Chapter 4 | Applications of Derivatives
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Figure 4.26 The value g ( x ) is the vertical difference between the point ( x , f ( x )) and the point ( x , y ) on the secant line connecting ( a , f ( a )) and ( b , f ( b )).
Since the graph of f intersects the secant line when x = a and x = b , we see that g ( a ) =0= g ( b ). Since f is a differentiable function over ( a , b ), g is also a differentiable function over ( a , b ). Furthermore, since f is continuous over [ a , b ], g is also continuous over [ a , b ]. Therefore, g satisfies the criteria of Rolle’s theorem. Consequently, there exists a point c ∈ ( a , b ) such that g ′( c ) =0. Since g ′( x ) = f ′( x )− f ( b )− f ( a ) b − a , we see that g ′( c ) = f ′( c )− f ( b )− f ( a ) b − a . Since g ′( c ) =0, we conclude that f ′( c ) = f ( b )− f ( a ) b − a . □ In the next example, we show how the Mean Value Theorem can be applied to the function f ( x ) = x over the interval [0, 9]. The method is the same for other functions, although sometimes with more interesting consequences.
Example 4.15 Verifying that the Mean Value Theorem Applies
For f ( x ) = x over the interval [0, 9], show that f satisfies the hypothesis of the Mean Value Theorem, and therefore there exists at least one value c ∈ (0, 9) such that f ′( c ) is equal to the slope of the line connecting ⎛ ⎝ 0, f (0) ⎞ ⎠ and ⎛ ⎝ 9, f (9) ⎞ ⎠ . Find these values c guaranteed by the Mean Value Theorem. Solution We know that f ( x ) = x is continuous over [0, 9] and differentiable over (0, 9). Therefore, f satisfies the hypotheses of the Mean Value Theorem, and there must exist at least one value c ∈ (0, 9) such that f ′( c ) is equal to the slope of the line connecting ⎛ ⎝ 0, f (0) ⎞ ⎠ and ⎛ ⎝ 9, f (9) ⎞ ⎠ ( Figure 4.27 ). To determine which value(s)
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