Linear function (calculus): Difference between revisions

Revision as of 13:26, 28 June 2013

Revision as of 13:36, 28 June 2013

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==Slope==

[[File:Slope picture.svg|thumb|right|100px|The slope of a line is the ratio between a change in {{math|''x''}}, denoted {{math|Δ''x''}}, and the corresponding change in {{math|''y''}}, denoted {{math|Δ''y''}}]]

The [[slope (mathematics)|slope]] of a nonvertical line is a number that measures how steeply the line is slanted. The [[first derivative]] of a linear function, in the sense of calculus, is exactly this slope of the graph of the function. For {{math|<nowiki>1=''f''(''x'') = ''ax'' + b</nowiki>}}, this slope and derivative is given by the constant {{math|1=a}}. Linear functions can be characterized as the only real-valued functions that are defined on the entire real line and have a constant derivative.

The derivative of a function, in general, measures its rate of change. Because a linear function {{math|1=''f''(''x'') = ''ax'' + b}} has a constant rate of change {{math|''a''}}, it has the property that whenever the input {{mvar|x}} is increased by one unit, the output changes by {{mvar|a}} units. If {{mvar|a}} is positive, this will cause the value of the function to increase, while if {{mvar|a}} is negative it will cause the value to decrease. More generally, if the input increases by some other amount, {{mvar|c}}, the output changes by&nbsp;{{math|''ca''}}.

==Relationship with linear equations==