121616949-math.150 - 136 Chapter 6 Applications of the...

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136 Chapter 6 Applications of the Derivative EXAMPLE 6.27 Describe all functions that have derivative 5 x - 3. It’s easy to find one: g ( x ) = (5 / 2) x 2 - 3 x has g ( x ) = 5 x - 3. The only other functions with the same derivative are therefore of the form f ( x ) = (5 / 2) x 2 - 3 x + k . Alternately, though not obviously, you might have first noticed that g ( x ) = (5 / 2) x 2 - 3 x + 47 has g ( x ) = 5 x - 3. Then every other function with the same derivative must have the form f ( x ) = (5 / 2) x 2 - 3 x + 47 + k . This looks different, but it really isn’t. The functions of the form f ( x ) = (5 / 2) x 2 - 3 x + k are exactly the same as the ones of the form f ( x ) = (5 / 2) x 2 - 3 x + 47 + k . For example, (5 / 2) x 2 - 3 x + 10 is the same as (5 / 2) x 2 - 3 x + 47 + ( - 37), and the first is of the first “form” while the second has the second form. This is worth calling a theorem: THEOREM 6.28 If f ( x ) = g ( x ) for every x ( a, b ), then for some constant k , f ( x ) = g ( x ) + k on the interval ( a, b ). EXAMPLE 6.29 Describe all functions with derivative sin x + e x . One such function is - cos x + e x , so all such functions have the form - cos x + e x + k .
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