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Chapter 4 | Applications of Derivatives
The Mean Value Theorem and Its Meaning Rolle’s theorem is a special case of the Mean Value Theorem. In Rolle’s theorem, we consider differentiable functions f defined on a closed interval [ a , b ] with f ⎛ ⎝ a ⎞ ⎠ = f ⎛ ⎝ b ⎞ ⎠ . The Mean Value Theorem generalizes Rolle’s theorem by considering functions that do not necessarily have equal value at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem ( Figure 4.25 ). The Mean Value Theorem states that if f is continuous over the closed interval [ a , b ] and differentiable over the open interval ( a , b ), then there exists a point c ∈ ( a , b ) such that the tangent line to the graph of f at c is parallel to the secant line connecting ⎛ ⎝ a , f ( a ) ⎞ ⎠ and ⎛ ⎝ b , f ( b ) ⎞ ⎠ .
Figure 4.25 The Mean Value Theorem says that for a function that meets its conditions, at some point the tangent line has the same slope as the secant line between the ends. For this function, there are two values c 1 and c 2 such that the tangent line to f at c 1 and c 2 has the same slope as the secant line.
Theorem 4.5: Mean Value Theorem Let f be continuous over the closed interval [ a , b ] and differentiable over the open interval ( a , b ). Then, there exists at least one point c ∈ ( a , b ) such that f ′( c ) = f ( b )− f ( a ) b − a . Proof The proof follows from Rolle’s theorem by introducing an appropriate function that satisfies the criteria of Rolle’s theorem. Consider the line connecting ⎛ ⎝ a , f ( a ) ⎞ ⎠ and ⎛ ⎝ b , f ( b ) ⎞ ⎠ . Since the slope of that line is f ( b )− f ( a ) b − a and the line passes through the point ⎛ ⎝ a , f ( a ) ⎞ ⎠ , the equation of that line can be written as y = f ( b )− f ( a ) b − a ( x − a )+ f ( a ). Let g ( x ) denote the vertical difference between the point ⎛ ⎝ x , f ( x ) ⎞ ⎠ and the point ( x , y ) on that line. Therefore, g ( x ) = f ( x )− ⎡ ⎣ f ( b )− f ( a ) b − a ( x − a )+ f ( a ) ⎤ ⎦ .
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