Suppose that f is continuous on [ a, b ] and differentiable on ( a, b ) and that m ≤ f ′( x ) ≤ M on ( a, b ). Use the Racetrack Principle to prove that f ( x )− f ( a ) ≤ M ( x − a ) for all x in [ a, b ], and that m ( x − a ) ≤ f ( x )− f ( a ) for all x in [ a, b ]. Conclude that m ≤ ( f ( b )− f ( a ))∕( b − a ) ≤ M . This is called the Mean Value Inequality. In words: If the instantaneous rate of change of f is between m and M on an interval, so is the average rate of change of f over the interval.
Suppose that f is continuous on [ a, b ] and differentiable on ( a, b ) and that m ≤ f ′( x ) ≤ M on ( a, b ). Use the Racetrack Principle to prove that f ( x )− f ( a ) ≤ M ( x − a ) for all x in [ a, b ], and that m ( x − a ) ≤ f ( x )− f ( a ) for all x in [ a, b ]. Conclude that m ≤ ( f ( b )− f ( a ))∕( b − a ) ≤ M . This is called the Mean Value Inequality. In words: If the instantaneous rate of change of f is between m and M on an interval, so is the average rate of change of f over the interval.
Suppose that f is continuous on [a, b] and differentiable on (a, b) and that m ≤ f′(x) ≤ M on (a, b). Use the Racetrack Principle to prove that f(x)−f(a) ≤ M(x−a) for all x in [a, b], and that m(x−a) ≤ f(x)−f(a) for all x in [a, b]. Conclude that m ≤ (f(b)−f(a))∕(b−a) ≤ M. This is called the Mean Value Inequality. In words: If the instantaneous rate of change of f is between m and M on an interval, so is the average rate of change of f over the interval.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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