121616949-math.149

121616949-math.149 - 6.5 The Mean Value Theorem 135 So the...

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6.5 The Mean Value Theorem 135 So the height of g ( x ) is the same at both endpoints. This means, by Rolle’s Theorem, that at some c , g ( c ) = 0. But we know that g ( c ) = f ( c ) - m , so 0 = f ( c ) - m = f ( c ) - f ( b ) - f ( a ) b - a , which turns into f ( c ) = f ( b ) - f ( a ) b - a , exactly what we want. Returning to the original formulation of question (2), we see that if f ( t ) gives the position of your car at time t , then the Mean Value Theorem says that at some time c , f ( c ) = 70, that is, at some time you must have been traveling at exactly your average speed for the trip, and that indeed you exceeded the speed limit. Now let’s return to question (1). Suppose, for example, that two functions are known to have derivative equal to 5 everywhere, f ( x ) = g ( x ) = 5. It is easy to find such functions: 5 x , 5 x + 47, 5 x - 132, etc. Are there other, more complicated, examples? No—the only functions that work are the “obvious” ones, namely, 5 x plus some constant. How can we see that this is true?
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• Fall '07
• JonathanRogawski

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