Editing Differential calculus (section)

=== Mean value theorem ===
{{Main|Mean value theorem}}

[[File:Mvt2.svg|thumb|The mean value theorem: For each differentiable function <math>f:[a,b]\to\R</math> with <math>a<b</math> there is a <math>c\in(a,b)</math> with <math>f'(c) = \tfrac{f(b) - f(a)}{b - a}</math>.]]
The mean value theorem gives a relationship between values of the derivative and values of the original function. If {{math|''f''(''x'')}} is a real-valued function and {{math|''a''}} and {{math|''b''}} are numbers with {{math|''a'' < ''b''}}, then the mean value theorem says that under mild hypotheses, the slope between the two points {{math|(''a'', ''f''(''a''))}} and {{math|(''b'', ''f''(''b''))}} is equal to the slope of the tangent line to {{math|''f''}} at some point {{math|''c''}} between {{math|''a''}} and {{math|''b''}}. In other words,
:<math>f'(c) = \frac{f(b) - f(a)}{b - a}.</math>
In practice, what the mean value theorem does is control a function in terms of its derivative. For instance, suppose that {{math|''f''}} has derivative equal to zero at each point. This means that its tangent line is horizontal at every point, so the function should also be horizontal. The mean value theorem proves that this must be true: The slope between any two points on the graph of {{math|''f''}} must equal the slope of one of the tangent lines of {{math|''f''}}. All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero. But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function.