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.
Let f be a real valued function defined on [a, b] that is differentiable on (a, b). Suppose f′(x) ≤ 0, ∀x ∈ (a, b). Prove f is monotonically decreasing on (a, b).
Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.
Prove that f(x) and xf(x) are linearly independent on II.
Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a)
and f(b) < g(b). Prove that there exists c e (a,b) such that f(c) = g(c).
Precalculus: Mathematics for Calculus - 6th Edition
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