Linear function (calculus): Difference between revisions

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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|1=''f''(''x'') = ''ax'' + ''b''}}, this slope and derivative is given by the constant {{mvar|a}}. Linear functions can be characterized as the only real-valued functions that are defined on the entire real line and have a [[constant function|constant]] derivative.

TheIn calculus, the derivative of a function, in general, function measures its rate of change. Because a linear function {{math|1=''f''(''x'') = ''ax'' + ''b''}} has a constant rate of change {{mvar|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 {{math|''ca''}}.

The fundamental idea of differential calculus is that any [[differentiable function|smooth]] function <math>f(x)</math> can be closely [[linear approximation|approximated]] by a unique linear function near a given point <math>x=c</math>. The derivative <math>f'(a)</math> is the slope of this linear function, and the approximation is: <math>f(x) \approx f'(c)(x{-}c)+f(c)</math> for <math>x\approx c</math>.

==Relationship with linear equations==