two ways:<ref>{{cite book|last1=Adams |first1=Robert A. |title=Calculus: A Complete Course|edition=3rd |year=1995 |publisher=Addison-Wesley Publishers Ltd. |isbn=0-201-82823-5 |page=129}}</ref>', 10 => ':<math>\begin{align}', 11 => ' \mbox{Average rate of change} &= \frac{f(a + h) - f(a)}{h}\\', 12 => ' \mbox{Instantaneous rate of change} &= \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}', 13 => '\end{align}</math>', 14 => '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]].', 15 => false, 16 => 'An example to contrast the differences between the ''average'' and ''instantaneous'' definitions: the [[speed]] of a car can be calculated:', 17 => '#An average rate can be calculated using the total distance travelled between ''a'' and ''b'', divided by the travel time', 18 => '#An instantaneous rate can be determined by viewing a [[speedometer]].', 19 => false, 20 => '==Terms based on rates==', 21 => false, 22 => 'In chemistry and physics:', 23 => '*[[Speed]], being the distance covered per unit time; e.g., miles per hour and [[meters per second]]', 24 => '*[[Acceleration]], the rate of change in speed, or the change in speed per unit time', 25 => '*[[Radioactive decay]], the amount of radioactive material in which one nucleus decays per second, measured in [[Becquerel]]s', 26 => '*[[Reaction rate]], the speed at which chemical reactions occur', 27 => '*[[Volumetric flow rate]], the volume of fluid which passes through a given surface per unit time; e.g., [[cubic meters per second]]', 28 => false, 29 => 'In computing:', 30 => '*[[Bit rate]], the number of bits that are conveyed or processed by a computer per unit of time', 31 => '*[[Symbol rate]], the number of symbol changes (signalling events) made to the transmission medium per second', 32 => '*[[Sampling rate]], the number of samples (signal measurements) per second', 33 => false, 34 => 'In finance:', 35 => '*[[Interest rate]], the price a borrower pays for the use of money they do not own, usually expressed as a percentage rate over the period of one year; see also for related rates', 36 => '*[[Exchange rate]], how much one currency is worth in terms of the other', 37 => '*[[Inflation rate]], a measure of inflation change per year', 38 => '*[[Rate of return]], the ratio of money gained or lost on an investment relative to the amount of money invested', 39 => '*[[Tax rate]], the tax amount divided by the taxable income', 40 => false, 41 => 'Miscellaneous definitions:', 42 => '*[[Rate of reinforcement]], number of reinforcements per time, usually per minute', 43 => '*[[Heart rate]], usually measured in beats per minute', 44 => '*[[Unemployment rate]], a ratio between those in the labor force to those who are unemployed', 45 => '*[[Birth rate]] and [[mortality rate]], the number of births or deaths scaled to the size of that population, per unit time', 46 => '*[[Literacy rate]], the proportion of the population over age fifteen that can read and write', 47 => false, 48 => '==References==', 49 => '{{Reflist}}', 50 => false, 51 => '==See also==', 52 => '*[[Derivative]]', 53 => '*[[Flux]]', 54 => '*[[Gradient]]', 55 => '*[[Slope]]', 56 => false, 57 => '{{DEFAULTSORT:Rate (Mathematics)}}', 58 => '[[Category:Interest rates]]', 59 => '[[Category:Measurement]]', 60 => false, 61 => '[[de:Rate]]' ]